UNIVERSITY OF CAPE COAST MATHEMATICAL MODEL OF IMMUNE RESPONSE TO HEPATITIS B VIRUS AND LIVER CANCER CO-EXISTENCE DYNAMICS IN THE PRESENCE OF TREATMENT BY PAUL CHATAA Thesis submitted to the Department of Mathematics of the School of Physical Sciences, College of Agriculture and Natural Sciences, University of Cape Coast, in partial fulfilment of the requirements for the award of Doctor of Philosophy degree in Mathematics JULY 2024 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library ©Paul Chataa University of Cape Coast University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library DECLARATION Candidate’s Declaration I hereby declare that this thesis is the result of my own original research and that no part of it has been presented for another degree in this university or elsewhere. Candidate’s Signature ......................................... Date ...................... Name: Paul Chataa Supervisors’ Declaration We hereby declare that the preparation and presentation of the thesis were super- vised in accordance with the guidelines on supervision of thesis laid down by the University of Cape Coast. Principal Supervisor’s Signature ............................ Date ...................... Name: Dr. Samuel Mindakifoe Naandam Co-Supervisor’s Signature ..................................... Date: ...................... Name: Dr. Francis Tanam Djankpa ii University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library ABSTRACT The principal method for modeling the spread of infectious diseases generally in- volves the application of ordinary differential equations. Studies have demonstrated that an effective strategy for refining certain mathematical models is the integration of fractional-order differential equations. To gain a more profound understanding of the interactions between the hepatitis B virus (HBV), liver cancer, and immune sys- tem cells, a mathematical model that combined both ordinary and fractional differ- ential equations was investigated. This model was closely aligned with experimental data on viral DNA load. The work concentrated on four qualitative scenarios: the innate immune response, adaptive immune response, cytokine response, and the co- existence of infection dynamics. Unlike earlier models, liver cells were classified into distinct stages of infection. For populations of non-pathogenic macrophages in the presence and absence of malignant cells, the study calculated the invasion prob- ability for transmission dynamics, represented by the control reproduction number, Rc. The iterated two-step Adams-Bashforth method was employed for numerical simulations using the ABC fractional derivative in the Caputo sense, while the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCC) tech- niques were utilized for parameter sensitivity analysis. The work identified the key transmission mechanism of viral load and proposed an optimal therapeutic method for viral treatment. Model parameters were estimated using nonlinear least squares fitting of longitudinal data (serum HBV DNA viral load) from existing literature. Finally, the study compared the classical-order model system with the ABC frac- tional differential equations model system to determine which offered superior per- formance. Both methods were evaluated using simulation results of the state vari- ables, revealing that the fractional model provides more detailed results than the classical model. iii University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library KEY WORDS Chronicity Iterative Method Interleukin-10 Lagrange Interpolation Macrophages Ulam-Hyers Stability iv University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library ACKNOWLEDGEMENTS I wish to convey my profoundest appreciation to my supervisors, Dr. Samuel Mindakifoe Naandam and Dr. Francis Tanam Djankpa, for acquainting me with the discipline of mathematical biology and allowing me to work on various fascinating projects. I am immensely thankful for their insightful feedback, guidance, encour- agement, and dedication. May God bless them with good health and enrich their lives. I would additionally like to articulate my utmost and most profound gratitude to Professor Bismark Oduro for his invaluable guidance, mentorship, and support throughout the course of my dissertation. His insightful advice, encouragement, and patience have been essential to my development as a researcher. My sincere thanks also go to Professor Ernest Yankson, Dean of the School of Physical Sciences, for his valuable academic contributions throughout this research. Additionally, I extend my heartfelt thanks to the staff and lecturers of the Mathematics Department. I am particularly grateful to Nicholas Opoku for his comments, feedback, and support throughout this dissertation. This dissertation would have been inconceivable without the extraordinary support I received from my family, particularly my brother, Mr. Yaw Chataa. I am profoundly grateful for his steadfast assistance, relentless encouragement, and fervent prayers. v University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library DEDICATION To my family vi University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library TABLE OF CONTENTS Page DECLARATION ii ABSTRACT iii KEY WORDS iv ACKNOWLEDGEMENTS v DEDICATION vi LIST OF TABLES xi LIST OF FIGURES xii LIST OF ABBREVIATIONS xvi CHAPTER ONE: INTRODUCTION Background to the Study 1 Statement of the Problem 7 Research Objectives 10 Significance of the Study 11 Delimitation 11 Limitation 12 Definitions of Terms 12 Organisation of the Study 16 CHAPTER TWO: LITERATURE REVIEW Introduction 18 Immune System and its Constituents 18 Immune System Response to Hepatitis B Virus Infected Cells 20 Immune System Response to Cancer Infected Cells 23 Modelling HBV and Liver Cancer via Classical Differential Operators 27 vii University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Modelling HBV and Liver Cancer via Fractional Differential Operators 30 Chapter Summary 34 CHAPTER THREE: RESEARCH METHODS Introduction 35 Classical-order Model 35 Model Formulation 35 Equations of the Model 46 Basic Model Properties 47 Positivity of Solutions 47 Boundedness of Solutions 53 Model Analysis 60 Existence and Uniqueness of Solutions 60 Virus-free Equilibrium State (VFE) 69 Basic Reproduction Number; R0 70 Stability Analysis of Virus-free Equilibrium 74 Local Stability of the Virus-free Equilibrium 74 Global Stability of Virus-free Equilibrium 77 Virus-persistence Equilibrium (VPE) and Stability Analysis 81 Fractional-order Model 102 Immune Response in HBV and Liver Cancer Model via ABC Operator 103 Basic Model Properties 104 Model Analysis 116 Virus-free Equilibrium State (VFE) 117 Basic Reproduction Number; R0 117 Existence and Uniqueness of Solutions 120 Virus-persistence Equilibrium Analysis (VPE) 156 Local Stability Analysis of Virus-free Equilibrium 163 Global Stability Analysis of Virus-free Equilibrium 167 Ulam-Hyers (UH) Stability Scheme 168 viii University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Fractional-order Model With Treatment 173 Control Reproduction Number, Rc 175 Chapter Summary 177 CHAPTER FOUR: RESULTS AND DISCUSSIONS Introduction 179 Model Fitting Procedure 179 Non-linear Least Square Method 180 Model Fitting Results 181 Parameter Estimation for the Classical-order Model 186 Numerical Results of Classical-order Model 188 Quantitative Findings at the Virus-free Equilibrium 188 Quantitative Findings at the Virus-persistence Equilibrium 191 Impact of Natural Killer Cells on Infected Macrophages Cells 194 Impact of Effector B Cells on Infected Macrophages Cells 195 Impact of Cytotoxic T Cells on Hepatitis B Virus (HBV) Cells 197 Impact of Antibodies Cells on Hepatitis B Virus (HBV) Cells 199 Time to Chronicity 201 Sensitivity Analysis 206 Numerical Simulation of Fractional-order Model 210 Numerical Scheme of the Fractional Model System 214 Quantitative Findings for the Fractional-order Model 226 Numerical Simulations of State Trajectories at Varying Fractional Orders 228 Impact of Antiviral Treatment on HBV and Liver Cancer 232 Discussions 234 Chapter Summary 240 CHAPTER FIVE: SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Overview 241 Summary 241 Conclusions 242 ix University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Recommendations 243 REFERENCES 244 APPENDIX: PATIENT DATA 263 x University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library LIST OF TABLES Page 1 Model State Variables and their Description 43 2 Parameters of Model and their Description 44 3 Parameters of Model and their Description 45 4 Estimated Parameter Values for Seven Patients 185 5 Parameters and their Values for the Classical-order Model 186 6 Parameters and their Values for the Classical-order Model 187 7 Effects of NK Cell Clearance Rates on Infected Cells 195 8 Effects of Plasma Cell Clearance Rates on Infected Cells 197 9 Effects of Cytotoxic Cells Viral Removal Rate on HBV Cells 199 10 Effects Antibodies Viral Clearance Rate on HBV Cells 201 11 Parameters and their Relationship With R0 210 12 Parameters and their Values for the Fractional-order Model 226 13 Parameters and their Values for the Fractional-order Model 227 14 HBV DNA Levels for Patient 1 at Various Time Points 263 15 HBV DNA Levels for Patient 2 at Various Time Points 263 16 HBV DNA Levels for Patient 3 at Various Time Points 264 17 HBV DNA Levels for Patient 4 at Various Time Points 264 18 HBV DNA Levels for Patient 5 at Various Time Points 264 19 HBV DNA Levels for Patient 6 at Various Time Points 265 20 HBV DNA Levels for Patient 7 at Various Time Points 266 xi University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library LIST OF FIGURES Page 1 Illustrate the Schematic Representation of the Immunological Re- action to the Concurrent Presence of HBV and Hepatic Carcinoma (Author’s construct, 2022) 42 2 The Optimal Representation of V Provided by Model 1 (Solid Red Lines) is Compared to Patient Data (◦), Where (a) Represents the Best Fit for Patient 1 Data, (b) Represents the Best Fit for Patient 2 Data, (c) Represents the Best Fit for Patient 3 Data, and (d) Rep- resents the Best Fit for Patient 4 Data (Source: Author’s construct, 2023) 181 3 The Optimal Representation of V Provided by Model 1 (Solid Red Lines) is Compared to Patient Data (◦), Where (a) Represents the Best Fit for Patient 5 Data, (b) Represents the Best Fit for Patient 6 Data, and (c) Represents the Best Fit for Patient 7 Data (Source: Author’s construct, 2023) 182 4 Simulation Results Illustrating the Dynamics of State Variables at the Virus-Free Equilibrium Point, Where (a) Represents the Unac- tivated T Cell Class, (b) Represents the T Helper 1 Cell Class, (c) Represents the Killer T Cell Class, (d) Represents the Uninfected Macrophage Free of Cancer Cells Class, (e) Represents the Unin- fected Macrophage Containing Cancer Cells Class, and (f) Repre- sents the Infected Macrophage Containing Cancer Cells Class (Source: Author’s construct, 2023) 189 xii University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 5 Simulation Results Illustrating the Dynamics of State Variables at the Virus-Free Equilibrium Point, Where (a) Represents the Hep- atitis B Virus Particles Class, (b) Represents the Antibodies Class, (c) Represents the Effector B Cell Class, (d) Represents the Natural Killer Cell Class, (e) Represents the Interleukin-10 Cytokine Class, and (f) Represents the Interferon Alpha and Beta Class (Source: Au- thor’s construct, 2023) 190 6 Simulation Outcomes Demonstrating the Dynamics of the F2 Cy- tokine Class at the Virus-Free Equilibrium (Source: Author’s con- struct, 2023) 190 7 Simulation Results Illustrating the Dynamics of State Variables at the Endemic Equilibrium Point, Where (b) Represents the Unacti- vated T Cell Class, (b) Represents the T Helper 1 Cell Class, (c) Represents the Killer T Cell Class, (d) Represents the Uninfected Macrophage Free of Cancer Cells Class, (e) Represents the