Analysis of Malaria Model with Two Infectious Classes and Seasonal Mosquito Birth Rate

Abstract

Malaria is a complex and deadly infectious disease that puts approximately 3.3 billion people at risk in 109 countries and territories around the world. A mathematical model for malaria was analysed to better understand the transmission and spread of this disease. The goal was to use this model to estimate human malaria infection for a given data set. The model was analysed for the existence and stability of disease-free and endemic (malaria persisting in the population) equilibria. In some cases essential parameters (i.e. parameters with significant influence on the endemic equilibrium) was determined and also bifurcation parameter K was calculated. It was shown that a scaled version of the model has a unique endemic equilibrium for certain values of the ratio of mosquito to human population, which is always a global attractor. Otherwise, there is no endemic equilibrium and the disease-free equilibrium is a global attractor. When the ratio of mosquito to human population changes the endemic equilibrium changes and forms a curve Ce in the phase space parameterized by this ratio. For a certain range of the rate of human population entering the susceptible class (either by birth or migration) the original model has an equilibrium on the curve Ce. This equilibrium is a saddle with a four-dimensional stable and a one-dimensional unstable manifold. The unstable manifold is well approximated by this curve. A new version of periodicity was introduced and applied it to two malaria models. The results of the analyses of these models with the periodicity on how they estimate human malaria infection was compared with the results of the models without periodicity.