Deterministic, delay and stochastic in-host models for human malaria dynamics

Abstract

Malaria is one of the major public health hazards in the developing world in terms of infection, morbidity and mortality, with at least 300 million acute cases of malaria each year globally, resulting in more than a million deaths. The most vulnerable populations are pregnant women, their unborn babies and children under five years of age. Plasmodium falciparum, a parasite spread by the female anopheles mosquito, is the most common cause of malaria in humans and is responsible for almost all deaths associated with this case, followed by rare plasmodium vivax, popularly known for malaria relapse cases. In this thesis, we study the in-host dynamics of malaria (plasmodium falciparum and plasmodium vivax) based on the early work of Anderson et al's inhost models. We begin our research by reviewing Anderson et al (1989)'s model and incorporate treatment to the model. A drug efficacy threshold approximately equal to 0.9952 for which the parasite is cleared from the host's blood was determined using numerical simulations. We model the transplacental transmission of plasmodium falciparum (P:f alciparum) malaria in pregnant mothers. A treatment model for the transplacental transmission of P:f alciparum malaria in pregnant mothers with and without time delay was developed. The model is considered, first without delay and intervention and then PhD Thesis xv with treatment of the infected mother with Artemisinin-based combination therapies (ACTs) and injectable artesunate (AS). The model without delay, (4.8)-(4.13), is shown to possess three infected states, that is, an infected state where the erythrocytic infection in the host is active but latent in the foetus that is locally stable for R0at > 1 and R0ft < 1; a state where the erythrocytic infections are active in both the mother and the foetus which is locally stable for R0at > 1 and R0ft > 1; and lastly, a state where the erythrocytic infections are under control in the mother but active in the foetus which exist for for R0at < 1 and R0ft > 1: For the model incorporating treatment, the model reproduction numbers, R0ft and R0at are computed and numerical simulations carried out show that administering antimalarial drugs with a drug efficacy level of between 0.982 and 0.983, exclusively, will help in completely wiping out the malaria parasite in both the mother and foetus at an estimated placental out the malaria parasite in both the mother and foetus at an estimated placental drug transfer permiability factor of at least = 0:97: For the model with delay, we investigate the effect of intracellular delay on the stability of the parasite present equilibrium state. A critical condition is given to ensure that the parasite-present steady state is asymptotically stable for all delays. We also considers how the stability of the basic malaria model is altered by a Brownian diffusion structure. We first consider a model with constant diffusion matrix and show that for this diffusion structure, the revised Anderson et al model possesses two steady states, the parasite-free and parasite-present steady states whose stability is degraded by the diffusion term. This type of diffusion can be used to study infections whose states can switch from parasite-free to parasite-present and vice versa, such as P:f alciparum. Secondly, we consider models with a variable diffusion matrix, and that models with this diffusion structure only possess parasite-present steady states and can be used to study infections which maintain unstable parasite-present states with relapse tendencies, like P:vivax.