Unin- fected Macrophage Containing Cancer Cells Class, and (f) Repre- sents the Infected Macrophage Containing Cancer Cells Class (Source: Author’s construct, 2023) 192 8 Simulation Results Illustrating the Dynamics of State Variables at the Endemic Equilibrium Point, Where (a) Represents the Hepati- tis B Virus Particles Class, (b) Represents the Antibodies Class, (c) Represents the Effector B Cell Class, (d) Represents the Natural Killer Cell Class, (e) Represents the Interleukin-10 Cytokine Class, and (f) Represents the Interferon Alpha and Beta Class (Source: Au- thor’s construct, 2023) 193 9 Simulation Outcomes Demonstrating the Dynamics of the F2 Cy- tokine Class at the Endemic Equilibrium (Source: Author’s con- struct, 2023) 193 10 Effects of NK Cell Clearance Rates on Infected Cells (Source: Au- thor’s construct, 2023) 195 xiii University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 11 Effects of Plasma Cell Clearance Rates on Infected Cells (Source: Author’s construct, 2023) 197 12 Effects of Cytotoxic T Cells on HBV Cells (Source: Author’s con- struct, 2023) 199 13 Effects of Antibodies on Hepatitis B Virus Cells (Source: Author’s construct, 2023) 200 14 The Graph Represents the Duration to Chronicity at ρ3 = 0.6697 (Source: Author’s construct, 2023) 203 15 The Graph Represents the Duration to Chronicity at ρ3 = 1.0046 (Source: Author’s construct, 2023) 203 16 The Graph Represents the Duration to Chronicity at ρ3 = 1.1720 (Source: Author’s construct, 2023) 204 17 The Graph Represents the Duration to Chronicity at ρ3 = 1.3394 (Source: Author’s construct, 2023) 204 18 Graph Showing HBV Load Fluctuations Over Time to Chronicity (Source: Author’s construct, 2023) 206 19 Partial Rank Correlation Coefficients Demonstrating the Effect of Twelve Model Parameters on the Basic Reproduction Number (R0) of the Model Framework (Source: Author’s construct, 2023) 209 20 Simulation Results Showing the Trajectories of the State Variables at Different Fractional Orders, Where (a) Represents the Trajectory of T0 Cell Class, (b) Represents the Trajectory of T1 Cell Class, (c) Represents the Trajectory of T2 Cell Class, (d) Represents the Tra- jectory of M0 Cell Class, (e) Represents the Trajectory of M1 Cell Class, and (f) Represents the Trajectory of IM Cell Class (Source: Author’s construct, 2023) 230 xiv University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 21 Simulation Results Showing the Trajectories of the State Variables at Different Fractional Orders, Where (a) Represents the Trajectory of V Cell Class, (b) Represents the Trajectory of A Cell Class, (c) Represents the Trajectory of E Cell Class, (d) Represents the Tra- jectory of NK Cell Class, (e) Represents the Trajectory of I10 Cell Class, and (f) Represents the Trajectory of F1 Cell Class (Source: Author’s construct, 2023) 231 22 Simulation Results Showing the Trajectories of (F2) Cells at Differ- ent Fractional Orders (Source: Author’s construct, 2023) 231 23 Effect of Antiviral Treatments on HBV and Liver Cancer (Source: Author’s construct, 2023) 233 24 Optimal Control Strategy (Source: Author’s construct, 2023) 233 xv University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library LIST OF ABBREVIATIONS ESLD End-Stage Liver Disease NK Natural Killer DNA Deoxyribonucleic Acid HBV Hepatitis B Virus HCC Hepatocellular Carcinoma LHBs Large Hepatitis B surface Protein RNA Ribonucleic Acid MHC Major Histocompatibility Complex WHO World Health Organisation LHS Latin Hypercube Sampling PRCC Partial Rank Correlation Coefficients UH Ulam-Hyers Stability AUC Area Under the Curve PC Percentage Change xvi University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library CHAPTER ONE INTRODUCTION This dissertation investigates the immune response to the simultaneous pres- ence of hepatitis B virus (HBV) and liver carcinoma through a deterministic mod- eling framework utilizing both classical-order and fractional-order analyses. Our objective is to formulate and evaluate an epidemiological model that integrates im- munological dynamics and cancer progression, employing both fractional-order and classical differential and integral operators novel areas in the realm of mathematical biology. The research encompasses an introduction to the topic, a delineation of the problem, the aims and significance of the study, and a comprehensive outline of the thesis structure. Background to the Study Hepatitis B, a hepatic viral infection, can evolve into both acute and chronic conditions, resulting in significant health repercussions. As detailed by Ciupe et al. (2014), this ailment causes approximately 750, 000 fatalities annually, with around 300, 000 of these deaths attributable to hepatic cirrhosis and hepatocellular carci- noma (HCC) (Guidotti & Chisari, 2001). This pathogen is presumed to impact ap- proximately one-third of the worldwide populace, rendering it not merely one of the most primordial but also among the most hazardous viral menaces to human well- being (Ciupe, Ribeiro, Nelson, Dusheiko, & Perelson, 2007). Based on projections from the World Health Organization (WHO), approximately 1.5 million novel cases of hepatitis B are documented annually, resulting in a prevalence of 296 million in- dividuals enduring chronic infection (World Health Organization, 2019). In 2019 alone, hepatitis B accounted for 820,000 fatalities, primarily attributable to hepatic fibrosis and hepatocellular malignancy, the most prevalent form of liver neoplasm (World Health Organization, 2019). The global mortality from liver cancer is sub- stantial, with projections indicating that this figure will continue to escalate. Among the individuals affected, 6.3% are also co-infected with hepatitis B. Conversely, hep- atitis B prevalence is relatively low in Europe and North America, with rates below 1 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 1%. Nonetheless, the disease remains a major health challenge in Asia and Africa, where the incidence of chronic infection is between 5 and 10 per 100 adults. The hepatitis B virus (HBV), classified within the hepadnaviridae family, is the etiological agent responsible for hepatitis B infection. Due to its predominantly double-stranded DNA structure, once an individual is infected, the clearance of the hepatitis B virus becomes particularly arduous (Guidotti & Chisari, 2001). The in- cubation period for HBV ranges from 30 to 180 days. The virus can be detected approximately 30–60 days post-infection, and if transmission occurs during infancy or early childhood, it frequently establishes a persistent infection, leading to chronic hepatitis B in the majority of cases (around 90%) (Babiker et al., 2012; Nowak et al., 1996). Maternal-to-child transmission, primarily occurring during parturition, con- stitutes the foremost mechanism of infection in these initial developmental phases, leading to repetitive infection cycles and elevated chronicity rates. Horizontal trans- mission routes, including sexual contact among adults, intravenous drug use, expo- sure to contaminated blood products, and unhygienic practices, represent another vector of HBV dissemination. However, only 5–10% of adults subjected to these transmission routes develop chronic infection (Babiker et al., 2012; Nowak et al., 1996). Adults generally recover from HBV infection when the immune system mounts an adequate and multifaceted response. This includes the activation of potent and diverse CD4+ (T-helper 1) and CD8+ (cytotoxic T lymphocyte) T-cell responses, production of neutralizing antibodies specific to the HBV surface antigen (HB- sAg), and the secretion of antiviral cytokines within the liver, such as interferon- gamma (type-2 interferons) and tumor necrosis factor-alpha (TNF-α) (DeVico & Gallo, 2004; Nowak et al., 1996). These responses contribute to the generation of refractory hepatocytes that become resistant to reinfection (Su et al., 2012; Busca & Kumar, 2014). However, the precise roles of various immune system compo- nents, particularly the regulatory cytokine interleukin-10, in the initiation and reso- lution of HBV infections, remain inadequately understood. Unvaccinated neonates and immunosuppressed adults are more susceptible to progressing to chronic HBV 2 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library infection (Guidotti et al., 1994). In these individuals, both humoral and cellular immune responses are insufficiently effective, leading to persistent viral replication and continuous expression of HBsAg in the bloodstream (Guidotti & Chisari, 2001; Herbein & O’brien, 2000). The proportional influences of the immune system’s het- erogeneous constituents, especially the role of anti-inflammatory cytokines such as interleukin-10, in the advancement and chronicity of HBV infection, remain incom- pletely delineated. The empirical investigation of the antibody-mediated immune response to hep- atitis B virus (HBV) infection poses significant challenges. This complexity arises from the fact that free surface antigen antibodies are not discerned until subsequent to the resolution of HBV infection (Pawelek et al., 2012). However, both acute and chronic HBV infections are associated with circulating immune complexes that consist of antibodies bound to hepatitis B surface antigen (HBsAg), suggesting that antibody production occurs much earlier in the infection process than detectable and may contribute to the attenuation disease transmission dynamics (Ciupe, Ribeiro, Nelson, Dusheiko, & Perelson, 2007; Fatehi et al., 2022). Immune protection against HBV is conferred by HBsAg-specific antibodies, which possess neutralizing capa- bilities (Fatehi et al., 2022). Interestingly, the production of “subviral particles” (SVPs) increases exponentially in HBV infection compared to the generation of complete HBV virions (J. Hu & Liu, 2017; Fatehi et al., 2022). Sub-viral particles are viral particles that display HBV envelope proteins but lack the nucleocapsid and viral genetic material, rendering them non-infectious, and are produced by HBV- infected hepatocytes (Fatehi et al., 2020). Ongoing research aims to elucidate the mechanisms underlying their excessive production and their role in HBV pathogene- sis (Murray & Goyal, 2015). Sub-viral particles may influence the immune response to HBV infection, potentially inducing immunological acquiescence during neona- tal infection by delaying the generation of neutralizing antibodies. Furthermore, an abundance of SVPs can act as immunological decoys by sequestering neutralizing antibodies, thereby slowing the clearance of the virus from the host. The hepatitis B virus employs host cellular machinery to facilitate its survival 3 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library and replication during infection. Once intracellular, the virus evades detection by immune surveillance, effectively shielding itself from the host’s immunocompetent cells. In order to counteract this evasion, various components of the immune system synergistically mount a multi-layered defense across different stages of the infection. At the initial phase of infection, the innate immune response, involving natural killer (NK) cells and cytokine release, is pivotal in the host defense against a spectrum of viral infections, including SARS-CoV-2 (Covid-19), influenza, HIV, lymphocytic choriomeningitis virus (LCMV), and HBV. These innate immune mechanisms act to limit viral dissemination while simultaneously promoting the establishment of an adaptive immunological response. The NK cells specifically recognize virally infected cells by identifying reduced display of major histocompatibility complex (MHC) molecules on the cellular membrane, subsequently triggering the destruction of these compromised cells. Both NK cells and cytotoxic T lymphocytes (CTLs) harbor cytolytic effector molecules within granules that are mobilized and released upon interaction with an infected cell. Among these mediators, perforin, a pore- forming protein, disrupts