Global Dynamics of a Seasonal Mathematical Model of Schistosomiasis Transmission With General Incidence Function

Abstract

In this paper, we investigate a nonautonomous and an autonomous model of schistosomiasis transmission with a general incidence function. Firstly, we formulate the nonautonomous model by taking into account the effect of climate change on the transmission. Through rigorous analysis via theories and methods of dynamical systems, we show that the nonautonomous model has a globally asymptotically stable disease-free periodic equilibrium when the associated basic reproduction ratio [Formula: see text] is less than unity. Otherwise, the system admits at least one positive periodic solutions if [Formula: see text] is greater than unity. Secondly, using the average of periodic functions, we further derive the autonomous model associated with the nonautonomous model. Therefore, we show that the disease-free equilibrium of the autonomous model is locally and globally asymptotically stable when the associated reproduction ratio [Formula: see text] is less than unity. When [Formula: see text] is greater than unity, the existence and global asymptotic stability of the endemic equilibrium is established under certain conditions. Finally, using linear and nonlinear specific incidence function, we perform some numerical simulations to illustrate our theoretical results.

A mathematical analysis of Hopf-bifurcation in a prey-predator model with nonlinear functional response

“…The trivial stationary states of system (2.3) are given in the following proposition [8,[11][12][13].…”

“…The following proposition gives the necessary and sufficient conditions of stability in case there is more than one equilibrium point [13,19,22,27,30].…”

“…These different types of functional responses present a key element for understanding the dynamics of these populations. The main questions concerning population dynamics concern the effects of interaction in the regulation of natural populations, the reduction of their size, the reduction of their natural fluctuations, or the destabilization of the equilibria in oscillations of the states of the population [8][9][10][11][12][13]. The predator-prey relationship is important to maintain the balance between different animal species.…”

Mathematical analysis of mosquito population global dynamics using delayed-logistic growth

“…It usually takes about 1-2 weeks before mosquitoes mature to adulthood, a time frame which is largely relative to the average lifespan of the mosquito. To account for this delay, delay-differential equation models with delay in recruitment are composed [11,24]. Indeed, the three first stages are all aquatic and the adult stage is aerial.…”

“…In [12], Koutou et al have proposed an autonomous mathematical model of mosquito growth dynamics including the immature stages of the vectors. And more recently, by considering the climate effects and applying the theory of uniform persistence and the Floquet theory, Traoré et al [24] have extended the study proposed by Koutou et al Due to the complexity of the dynamics of mosquito populations, and since only the adult female mosquitoes are responsible for transmitting diseases, therefore in general, models only focus on describing the dynamics of adult female mosquitoes, [7,17]. Usually, the total size of the mosquito population is treated as a constant value or the simplest model of population growth due to Malthus is used.…”