Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment

Abstract

A family of deterministic SEIRS epidemic dynamic models for malaria is presented. The family type is determined by a general functional response for the nonlinear incidence rate of the disease. Furthermore, the malaria models exhibit three random delays — the incubation periods of the plasmodium inside the female mosquito and human hosts, and also the period of effective acquired natural immunity against the disease. Insights about the effects of the delays and the nonlinear incidence rate of the disease on (1) eradication and (2) persistence of malaria in the human population are obtained via analyzing and interpreting the global asymptotic stability results of the disease-free and endemic equilibrium of the system. The basic reproduction numbers and other threshold values for malaria are calculated, and superior threshold conditions for the stability of the equilibria are found. Numerical simulation results are presented.

The stochastic extinction and stability conditions for nonlinear malaria epidemics

“…The results in [Theorem 5.1, Wanduku [51]] also show that when R 0 > 1, and the expected survival probability rate E(e −µ(T 1 +T 2 ) ) of the parasites over their complete life cycle is significant, the deterministic system (2.9) establishes a unique endemic equilibrium state denoted by E 1 = (S * 1 , E * 1 , I * 1 ). The current paper extends the previous study Wanduku [51], by incorporating the independent white noise perturbations of the effective disease transmission rate β, and the natural deathrates of the susceptible, exposed, infectious and removal populations. The primary focus of this study is to investigate the extinction of malaria in a class of stochastic models for vector-borne diseases in a very noisy environment comprising of variability from the disease transmission and natural death rates.…”

“…It is assumed in the formulation of this incidence rate that the number of infectious vectors at time t interacting and effectively transmitting infection to susceptible individuals, S, after β number of effective contacts per unit time per infective is proportional to the infectious human population, I, at earlier time t − T . Cook's method of effectively studying the dynamics of a vector-borne disease in a human population without directly including the vector population dynamics has been utilized by several other authors, for example [49,39,55,33,51]. demic dynamic models for malaria with three distributed delays:…”

“…The nonlinear incidence function G which signifies the response to disease transmission by the susceptible class as malaria increases in the population, satisfies the following assumptions More details about the derivation of the model in (2.9) is given in Wanduku [51]. The study…”

“…Wanduku [51] provides a suitable platform to investigate the dynamics of mosquito-borne diseases of humans with similar general structure as malaria, for instance, dengue fever, yellow fever, zika fever, lymphatic filariasis, and the different types of encephalitis etc. which are all transmitted by the mosquito.…”

“…In the analysis of the deterministic malaria model (1.1) with initial conditions in (1.2)-(1.3) in Wanduku [51], the threshold values for disease eradication such as the basic reproduction number for the disease when the system is in steady state are obtained in both cases where the delays in the system T 1 , T 2 and T 3 are constant, and also arbitrarily distributed.…”