the integrity of target cell membranes, enabling the entry of other cytotoxic molecules. The granules also contain enzymes known as granzymes, which infiltrate the target cell through perforin-created pores, inducing apoptosis and eliminating the infected cell. Contrary to the typical progression of viral infections, the initial stages of hepatitis B virus (HBV) infection are characterized by a subdued production of interferon-α/β (IFNα/β) and a relatively slow rate of viral replication. Several hypotheses have been proposed to elucidate this anomaly. One theory posits that the virus may not immediately infiltrate hepatocytes but instead resides transiently in other tissues, while another suggests that the virus replicates at a significantly re- duced rate during this phase (Perelson et al., 1996; Su et al., 2012). However, the pre- cise biological transformations occurring during this period remain largely elusive. As HBV enters its exponential replication phase, the innate immune system, along with cytokine activity, is triggered (Busca & Kumar, 2014). This activation subse- quently initiates the adaptive immune response, which includes antibody-mediated 4 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library neutralization of HBV surface antigen (HBsAg), thereby preventing the reinfection of hepatocytes. Furthermore, cytotoxic lymphocytes play a crucial role in eradicat- ing infected cells while concurrently inducing cytokine production. Interestingly, cytotoxic lymphocytes not only eliminate HBV-infected hepatocytes but also facil- itate non-cytolytic clearance of these cells (Busca & Kumar, 2014; Guidotti et al., 1994; Herbein & O’brien, 2000). In the evolving immune response to HBV, cy- tokines serve a pivotal function, inhibiting viral replication (Pawelek et al., 2012), stimulating antibody production, and activating natural killer (NK) cells and cyto- toxic T lymphocytes (CTLs) (Ciupe, Ribeiro, Nelson, Dusheiko, & Perelson, 2007). Moreover, they aid in inducing immunity within target cells that have yet to be in- fected (Ramsay et al., 1993). Tumorigenesis in the liver commences when a single hepatocyte undergoes aberrant growth. Based on its potential for metastasis, a tumor can be classified as benign, indicating it is non-cancerous with no propensity for metastasis; pre- malignant, suggesting it may develop into cancer; or malignant, indicating an ag- gressive capacity for rapid, uncontrolled growth and metastasis (Yang et al., 2020; Dehingia et al., 2021). Hepatocellular carcinoma (HCC), the medical term for pri- mary liver cancer, represents a malignant neoplasm of the liver, which is the second- largest organ in the human body (Engelhart et al., 2011). Liver cancer ranks as the fifth leading cause of cancer-related mortality globally, constituting a significant public health concern, especially in low-resource settings where it receives insuffi- cient attention. The scarcity of reliable epidemiological data on the incidence and prevalence of liver cancer impedes accurate projections of the HCC burden in de- veloping countries. Consequently, the policy framework for the management and control of HCC is inadequate. This inadequacy is also evident in Ghana’s approach to the management and control of HCC. In developing regions, the majority of patients afflicted with hepatocellular carcinoma (HCC) present at medical facilities during advanced stages of the dis- ease, resulting in exceedingly high mortality rates. Liver cancer is unique in that it can arise from both infectious and non-infectious etiologies. Key risk factors in- 5 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library clude chronic alcohol abuse and tobacco use. Due to its asymptomatic nature in early stages, HCC is often referred to as a “silent killer.” Planning for the impact of HCC is challenging, as data for accurately estimating the incidence and prevalence of the disease within populations remains scarce. Proper estimation of the HCC bur- den is critical for effective allocation of healthcare resources, mobilization efforts, and the implementation of preventive strategies. Projections indicate that by 2030, liver and pancreatic cancers are expected to surpass breast, prostate, and colorec- tal cancers as the second and third leading causes of cancer-related mortality in the United States, respectively (Engelhart et al., 2011). The evolving epidemiological landscape, exacerbated by factors such as undetermined causes, inadequate preven- tion policies, insufficiently trained medical personnel, suboptimal monitoring, and a dearth of research, has rendered HCC a neglected non-communicable disease (NCD) in developing nations. The application of advanced calculus has garnered significant scholarly atten- tion and has been extensively utilized across various scientific domains. Researchers have employed modern calculus to construct mathematical models for a wide ar- ray of viral diseases. The majority of these mathematical modeling methodologies hinge on systems of differential equations, which may be linear or nonlinear, or on classical-order integral and differential equations. However, over the past three decades, fractional-order differential equations (FDEs) have been increasingly uti- lized to model complex real-world problems with greater precision and accuracy. The traditional integer-order mathematical framework may not always be suitable for capturing the intricacies of all systems. Consequently, fractional calculus has exerted a profound influence across multiple disciplines, including mechanics, chemistry, biology, and image processing. The utilization of fractional calculus enables the resolution of various physical challenges. In many scenarios, fractional-order differ- ential equation (FODE) models exhibit superior alignment with real-world phenom- ena compared to their integer-order counterparts. This can be attributed to the fact that fractional derivatives and integrals allow for the incorporation of memory and hereditary effects, which are characteristic of most biological systems and infectious 6 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library disease processes. Additionally, FDEs provide a broader domain of stability relative to traditional differential equations. Moreover, unlike classical derivatives, which function as local operators, fractional derivatives serve as non-local operators. This non-locality permits fractional differential equations to represent epidemic models in a manner that is more realistic and comprehensive, accounting for both historical and current conditions. The probability of eradicating hepatocellular carcinoma (HCC) diminishes significantly when hepatitis B virus (HBV) infection concomitantly occurs with hep- atic carcinoma. Consequently, it is essential for the research consortium to conduct thorough inquiries into both hepatitis B and hepatic malignancy via clinical exam- inations and conceptual evaluations. The formulation of efficacious therapeutic in- terventions for hepatitis B and HCC constitutes a critical domain within medical research that will significantly enhance our understanding of tumor-immune system interactions. Despite the fact that a substantial proportion of individuals (approx- imately 90%) mount a robust and protective cell-mediated immune response that prevents the progression to chronic HBV infection, which is a precursor to HCC, hepatitis B remains one of the foremost causes of mortality worldwide, contributing to an estimated 3 million deaths annually (Guidotti & Chisari, 2001). The objective of this research was to explore the progression from acute HBV infection to chronic HBV infection, and ultimately to HCC, while examining the immune system’s re- sponse to each of these stages. A more profound comprehension of the mechanisms governing immune response dynamics cells would enhance the immune system’s ca- pacity to control HBV in the early phases of infection and would elucidate how the virus evades both adaptive immunity and innate cytokine-mediated defenses. Conse- quently, we propose and analyze fractional and classical-order deterministic models to assess the impact of various immune response components on the co-dynamics of HBV and HCC in the context of therapeutic interventions. Statement of the Problem The World Health Organization (WHO) has instituted an ambitious target to eradicate hepatocellular carcinoma and hepatitis B virus (HBV) infection by the year 7 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 2030 (Vos et al., 2015; Fatehi et al., 2022). Current strategies for controlling the spread of HBV include the medical treatment of infected individuals and the immu- nization of neonates and vulnerable adults through the administration of the hepatitis B vaccine. Additionally, public health campaigns designed to raise awareness about the detrimental effects of alcohol consumption and smoking are employed as pre- ventive measures to mitigate the risk of liver cancer and reduce the transmission of the virus. These prophylactic interventions aim to avert acute HBV infections and to prevent the recurrence of liver cirrhosis in individuals with pre-existing hepatic conditions. However, despite these efforts, the occurrence of hepatitis B and hepatic neoplasia remains profoundly elevated, with millions of novel instances and fatalities recorded annually. Although progress has been made toward the global elimination of HBV, the virus continues to pose a significant public health threat. As of 2019, an estimated 316 million individuals were living with chronic HBV infection globally (Sheena et al., 2022), reflecting a chronic HBV prevalence rate of approximately 4.1%. Hepatocellular carcinoma (HCC), a predominant type of liver cancer, along with cirrhosis, constitutes the leading causes of mortality related to hepatitis B in- fection, according to the WHO (World Health Organization, 2019). Projections in- dicate that these diseases are likely to result in millions of additional fatalities in the coming years, underscoring the critical need for long-term strategies to manage HBV infection and liver cancer. Nevertheless, significant gaps remain in our under- standing of the epidemiology of HBV and its disease course. There are unresolved questions about the virus’s capacity to evade the immune system and progress to advanced stages, such as liver cirrhosis, HCC, and liver cancer. Despite these un- certainties, it is widely acknowledged that the application of mathematical modeling offers a robust approach for elucidating the epidemiological dynamics of HBV and liver cancer. This aligns with the notable successes observed through the integration of theoretical and empirical investigations in the realms of clinical and biological sciences. Researchers have recognized the critical role that mathematical models can play in epidemiological research. Consequently, gaining a deeper comprehen- 8 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library sion of the co-occurrence and interplay of HBV and liver cancer is essential for the development of effective fractional and classical order mathematical models that can accurately capture the dynamics of these disease processes. During the acute phase of hepatitis B virus (HBV) infection, viral titers can surge to approximately 1010 HBV DNA copies per milliliter of plasma, persisting at elevated levels for several weeks as numerous hepatocytes become infected. Over time, viral titers diminish, and in roughly 85%− 95% of acutely infected adults, the virus is successfully eradicated. Individuals who achieve viral clearance generally exhibit more robust and diverse CD4+ and CD8+ T lymphocyte responses. Addi- tionally, viral clearance in these patients is typically accompanied by an elevation in alanine aminotransferase (ALT) levels, indicative of hepatic injury and an active immune response mediated by cytotoxic cells. Although the immune system plays a crucial role in reducing viral titers, the