Development and analysis of a malaria transmission mathematical model with seasonal mosquito life‐history traits

Abstract

In this paper, we develop and analyze a malaria model with seasonality of mosquito life-history traits: periodicmosquitoes per capita birth rate, -mosquitoes death rate, -probability of mosquito to human disease transmission, -probability of human to mosquito disease transmission, and -mosquitoes biting rate. All these parameters are assumed to be time dependent leading to a nonautonomous differential equation system. We provide a global analysis of the model depending on two threshold parameters  0 and  0 < 1 (with  0 ≤  0 ). When  0 < 1, then the disease-free stationary state is locally asymptotically stable. In the presence of the human disease-induced mortality, the global stability of the disease-free stationary state is guarantied when  0 < 1. On the contrary, if  0 > 1, the disease persists in the host population in the long term and the model admits at least one positive periodic solution.Moreover, by a numerical simulation, we show that a sub-critical (backward) bifurcation is possible at  0 = 1. Finally, the simulation results are in accordance with the seasonal variation of the reported cases of a malariaepidemic region in Mpumalanga province in South Africa. K E Y W O R D S basic reproduction number, global stability, periodic solution, seasonal pattern, uniform persistence Stud Appl Math. 2020;144:389-411.wileyonlinelibrary.com/journal/sapm

A voluntary use of insecticide treated nets can stop the vector transmission of Chagas disease

“…One can include disease related mortality which was omitted here for the sake of simplicity. A disease related mortality could cause a backward bifurcation and an existence of endemic equilibria even for [ 80 , 81 ]. One can also relax the assumption about vector migration and allow the vectors to migrate from the settlement.…”

Human-vector malaria transmission model structured by age, time since infection and waning immunity

“…By setting R 0 := R 2 0 , it follows that the necessary and sufficient condition (38), with N * h , s * h respectively given by ( 41)-( 42), becomes The latter condition has only i * h as unknown, but is though difficult to exploit because i * h depends both on a and τ . Let us recall the constant C bif given by (15). We can now give details on the proof of Theorem 3.6 in several steps.…”

“…[4] and the references therein. Most mathematical models about the transmission dynamic of malaria use ordinary differential equations [6,7,8,15,19,22,24,36,37,46,52,55,56,60].…”

Threshold dynamics in mathematical models for mosquito- and rodent-borne diseases with seasonality

“…Periodicity of weather and climate change are very important factors in the life cycle of the parasites and the mosquitoes transmitting them. Hence, it is of crucial importance to understand how changes in weather affect the spread of malaria [38]. Mordecai et al [73] formulated a nonlinear thermal-response model to explain the role of temperature changes in the spread of malaria.…”

“…In the case of a disease like malaria, which depends on the abundance of mosquitoes, which, in turn, is highly dependent on the periodically changing weather, it is especially important to include this seasonality in our models. Several papers [13,33,38,79,94,102,104] study the spread of malaria transmission with periodically changing mosquito birth, death and biting rates.…”