exact mechanisms underlying this process remain incompletely elucidated. Numerous mathematical models have been devised to investigate the transmis- sion dynamics of HBV and hepatocellular carcinoma, as well as strategies for the prevention, treatment, and control of these diseases across various geographic re- gions. Utilizing systems of ordinary and partial differential equations, these models have yielded valuable insights into the impact of vaccination and therapeutic inter- ventions as control measures for HBV infection and the management of liver cancer. Despite these contributions, existing modeling research has yet to result in the de- velopment of highly effective therapies for HBV and hepatocellular carcinoma. One conceivable rationale for this deficiency is that the majority of frameworks address- ing HBV and hepatic malignancy neglect to integrate the impact of interleukin-10 and the differentiation pathways of naive T lymphocytes, which we have recognized as crucial components in devising effective therapeutic approaches for HBV and hepatic carcinoma. Additionally, it is vital to acknowledge the paramount role that the immune system occupies in the evolution of hepatitis B virus (HBV) and hepatocellular car- cinoma (HCC) infections. Interleukin-10 (I10), a critical cytokine, regulates both 9 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library innate and adaptive immune responses, predominantly by imparting immunosup- pressive and anti-inflammatory effects. Therefore, the development of a sophis- ticated model to analyze the regulatory functions of interleukin-10 and other im- mune mediators in orchestrating both cellular and humoral immunity would prove beneficial. This study formulated an advanced mathematical framework that incor- porates interleukin-10 (I10), naive T lymphocytes, antibodies, natural killer (NK) cells, effector B cells, and cytokines to investigate the immune system’s multifaceted response to HBV and HCC infections. The research focuses on four pivotal do- mains: optimizing therapeutic interventions, evaluating the efficacy of fractional and classical-order models, elucidating the roles of innate immunity, humoral immunity, and cytokine-mediated pathways, and examining their coexistence in the complex dynamics of infection. The ultimate objective was to mitigate viral proliferation and minimize hepatic damage. Our model integrates several immune strategies, such as interferon-mediated immunity (type I and type II) to hinder viral replication by depleting the substrates necessary for viral synthesis (i.e., healthy hepatocytes), cell- mediated immunity (T lymphocytes, NK cells, effector B cells, interleukin-10 cells) to target and eliminate the source of viral generation (i.e., infected hepatocytes), and adaptive immunity (anti-HBV antibodies) to reduce the effective viral concentration. A distinctive feature of the model was its ability to simulate the impact of HBV in- fections on hepatocytes that have already undergone malignant transformation and how this interaction affects the overall population dynamics of liver cells. Research Objectives General objective The primary aimed of this thesis is to mathematically examine the immune response in relation to the transmission dynamics of the hepatitis B virus and its co-existence with liver cancer. Specific objectives This research has the following specific objectives: (1) to develop a classical and fractional order models, which takes into account the 10 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library important role played by various branches of the immune system in clearing HBV and liver cancer infections. (2) to estimate the model parameters by fitting the model to HBV DNA viral load data and perform sensitivity analysis to determine the most sensitive parame- ters. (3) to conduct numerical simulations on the model system to determine the tran- sition time from acute infection to the chronic stage and to recommend a treat- ment strategy. (4) to evaluate the classical-order and ABC fractional-order model systems to de- termine which one yields better results. Significance of the Study We can use mathematical models to evaluate the efficacy of various control and prevention efforts and to acquire a thorough knowledge of the dynamics of disease transmission. As far as we are aware, no mathematical model has been created that addresses the coexistence of the hepatitis B virus and liver cancer while considering the influence of innate, humoral, adaptive, and cytokine responses of the immune system. This research would aid medical professionals in their understanding of the immune system’s defenses against the hepatitis B virus and liver cancer infection. The results of this research would give Ghanaian health officials a more comprehen- sive framework to create the best infectious disease policies. The study would also provide light on how medical facilities use their resources to hepatitis B and liver cancer patients who are receiving treatment. Finally, it would expand the body of knowledge on the hepatitis B virus and liver cancer infection and give researchers a platform to push the boundaries of knowledge in these two fields. Delimitation The thesis is limited to investigating the important role played by the immune system (i.e macrophages, T cells, NK cells, anti-bodies, effector B cells, interleukin- 10, interferon-alpha and beta, interferon-gamma) in fighting hepatitis B virus and 11 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library liver cancer infections. Limitation This thesis has a limitation of accurate estimation of parameter values since we largely rely on patients DNA viral load data for parameter values for the numerical simulations which may be influenced by other factors. Another significant limitation of this study is that the patient dataset used does not include individuals diagnosed with liver cancer. As such, the findings may not be directly applicable to patients with liver cancer, whose immune response and disease progression could differ substantially from those observed in our cohort. Future studies should aim to incorporate datasets that include patients with liver cancer to evaluate the generalizability of the results and to explore disease-specific dynamics. Definitions of Terms In this section, our focus lies on several significant definitions of factional derivatives concerning the Atangana-Baleanu and Caputo fractional operators, which are intended for application in the immune response to co-existing hepatitis B virus and liver cancer model systems. Definition 1 Gamma Function: The Gamma function, denoted by Γ(k), is a generalization of the factorial function when k is not an integer. Particularly, Γ(x) = (n− 1)! for x ∈ N. For k > 0, it is defined as Γ(k) = ∫ ∞ 0 tk−1e−kdt. Definition 2 Given a function g : R+ → R, then the fractional integral of order φ in Caputo sense is given as C 0 I φ t [g (t)] = 1 Γ (φ) ∫ t 0 (t− f)φ−1 g (f) df, where Γ denote the gamma function and φ stands for the fractional order. 12 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Definition 3 Given a function g ∈ Cn, then the Caputo derivative with order φ is given as C 0 D φ t [g (t)] = In−φDg (t) = 1 Γ (n− φ) ∫ t 0 gn (f) (t− f)φ−n+1df, which is defined for the continuous function in absolute terms and n− 1 < φ < n ∈ N . Here, C 0 D φ t [g (t)] approaches g′(t) as φ approaches 1. Definition 4 Sobolev Space: Let p ∈ [1,∞) and Ω be an open set of R, the Sobolev Space Hp (Ω) is defined by Hp (Ω) = {f ∈ L2 (Ω) : Dφf ∈ L2 (Ω) ∀, |φ| ≤ p}. Definition 5 Let h ∈ H ′ (a, b) , b > a be a function. Then the Caputo-Fabrizo derivative of order φ ∈ (0, 1) of a function h is defined as CF a Dφ t [h (t)] = P (φ) 1− φ ∫ t a h′ (x) exp [ −φ t− x 1− φ ] dx, where P (φ) is a normalization function satisfying P (0) = P (1) = 1. Definition 6 The Caputo-Fabrizo fractional integral of order φ ∈ (0, 1) of a function h is defined as CF a Iφt [h (t)] = 2 (1− φ) 2 (1− φ)P (φ) h(t) + 2φ (2− φ)P (φ) ∫ t a h (ψ) dψ, t ≥ 0. Definition 7 Let g ∈ H ′ (a, b) , a < b be a function and φ ∈ (0, 1). Then the Atangana- Baleanu derivative in Caputo sense of order φ of the function g is defined as ABC a Dφ t [g (t)] = G (φ) 1− φ ∫ t a g′ (x)Eφ [ −φ(t− x) 1− φ φ] dx, where G(φ) is a normalization function satisfying G(0) = G(1) = 1 and Eφ is the Mittag-Leffler function. 13 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Definition 8 Let g ∈ H ′ (a, b) , a < b be a function and φ ∈ (0, 1). Then the Atangana- Baleanu derivative in Riemann-Liouville sense of order φ of the function g is defined as ABR a Dφ t [g (t)] = G (φ) 1− φ d dt ∫ t a g (x)Eφ [ −φ(t− x) 1− φ φ] dx. Definition 9 The Atangana-Baleanu fractional integral in Caputo sense is defined by ABC a Iφt [g (t)] = (1− φ) G (φ) g(t) + φ G(φ)Γ(φ) ∫ t a g (ψ) (t− ψ)φ−1 dψ. Definition 10 The Laplace transform of the ABC derivative of the function g(t) of order φ > 0 is defined as L{ABC a Dα t g(t)}(s) = B(φ) sφL{g(t)}(s)− sφ−1g(0) (1− φ) sφ + φ . Theorem 1 For a continuous function g on [a, b], the inequality given below holds on [a, b]: ||ABR a Dφ t [g (t)] || ≤ G (φ) 1− φ ||g(t)||, where ||g(t)|| = max a≤t≤b |g(t)| . Theorem 2 The Atangana-Baleanu derivative in both Caputo and Riemann-Liouville sense satisfy Lipschitz condition ||ABC a Dφ t [g (t)]−ABC a Dφ t [h (t)] || ≤ k||g(t)− h(t)|| and ||ABR a Dφ t [g (t)]−ABR a Dφ t [h (t)] || ≤ k||g(t)− h(t)||. 14 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library The Mittag-Leffler function’s one-parameter representations can be defined through power series Eα(t) = ∞∑ k=0 tk Γ (αk + 1) , α > 0, (1) while the two-parameter Mittag-Leffler is defined as Eα,β(t) = ∞∑ k=0 tk Γ (αk + β) , α > 0, β > 0. (2) The exponential series defined by Eq. (2) gives a generalization of equation Eq. (1). As a result of the definition given in Eq. (2), the following relations hold: Eα,β(t) = 1 Γ(β) + tEα,α+β(t) and (3) Eα,β(t) = βEα,β+1(t) + αt d dx Eα,β+1(t). (4) Observing from Eq. (4) implies that d dx Eα,β+1(t) = 1 αt [Eα,β−1(t)− βEα,β+1(t)] . So d dx Eα,β(t) = 1 αt [Eα,β−1(t)− (β − 1)Eα,β(t)] . Definition 11 The Laplace Transform can be used in differential equations to reduce a lin- ear differential equations to an algebraic one which can then be solved by rules of algebra. The transform is written as follows L{f(t)} = F (s) = ∫ ∞ 0 e−stf(t)dt. 15 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Whereas the Laplace transform of Caputo fractional derivatives is given by L [Dαf(t)] = sαF (s)− n−1∑ k=0 fk(0)sα−k−1. Lemma 1 Proposed by (Podlubny, 1998; Magin, 2004; Baleanu et al., 2012) Let α > 0, β > 0 and m ∈ C. We define f(t) = tβ−1Eα,β (±mtα) , where Eα,β(·) represents the two-parameter Mittag-Leffler function. Its Laplace transform is defined as L{f(t)} = sα−β sα ±m . Lemma 2 The solution of the following initial value problem for φ ∈ (0, 1]  ABC 0 Dφ t Φ(t) = Ψ(t), Φ(0) = Φ0, is given by Φ(t) = Φ0 + 1− φ N(φ) Ψ(t) + φ N(φ) 1 Γ(φ) ∫ t 0 (t− θ)φ−1 Ψ(θ)dθ. Organisation of the Study The thesis is organized into five chapters. CHAPTER ONE provides the back- ground of the study. In CHAPTER TWO, we present related literature on both frac- tional and classical mathematical models of hepatitis B and liver cancer transmis- sion dynamics. CHAPTER THREE is devoted to the formulation and analysis of the classical-order model. We also extend the classical-order model to a fractional- order model and analyze its basic properties. Parameter estimation, numerical simu- 16 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library lations, and sensitivity analysis of the proposed model are carried out and discussed in CHAPTER FOUR. CHAPTER FIVE is dedicated to conclusions and recommen- dations. 17 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library CHAPTER TWO LITERATURE REVIEW Introduction Hepatitis B and liver cancer are among the most insidious and lethal afflictions globally. This is largely due to the fact that many individuals infected with the hepatitis B virus are unaware of their condition. It is therefore crucial to investigate the patterns of their global dissemination. To achieve this effectively, it is essential to explore how various researchers have utilized mathematical modeling to tackle these issues and how their models can be adapted to aid in the eradication of liver cancer and hepatitis B. Our review indicates that numerous policymakers, analysts, and scholars have employed mathematical models to analyze the dynamics, outcomes, and impacts of liver cancer and hepatitis B over time within human populations, as well as to assess how these models address the response of health systems to such infections. Their primary objective has been to devise strategies for the prevention, diagnosis, and management of these severe diseases. Consequently, this chapter is dedicated to examining the empirical literature related to the pathophysiology of hepatitis B and liver cancer, the evolution of mathematical modeling as a tool for investigating persistent diseases, the integration of diverse therapeutic approaches as preventive strategies, and the response of health systems to these infections. Immune System and its Constituents The immune system is conceptualized as an intricate ensemble of biological mechanisms and structural entities within an organism, meticulously designed to counteract and neutralize potentially pathogenic entities. It is widely acknowledged that the immune system plays a pivotal role in maintaining human health by pro- viding defense against deleterious agents such as bacteria, viruses, and other mi- croorganisms (Guidotti & Chisari, 2001). This sophisticated network comprises a multitude of interconnected components that collectively safeguard the body against various pathogens, including bacteria, fungi, and viruses. Central to this defensive system are the thymus and bone marrow, which are instrumental in the development 18 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library and maturation of immune cells. The medullary cavity of the bones operates as the principal locus of genesis for both T and B lymphocytes, as well as other hematopoi- etic progenitor cells, thereby fulfilling a fundamental role in immunological efficacy. T lymphocytes migrate to the thymus for further maturation, while B lymphocytes remain within the bone marrow to undergo their developmental processes. The thymus, located superior to the heart, posterior to the sternum, and be- tween the pulmonary organs, consists of two lobe-shaped glands. Post-puberty, the thymus undergoes a progressive involution, gradually being substituted by connec- tive tissue and adipose tissue. Thymosin, a hormone secreted by the thymus, fa- cilitates the proliferation and differentiation of T lymphocytes. Within the thymic environment, T cells proliferate, acquire unique antigen receptors, and differentiate into subsets such as Helper T cells (T1) and Cytotoxic T cells (T2). These T cells express various surface proteins, including CD4+ and CD8+, essential for their functional roles. The thymus is responsible for generating the full complement of T cells necessary for an individual throughout puberty. Upon completion of their maturation in the thymus and bone marrow, T and B lymphocytes transit to secondary lymphoid organs such as the spleen and lymph nodes, where they persist until activated by antigenic stimuli. The spleen, positioned in the upper left quadrant of the abdomen, posterior to the stomach and diaphragm, primarily functions in blood filtration. Lymph nodes, distributed throughout the body, serve as crucial sites for immune surveillance. The spleen also plays a role in the degradation of senescent erythrocytes by macrophages, large phagocytic cells specialized in engulfing pathogens, cellular debris, and foreign particles. Addition- ally, the spleen acts as a reservoir for white blood cells and platelets and contributes to the immune response by identifying potential pathogens. The immune system’s defensive mechanisms are categorized into two princi- pal systems: innate (non-specific) and adaptive (specific) immunity. These systems comprise a diverse array of leukocytes, also known as white blood cells. Granulo- cytes, a subset of leukocytes, contain cytoplasmic granules with enzymatic activity and include neutrophils, basophils, and eosinophils. Conversely, cells of the adap- 19 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library tive immune system, often referred to as immune effector cells, execute specialized functions in response to specific antigens. Examples include T lymphocytes and B lymphocytes, which, alongside natural killer cells, perform targeted immune func- tions. Macrophages and neutrophils patrol tissues and the bloodstream, seeking and responding to potential threats. Upon encountering pathogens, these cells engage in interactions with other immune cells and perform phagocytosis. Furthermore, effector cells are instrumental in combating cancer cells, with activated B cells pro- ducing antibodies that facilitate the immune response, and activated T lymphocytes eliminating pathogens through cell-mediated mechanisms. Antigen-presenting cells (APCs), such as myeloid-derived suppressor cells, tumor-associated macrophages, regulatory T cells, and dendritic cells, are classified as non-effector cells. While non-effector cells do not possess direct tumor-killing capabilities, they play a role in modulating the immune response and can inadvertently support tumor growth during oncogenic processes. Immune System Response to Hepatitis B Virus Infected Cells It is broadly recognized that the immune system is crucial for human health as it provides defense against pathogenic entities such as bacteria, viruses, and other microorganisms (Guidotti & Chisari, 2001). Hepatitis B virus infection typically evolves through two principal stages: acute and chronic. An acute hepatitis B in- fection is marked by the transient presence (less than six months) of viral DNA, hepatitis B surface antigen (HBsAg), and hepatitis B envelope antigen (HBeAg), followed by their clearance and subsequent seroconversion to anti-HBsAg and anti- HBeAg. During convalescence, IgM antibodies to the hepatitis B core antigen (IgM anti-HBc) convert to IgG antibodies. This transition occurs within the initial six months of infection. Acute hepatitis B may resolve spontaneously within this pe- riod, with many patients asymptomatic and therefore not requiring treatment at this stage. As reported by Shepard et al. (2006), about 90% of adults will clear the in- fection naturally within the first six months (the acute phase) without necessitating therapy. If the immune system fails to eradicate the hepatitis B virus within this period, the infection becomes chronic. 20 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library At this stage, the virus may persist and not be eradicated spontaneously. Con- sequently, individuals with chronic hepatitis B will require sustained therapy to achieve a longer and healthier life (Wilson et al., 1998). Mahoney & Kane (1999) highlights that chronic hepatitis B leads to significant hepatic damage and con- tributes extensively to the global hepatitis burden. While some chronic hepatitis B patients may not display clinical or biochemical evidence of liver disease, oth- ers may experience symptoms such as chronic fatigue, anxiety, loss of appetite, and general malaise. This is in contrast to the acute phase, where symptoms may be ab- sent. The severity of chronic hepatitis B infection can vary from moderate to severe depending on the individual’s immune response. According to Hollinger & Liang (2001), chronic hepatitis B is associated with inflammatory liver conditions like cir- rhosis and hepatocellular carcinoma (HCC). Mahoney & Kane (1999) further notes that individuals at elevated risk for HCC include adult males with a history of hep- atitis B and chronic cirrhotic patients exposed to the virus during childhood. The risk of HCC is influenced by the individual’s age, gender, ethnicity, and geograph- ical location. Therefore, 90% of primary malignant liver tumors detected in adults are attributed to HCC (World Health Organization, 2002). Various researchers have investigated different facets of HBV dynamics and immune responses during infection through mathematical modeling. To examine acute HBV infection and the impact of temporal delays in effector cell activation and proliferation, Ciupe, Ribeiro, Nelson, Dusheiko, & Perelson (2007); Ciupe, Ribeiro, Nelson, & Perelson (2007) modified a conventional immune response model to in- clude delays in naive T cell recruitment. They subsequently analyzed the effects of pre-existing or vaccine-induced antibodies on HBV management (Ciupe et al., 2014). Instead of employing a mass-action model to address the limited liver capac- ity and susceptibility to HBV infection, Min et al. (2008) used a standard incidence function in their study of HBV transmission dynamics. Gourley et al. (2008) in- troduced a time-delayed version of the model proposed by Min et al. (2008). The model developed by Hews et al. (2010) provided a better fit to existing data and of- fered more accurate estimates of the basic reproduction number by incorporating a 21 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library standard incidence function and logistic growth for hepatocyte populations. In their study of HBV infection, Yousfi et al. (2011) explored potential limitations in the coordination among various branches of the adaptive immune response, particularly focusing on CTLs and antibodies. Regarding the role of cytokines in modulating immune responses, Wiah et al. (2011) posited that, alongside the actions of cytotoxic T lymphocytes (CTLs) and antibodies on hepatitis B virus (HBV) infection, interferon cytokines α and β also aid in the eradication of the virus by transforming susceptible hepatocytes into cells resistant to infection. This methodology has been similarly adopted by other scholars (Dahari et al., 2009; Lewin et al., 2001; Sypsa et al., 2005), who integrated a constant rate of non-cytolytic clearance into their treatment frameworks. Kim et al. (2012) modified an earlier hepatitis C model to implicitly incorporate cytokines by permitting effector cells to facilitate a non-cytolytic recovery of infected cells. An analytical framework for the antibody response post-hepatitis B virus (HBV) infection was formulated by (Ciupe et al., 2014) to investigate the mechanisms of immune protection. After identifying that the virus could evade neutralization by producing excessive quantities of non-infectious sub-viral particles bearing hepatitis B surface (HBs) proteins but lacking viral nucleic acids and nucleocapsid proteins they adjusted their model using data from seven adults identified during the acute infection phase. They demonstrated that viral clearance could be attained in individ- uals with high levels of anti-HBV antibodies, such as those who have been vacci- nated, under the following conditions: Hepatitis B sub-viral particles are generated at two distinct rates: (1) slowly; (2) rapidly if anti-HBV antibodies are produced rapidly, exhibit high affinity, or if substantial levels of pre-existing HBV-specific antibodies are present at the time of infection, potentially due to prior vaccination. Additionally, they showed that when a robust cellular immune response addresses early infection, viral clearance can occur with low steady-state levels of anti-HBV antibodies, similar to those in uninfected individuals. They concluded that achiev- ing HBV clearance necessitates higher antibody levels than those typically observed in clinical settings, as evidenced by many patients and most acutely infected indi- 22 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library viduals. Furthermore, they suggest that antibodies may function similarly to the CD8-antibody model, which implies that antibodies prevent re-infection and that cytotoxic effects are critical for initial viral control. The dynamics of their model correspond with the quantities and timing of antibodies seen in patients. Moreover, they assert that viral clearance remains attainable when pre-existing immunity from vaccination or cross-immunity results in elevated antibody levels early in infection, even if the virus generates a large quantity of sub-viral particles as a strategy to evade antibody-mediated defenses. Immune System Response to Cancer Infected Cells Hepatocellular carcinoma (HCC), commonly referred to as liver cancer, is a malignancy affecting the liver, the body’s second largest organ. This neoplasm is considered one of the most lethal and challenging cancers to manage clinically. Pro- jections indicate that by 2030, liver cancer is anticipated to surpass colorectal, breast, and prostate cancers as the second leading cause of cancer-related mortality in the United States, with pancreatic cancer following as the third (Rahib et al., 2014). Liver cancer is unique in that it originates from both infectious and non-infectious factors. According to the World Health Organization (WHO), cancer-related fatali- ties accounted for 8.8 million deaths in 2015 (Albertsen, 2003). By 2030, it is fore- casted that there will be approximately 21.4 million cases and 13.5 million deaths globally due to cancer (R. M. Anderson & May, 1991). This scenario has spurred ex- tensive research into the interplay between the immune system and tumor cells. The treatment of cancer remains one of the most formidable challenges in contemporary medicine. Therapeutic modalities include surgery, radiation, hormone therapy, vi- rotherapy, chemotherapy (Kuznetsov et al., 1994), and, more recently, immunother- apy (de Pillis et al., 2006). Cancer arises from mutations or environmental factors that disrupt normal cellular functions, leading to malignant transformation. These alterations can be triggered by environmental carcinogens such as tobacco, UV ra- diation, X-rays, or various toxins, or by genetic mutations. Mutations and environ- mental exposures can activate or inactivate specific genes, leading to the oncogenic conversion of normal cells. 23 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Typically, cancer therapies aim to eradicate malignant cells throughout the body while minimizing damage to healthy tissues. Chemotherapy remains a princi- pal strategy, involving pharmacological agents designed to target and destroy tumor cells. However, these agents often adversely affect normal cells, resulting in a range of side effects, including drug resistance and recurrence (Bellomo et al., 2008). Con- sequently, immunotherapy has emerged as a promising alternative. Immunotherapy enhances the immune system’s ability to identify and eliminate tumor cells through two principal approaches: active and passive immunotherapy. Passive immunother- apy involves the use of effector immune components to directly target tumor cells. This approach includes techniques such as chimeric antigen receptor [CAR]-T cell therapy and antibody-targeted treatments. Active immunotherapy, on the other hand, bolsters the immune system’s capacity to recognize and combat cancer cells by stim- ulating immune system activity. This category includes cytokine therapies, adoptive cell therapies, and cancer vaccines (Martins et al., 2007). Cancer vaccines enhance the responsiveness of cytotoxic T lymphocytes to tumor-specific antigens. Cytokines, essential signaling proteins produced by vari- ous cells including macrophages, B lymphocytes, and T lymphocytes, play a pivotal role in immune regulation. However, not all cytokines are approved for cancer treat- ment. The Food and Drug Administration (FDA) in the United States has authorized only two cytokines; interleukin-2 (IL-2) and interferon-alpha (IFN-α) due to their demonstrated clinical efficacy (Nagy, 2005). IL-2 primarily functions to activate T- cells and natural killer cells, which are instrumental in recognizing and eliminating tumor cells. Although IFN-α has a comparable response rate, it does not substan- tially enhance long-term survival (Mamat & Subiyanto, 2013; Sanmamed & Chen, 2018; Sharma & Samanta, 2013; de Pillis et al., 2008). Numerous engineering and physical phenomena can be adeptly characterized and scrutinized through mathematical modeling. Conditions such as aortic aneurysm development (Hao et al., 2017), management of HIV infections (Ogunlaran & Ouk- ouomi Noutchie, 2016), diabetes management (Shtylla et al., 2019), combating an- tibiotic resistant infections (Roberts et al., 2019), cardiac rhythms (Jones et al., 24 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library 2009), drug dispersion (Jones et al., 2009), hepatitis C virus (Elkaranshawy et al., 2019), and tumor proliferation and cancer treatment have all been elucidated using this technique. Scholars focusing on cancer biology primarily endeavor to develop mathematical models for the quantitative analysis of tumor growth and the response of cancer cells to various therapies. Theoretical exploration of liver cancer via mathematical models provides a valuable approach to enhancing our understanding of tumor-immune dynamics. De- spite substantial advancements in elucidating how tumors circumvent immune de- struction, the mechanisms by which the immune system protects against malignan- cies remain incompletely understood. Thus, it is increasingly imperative to investi- gate the interaction between the immune system and malignant cells in vivo (Coskun et al., 2010; Spalding et al., 2008). Several studies have concentrated on the dynamic characteristics of the tumor-immune response system. To advance immunotherapy strategies, including dendritic cell vaccines, Qomlaqi et al. (2017) proposed a math- ematical model delineating interactions between cancer cells and adaptive immune responses in murine models. Their model integrates interleukin-2 cytokines, tumor cells, natural killer cells, regulatory T cells, and both naive and mature helper T cells, as well as naive and mature cytotoxic T cells. Tumor growth due to inter- nal pressure was explored by (Jones et al., 2009), incorporating tumor size, internal pressure, and nutrient concentration within a system of partial differential equations in the absence of therapeutic interventions (Tao et al., 2010). A statistical model for cancer mortality was investigated by Ghosh & Samanta (2019). A framework for immunotherapy using mature dendritic cells was proposed by Arabameri et al. (2018). A revised model incorporating the effects of INF-α and IL-2 was suggested by Mamat & Subiyanto (2013). To evaluate the influence of CD4+ T cells, a model was initially developed by Eftimie et al. (2010) and subsequently refined by L. An- derson et al. (2015) and X. Hu & Jang (2018). CD4+ T cells, serving as helper cells to activate CD8+ T cells and safeguard against malignancies, also possess the ability to directly destroy tumor cells through cytokine secretion (L. Anderson et al., 2015; X. Hu & Jang, 2018; Mattes et al., 2003; Perez-Diez et al., 2007; S. Zhang et 25 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library al., 2009). A mathematical model combining chemotherapy and immunotherapy for tu- mor growth was introduced by de Pillis et al. (2006). A model for radiation ther- apy, which considers the proportional impact on normal cells, was developed by (Freedman & Pinho, 2009). The mathematical model for bacille Calmette-Guérin (BCG) immunotherapy for superficial bladder cancer was analyzed by (Bunimovich- Mendrazitsky et al., 2007). Ghosh & Banerjee (2018) explored the role of antibodies in regulating malignancy proliferation. The impact of natural killer cells (NK), circu- lating lymphocytes, CD8+ T cells, and CD4+ T cells on cancer cells was examined by Makhlouf et al. (2020), leading to the development of a treatment protocol that integrates chemotherapy, IL-2 cytokine therapy, and adoptive immunotherapy in- volving CD8+ and CD4+ T cells. They further argued that CD4+ T cells primarily eliminate tumor cells through cytokine production, in contrast to CD8+ T cells. Abernathy et al. (2020) investigated a modified nonlinear cancer model in- corporating virotherapy, optimizing the dose for effective tumor eradication. (Cho & Levy, 2020) devised a cancer growth model incorporating targeted therapies and chemotherapy to analyze the competitive dynamics between malignant and healthy cells. Schlicke et al. (2021) calibrated the model using data from three patients with non-small cell lung cancer, developing a framework that incorporates various treat- ment options and assesses their outcomes. To quantitatively assess the relationship between cancer and the immune system, while accounting for antibodies, Ghosh & Banerjee (2018) developed a model reflecting these parameters. The model pro- posed by (Ivanov et al., 2009) was based on clinical evidence of antibodies directly eliminating malignant cells. Kuznetsov et al. (1994) proposed a mathematical model for cytotoxic T lym- phocytes and their response to an immunogenic tumor. They examined how the immune system influences tumor growth, the mechanisms through which tumors evade detection, and the development of dormant tumor states. Kirschner & Panetta (1998) utilized mathematical modeling to elucidate interactions among tumor cells, immune effector cells, and interleukin-2 (IL-2), addressing both long-term relapses 26 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library and short-term tumor growth fluctuations. The periodic dynamics within immune system models were explored by (Bodnar & Foryś, 2000b). A generic model of immune system dynamics, proposed by (Bodnar & Foryś, 2000a) within Marchuk’s framework, illustrated various aspects of disease progression, oscillations, and re- covery, and provided insights into the qualitative behavior of the model solutions. The spontaneous regression and growth of malignant tumors were conceptualized as a prey-predator system, where tumor cells are the prey and cytotoxic T-lymphocytes are the predators, incorporating stochastic elements to account for random fluctua- tions around the positive equilibrium point (Sarkar & Banerjee, 2005). Modelling HBV and Liver Cancer via Classical Differential Operators The correlation between hepatitis and malignancy within a homogeneous pop- ulation, considering continuous influxes of cancer patients, was demonstrated by (Bhadauria, 2011) through the utilization of a nonlinear mathematical framework. Their study encompassed both vertical and horizontal transmission mechanisms of hepatitis within the community. By employing stability theory of differential equa- tions, the equilibrium stability of the model was analyzed. Sensitivity analysis was conducted to evaluate how variations in system parameters influenced the endemic equilibrium. Additionally, the fourth-order Runge-Kutta method was applied to per- form numerical simulations of the proposed model. A notable achievement of their model was quantifying the effect of hepatitis virus infections on cancer patients and its subsequent impact on increasing hepatitis outbreaks within the general popu- lace. Their findings indicated that an increase in hepatitis infection transmission rates corresponded with a rise in the endemic infective population levels. This effect was further exacerbated when there was an elevated risk of hepatitis infection among cancer patients. Their study also highlighted that hepatitis B and C infections, which progress to liver cancer, facilitate a heightened prevalence of cancer cases within the population. As part of oncolytic virotherapy, malignant tumor cells are targeted with viruses to induce infection and subsequent cell lysis while preserving healthy cells. This therapeutic approach also augments the immune system, enhancing the body’s de- 27 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library fense mechanisms. Oncolytic virotherapy operates without the necessity for chemo- therapy or radiation, simulating a virus’s natural infection process. Abernathy et al. (2020) incorporated this treatment modality into their work, proposing a system of four nonlinear differential equations to model the implementation of virotherapy as an anti-cancer strategy. Their model elucidates the interactions between virions, effector T cells, infected tumor cells, and non-infected tumor cells. To investigate the immune system dynamics and viral interactions during hep- atitis B virus (HBV) infections, Long & Qi (2009) developed a mathematical model grounded in the pathophysiology, virology, and immunology of HBV. Specifically, they calibrated the noncytopathic effects of the cellular immune response to be pro- portional to the count of cytotoxic T lymphocytes (CTLs), rather than the product of CTL numbers and infected hepatocyte counts. The model underscored that robust immune responses are crucial for combating HBV infections. It also illustrated that vertical transmission of HBV results in a less severe, asymptomatic form of hepati- tis, suggesting that both the absence of an antiviral immune response and the virus’s persistence post-mother-infant transmission could be attributed to neonatal immune tolerance. In 85%–95% of adults, HBV infection typically leads to acute hepati- tis followed by recovery, contingent upon the development of sufficient immune defenses. Recovery involves the generation of potent and diverse CD4 and CD8 T- cells, the synthesis of neutralizing antibodies against the HBV surface antigen (HB- sAg), the production of antiviral cytokines such as tumor necrosis factor-alpha and interferon-gamma, and the establishment of resistance to reinfection. However, the exact proportional contributions of various immune system components, particularly the role of neutralizing antibodies in infection development and progression, remain inadequately understood. Further research is necessary to elucidate the specific roles of these immune system components during HBV infection. To elucidate the intricacies of the immune system’s response to hepatitis B virus (HBV) infections, Chenar et al. (2018) has extended the immune response model proposed by (Long & Qi, 2009) by incorporating additional immune system dynamics. Their enhanced model integrates the roles of cytokines alongside innate 28 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library and adaptive immune responses, with the primary aim of examining the synergistic interactions among disparate constituents of the immune system during HBV infec- tions. Their research vividly illustrates the pivotal role of cytokines in modulating the immune system’s response and controlling viral replication. Through a compre- hensive stability analysis of multiple equilibrium states, the model was examined for conditions leading to periodic oscillations, infection eradication, or chronic in- fection persistence. The critical efficacy of therapeutic interventions, including inter- feron therapies and nucleoside analogs, was evaluated. Numerical simulations were employed to probe the interactions between cytokine-related parameters and the sta- bility of the endemic equilibrium, with the objective of refining the understanding of cytokine involvement in the system’s dynamics. Their results suggest that the endemic steady state can be stabilized by increasing the rates at which IFNα β and IFNγ augment the cytolytic activity of NK cells or by enhancing the rates at which these interferons inhibit new virion production. A notable feature of immunological dynamics is the non-instantaneous nature of several critical processes, such as the latency period between infection and cy- totoxic T lymphocyte (CTL) recruitment, the generation of new virions post-cell infection, and the duration of viral cellular invasion. Therefore, (Chenar et al., 2018) can be extended mathematically by incorporating discretely distributed time delays for each associated process, thereby enhancing the model’s realism and analytical complexity. It is well-established that antibodies bind to viral particles to form virus-antibody complexes rather than directly neutralizing the viruses. Since these complexes can dissociate and are not always stable, explicitly including them in the model could elucidate the dynamics more clearly. Despite their increased complex- ity, age-structured infectious disease models provide a more accurate depiction of disease transmission dynamics. Building upon the aforementioned analysis, Fatehi et al. (2022) advanced the work of (Chenar et al., 2018) by incorporating the age distribution of infected cell populations, thereby accounting for the contributions of both humoral and cell- mediated immunity in the context of HBV infection. The model was adapted to re- 29 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library flect chronic infection, infection clearance, and exacerbations by adjusting immune response parameters. Initial validation of the model was conducted using publicly available patient data from acute infection phases. According to prevailing theories, CD8+ T-cell depletion is associated with prolonged exposure to high antigen loads. The model was calibrated using viral load data from six patients recorded during the acute infection stage, with the initial viral load Vc(0) treated as a variable. It was observed that patient 2 exhibited an earlier peak viral load compared to other patients, suggesting a potentially higher initial viral load. For patients 1, 3, 4, 5, and 6, the initial viral load was consistent with a value of 0.33 virions per milliliter, aligning with most literature findings, whereas for patient 2, the initial viral load was substantially higher at 10 virions per milliliter. Analysis of the impact of non-infectious subviral particles and immune re- sponse exhaustion on immune response dynamics has been conducted. The com- parative evaluation of various therapeutic interventions within the framework of this model reveals that therapies targeting specific stages of the viral life cycle are more efficacious than fatigue mitigation therapies, which are designed to address immune response depletion. Further results from the model suggest that initiating antiviral treatment during a period of declining viral load is more advantageous than start- ing therapy during periods of increasing viral load. The model demonstrates that expedited antibody production consistently leads to viral eradication, highlighting the potential effectiveness of antibody-based therapeutics currently in clinical trials (Fatehi et al., 2022). Modeling HBV and Liver Cancer via Fractional Differential Operators Fractional calculus, an extension of classical calculus, focuses on the opera- tions of integration and differentiation of arbitrary orders (Nisar et al., 2022). During the 19th century, mathematicians developed fractional differential equations, frac- tional dynamics, and fractional geometry through the framework of fractional calcu- lus. Presently, fractional calculus is applied across nearly all scientific disciplines. It is utilized to model phenomena in physics, engineering, and biology where tra- ditional integer-order mathematical models prove inadequate. Consequently, disci- 30 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library plines such as mechanics, chemistry, biology, and image processing have signifi- cantly benefited from the adoption of fractional calculus. This approach addresses a range of issues, including historical and nonlocal effects, which are inadequately captured by integer-order derivatives. The Riemann-Liouville and Caputo fractional- order derivatives (RLFD) have been pivotal in the advancement of this research. The incubation periods for bacteria, viruses, and vectors to acquire infectious- ness exemplify how many biological systems exhibit memory effects or delayed re- sponses (Sun et al., 2022). Fractional differential equation modeling of these systems provides distinct advantages over conventional integer-order models, which neglect such effects. Studying the immune system’s response to cancer interactions is essen- tial (Sun et al., 2022). A primary objective among researchers is to accurately model these interactions without overly complicating the mathematical framework. Recently, there has been a growing focus on the issue of solitary kernels asso- ciated with fractional derivatives. To address this, (Amin et al., 2022) introduced a range of novel definitions. These new definitions, characterized by their nonsingu- lar kernels, have had a profound impact. The Caputo-Fabrizio, Atangana-Baleanu (Atangana, 2018), and Caputo fractional derivatives (Kai, 2010) differ primarily in their underlying definitions: the Caputo-Fabrizio derivative is defined by an expo- nential decay law, the Atangana-Baleanu derivative by the Mittag-Leffler (ML) law, and the Caputo fractional derivative by a power law . Consequently, these fractional differential operators have been employed by researchers to tackle various practi- cal challenges. For instance, Farman et al. (2021) explored the role of fractional time derivative operators in modeling anomalous diffusion. The study in (Ghan- bari & Gómez-Aguilar, 2018) expanded on the model of nutrients, phytoplankton, and zooplankton using variable-order fractional differential operators. Additionally, (J. Gómez-Aguilar et al., 2019) applied robust methods to investigate the dynamics of fractional calcium oscillation models, focusing on controlling nonlinear fractional systems. The influence of vitamins on tumor cell proliferation across various fractional orders was elucidated by (Ahmad et al., 2020), who explored a fractional-order 31 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library tumor-immune-vitamin model (TIVM) employing the Mittag-Leffler derivative. Uti- lizing adjusted parameters from the Caputo fractional-order model, Sweilam et al. (2020) developed a mathematical framework to control cancer cell dynamics. Their research incorporated optimal control strategies for a fractional-order delay model in oncological therapy. The primary aim of their study was to minimize tumor cell populations in the context of fractional optimal control problems (FOCPs) using the proposed paradigm. The fractional-order model proposed by Ahmad et al. (2020) incorporates two immune effectors interacting with cancer cells. Nevertheless, fractional derivatives often lead to singularities that are imprac- tical for modeling dynamic systems. To address this issue, the Caputo-Fabrizio (CF) operator was introduced as a novel fractional derivative (Caputo & Fabrizio, 2015). Despite its advantages, the Caputo-Fabrizio operator presents challenges, including issues related to kernel locality. Atangana & Baleanu (2016) proposed an innovative fractional derivative to resolve these issues, incorporating the generalized Mittag- Leffler function (MLF) derived from the power series analysis of the complex num- ber system as a non-local and non-singular kernel. Gao et al. (2021) introduced a new mathematical model for hepatitis B virus (HBV) incorporating the Caputo- Fabrizio fractional derivative with immune delay. The existence and uniqueness of solutions for this model were established using Laplace transforms and fixed point theorems. Building on this work, Sun et al. (2022) expanded the model to include cytotoxic T lymphocytes (CTL) and alanine aminotransferase (ALT) levels, which are indicative of liver damage. They highlighted the absence of studies analyzing HBV dynamics that integrate logistic proliferation, time delays, and CTL and ALT factors using fractional differential equations. The non-local nature of fractional differential equations (FDEs) encompasses not only the current state but also historical states. Given that the differentiation order, φ, is unique and can vary significantly based on model outcomes, converting models governed by integer differential equations (IDEs) to FDEs requires preci- sion. IDEs are limited in their capacity to model certain phenomena, whereas FDEs offer a more comprehensive approach. Due to their relevance in modeling mem- 32 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library ory and inherited traits, biological systems are predominantly modeled using FDEs. The use of fractional-order derivatives captures various issues, including historical and non-local effects, and has been applied to numerous physical process modeling scenarios. Ahmad et al. (2020) investigated a fractional-order model involving in- teractions between two immune effectors and cancer cells. Their study incorporated a comprehensive immune system model, considering multi-functional pathways and immune system cross-reactivity. The (AB) operator has been employed to investigate the tumor immune vita- mins model as documented by (Ahmad et al., 2020). A non-singular operator has been utilized to analyze the time-fractional Klein-Gordon equation, encompassing solitary and shock wave solutions (Saifullah et al., 2021). Leveraging a fractional- order operator, Alzahrani et al. (2021) developed a mathematical framework for Covid-19, incorporating effects such as isolation and quarantine. (Ahmed et al., 2012) examined a fractional-order model involving two immune effectors interacting with cancer cells. A novel mathematical paradigm was introduced by (Akman Yıldız et al., 2018) to elucidate the impact of obesity on malignant tumor growth following both immunotherapy and chemotherapy. They addressed an optimal control prob- lem aimed at eradicating the tumor population and optimizing drug dosage over a finite time horizon. The model, incorporating Caputo time fractional derivatives, includes variables such as tumor cells, immune cells, adipose cells, and drug con- centrations for both chemotherapy and immunotherapy. Analytical investigations were conducted to ascertain the stability and existence of equilibrium points, includ- ing coexisting and tumor-free equilibria. Additionally, they meticulously analyzed the correlation between the cost functional value, the differentiation order α, and the decay rate of chemotherapeutic agents. Their findings suggest that employing a fractional derivative with a nonsingular kernel, in conjunction with an optimal treatment strategy, is beneficial for devising the most effective immunotherapy and chemotherapy regimen. To the best of the existing knowledge of the subject matter, although exten- sive research has been conducted to address hepatitis B virus and liver cancer, no 33 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library mathematical modeling has been undertaken to explore the co-existence dynamics of these two diseases. This entails the assimilation of both fractional-order and classical-order models, which would enable the incorporation of the immune sys- tem’s function and the implementation of control strategies for addressing infected cells. Consequently, a convergence of this research with the aforementioned math- ematical frameworks related to the transmission dynamics of hepatitis B virus and hepatocellular carcinoma is expected to yield profound insights and bolster initia- tives aimed at the eradication of these widespread global maladies. Chapter Summary This chapter has encapsulated an exhaustive examination of various mathe- matical paradigms employed to elucidate the intricacies and mechanisms governing the propagation of the hepatitis B virus and hepatocellular carcinoma, along with the essential role of the immune system in regulating these infectious processes. Fur- thermore, a historical overview of the implementation of both fractional-order and classical-order mathematical models for the investigation and control of infectious diseases was presented. These models were devised with a meticulous consideration of the properties of the hepatitis B virus, the transmission dynamics of liver cancer, and the immunological responses provoked by these pathogens. Scholarly endeavors have concentrated on the conventional SIR model to rep- resent the dynamics of hepatitis B virus and liver cancer, while other research has employed modified versions of the SIR framework to explore the transmission dy- namics of these conditions and incorporate the immune system’s reaction. To deter- mine whether hepatitis B virus and liver cancer could establish endemicity within a population, the basic reproduction number, R0, was calculated for both classes of models. Consequently, this study adapted the classical SIR model initially in- troduced by (Kermack & McKendrick, 1927), utilizing both fractional-order and classical-order deterministic models to rigorously investigate the dynamics of hep- atitis B virus and liver cancer dissemination. 34 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library CHAPTER THREE RESEARCH METHODS Introduction In order to fulfill the goals set in Chapter one, a classical and fractional or- der mathematical models were developed in this chapter that uses a deterministic method to study the immune system’s role in combating the co-existence dynam- ics of hepatic carcinoma and the hepatitis B virus (HBV). The model’s predictive value (R0) was assessed based on how well it reflected the biological systems of the two infections. The transition period between acute and chronic infection was also determined. Classical-order Model Classical calculus is used to model infectious diseases by employing differ- ential equations to characterize the dynamics of variation within population strata, such as Susceptible (S), Infected (I), and Recovered (R), in models akin to the SIR paradigm. These equations capture how individuals transition between compart- ments based on factors like infection and recovery rates. For instance, the velocity at which susceptible individuals transition to infected states is frequently represented as commensurate with the number of prevailing infections. Solving these equations allows researchers to predict disease spread, estimate key thresholds such as the ba- sic reproduction number (R0), and assess the impact of control measures. This ap- proach provides a mathematical framework to understand and predict the dynamics of disease transmission over time. Model Formulation A compartmental mathematical model was developed to explore the complex interactions between hepatitis B pathogen and hepatic malignancy infections. This model relies on recent scientific insights into the core characteristics of HBV in- fections and hepatic carcinoma, as well as on theoretical constructs and empirical research conducted by other scholars (Nowak et al., 1996; Perelson et al., 1996; Ciupe, Ribeiro, Nelson, & Perelson, 2007; Long & Qi, 2009; Chenar et al., 2018; 35 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library Fatehi et al., 2022). The impact of HBV infection and the transition from acute in- fections to hepatic carcinoma is profoundly impacted by the functions of antibodies, cell-mediated immunity, innate immune mechanisms, and cytokines. Macrophages exposed to HBV infection exhibit dual functionality; they can act as both targets and defenders (Opoku & Mazandu, 2020). We modeled macro- phages (Kupffer cells) as the target cells in our mathematical model, rather than hepatocytes because of their central role in mediating immune responses and in- flammation in HBV infection and liver cancer progression. Although hepatocytes are the direct target of HBV, macrophages are critical in detecting HBV, releasing pro-inflammatory cytokines, and contributing to liver damage, fibrosis, and tumor progression Tacke (2017); Wan et al. (2014). Upon activation, these macrophages are capable of eradicating intracellular pathogens and participating in Type 1 and Type 2 T helper cell-mediated responses. Human hepatic cells (hepatocytes) are categorized into two discrete groups: unin- fected macrophages free from malignancy, labeled as M0, and macrophages com- promised by cancer, labeled as M1. These macrophages, whether or not they harbor cancer, are mobilized from the circulatory system to the locus of infection in reaction to the presence of HBV. Upon encountering the virus, these macrophages engage in phagocytosis, subsequently becoming contaminated at a specific moment t (Long & Qi, 2009; Chenar et al., 2018; Fatehi et al., 2022). The resultant macrophages harboring infection are labeled as IM . This phenomenon engenders the contamination of macrophages via interac- tions with HBV. It is conjectured that uninfected macrophages are synthesized at consistent rates ΛM0 and ΛM1 , while they undergo depletion at rates µM0 and µM1 , respectively. In our model, uninfected (cancer-affected) macrophages, M1, are syn- thesized at a constant rate ΛM1 , reflecting the steady progression of liver cancer due to external factors like excessive alcohol consumption and smoking. These fac- tors are independent of viral infection but contribute to the overall cancer burden. HBV entities infiltrate macrophages at a rate commensurate with the product of M0, M1, and V (specifically, β1VM0 and β2VM1), with proportionality coefficients β1 36 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library and β2 (Long & Qi, 2009; Chenar et al., 2018; Fatehi et al., 2022). Contaminated macrophages are eradicated at a constant rate of µIM per cell. Among the infected macrophages, some progress to cirrhosis and subsequently to hepatic carcinoma macrophages at a rate ζ (Bhadauria, 2011). Hepatic carcinoma macrophages may contract hepatitis through tainted blood transfusions or sexual contact with infected individuals. HBV particles V are generated at a rate ΛV , while infected macrophages produce additional free virions at a rate ω (Chenar et al., 2018; Fatehi et al., 2022). Virions undergo degradation at a constant rate µV within each cell. Figure 1 delin- eates the interactions among diverse cell types within the human liver. A primary role of the innate immune response is to promptly arrest the spread of the virus to adjacent cells following its entry into the liver cell (Laing, 2022). Innate immune cells are equipped to manage mild HBV infections without inducing severe inflammation once the pathogen is endocytosed and processed. However, in cases of severe infection, an adaptive immune response, characterized by pro- inflammatory macrophages and T cells, becomes essential. The adaptive immune response involves HBV-specific T lymphocytes, referred to as naive T cells T0, and HBsAg-specific antibodiesA that target and neutralize virions (Laing, 2022; Howard et al., 2022). Pathogen incursion is specifically mitigated by antibodies, which bind to and neutralize the pathogens. During HBV infections, cells compromised by the virus generate surplus sub-viral particles (SVPs) in quantities ranging from one thousand to one million times the number of infectious viral particles. These SVPs may pro- foundly impact the host’s immunological response to HBV infection. As elucidated by Ciupe et al. (2014); J. Hu & Liu (2017), SVPs function as subterfuges, divert- ing antiviral immunoglobulins away from targeting the hepatitis B virus. Further- more, they may contribute to immunological tolerance during neonatal contami- nation, thereby postponing the production of neutralizing antibodies and allowing HBV to circumvent immune surveillance. The response is orchestrated through the generation of macrophages, T lymphocytes, and cytokines. In the absence of infec- tion, HBsAg-specific antibodies (A) are synthesized at a steady rate ΛA and undergo 37 University of Cape Coast https://ir.ucc.edu.gh/xmlui Digitized by Sam Jonah Library degradation at a per capita rate µA. During infection, antibodies are produced at a pace ψ, which correlates with the viral burden, and they neutralize viral particles at a pace δ4 (Ciupe et al., 2014; Chenar et al., 2018; Fatehi et al., 2022). Macrophage T cells are produced at a pace ΛT0 and experience mortali