Search

In this paper, we propose and analyze a detailed mathematical model describing the dynamics of a prey-predator model under the influence of an SIS infectious disease by using nonlinear differential equations. We use the functional response of ratio-dependent Michaelis-Menten type to describe the predation strategy. In the presence of the disease, prey and predator population are divided into two disjointed classes, namely infected and susceptible. The first one is governed through due predation interaction, and the second one is governed through the propagation of disease in the prey and predator population via predation. Our aim is to analyze the effect of predation on the dynamic of the disease transmission. Important mathematical results resulting from the transmission of the disease under influence of predation are offered. First, results concerning boundedness, uniform persistence, existence and uniqueness of solutions have been developed. In addition, many thresholds have been computed and used to investigate local and global stability analysis by using Routh-Hurwitz criterion and Lyapunov principle. We also establish the Hopf bifurcation to highlight periodic fluctuation with persistence of the disease or without disease in the prey and predator population. Finally, numerical simulations are carried out to illustrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a mathematical model of malaria transmission dynamics with maturation delay of a vector population in a periodic environment. The incidence rate between vector and human hosts is modeled by a general nonlinear incidence function which satisfies a set of conditions. Thus, the model is formulated as a system of retarded functional differential equations. Furthermore, through dynamical systems theory, we rigorously analyze the global behavior of the model. Therefore, we prove that the basic reproduction number of the model denoted by % is the threshold between the uniform persistence and the extinction of malaria virus transmission. More precisely, we show that if % is less than unity, then the disease-free periodic solution is globally asymptotically stable. Otherwise, the system exhibits at least one positive periodic solution if % is greater than unity. Finally, we perform some numerical simulations to illustrate our mathematical results and to analyze the impact of the delay on the disease transmission. [ABSTRACT FROM AUTHOR]

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 ℛ 0 is less than unity. Otherwise, the system admits at least one positive periodic solutions if ℛ 0 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 ℛ 1 is less than unity. When ℛ 1 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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show with a functional Beddington-DeAngelis response, that cross-diffusion plays an important role in the phenomenon of toxin-produc- tion-phytoplankton (TPP) in the dynamics of zooplankton and phytoplankton system. The demonstration tools are mainly based on the theory of fixed point indices and analytical techniques. Two-dimensional numerical simulations have established the formation of spatial patterns and a release threshold of the toxin, above which we talk about the phytoplankton proliferation. [ABSTRACT FROM AUTHOR]

The aim of this paper is the formulation and the study of a prey-predator model to describe fish and plankton population dynamics, with three developmental stages of the fish population (larva, juvenile and adult). First, we develop a mathematical model based on the ODEs, describing the dynamics of the various classes for the fish population depending on the plankton in a general framework. Then, we are interested in the model in the case of a trophic chain predation for the fish population. Finally, we continue our study through numerical simulations of the model in different fshing areas. The obtained numerical results confrm the mathematical analysis and allow us to have an idea on the development of the fish population in a fshing area. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a mathematical model of nonautonomous ordinary differential equations describing the dynamics of malaria transmission with age structure for the vector population. The biting rate of mosquitoes is considered as a positive periodic function which depends on climatic factors. The basic reproduction ratio of the model is obtained and we show that it is the threshold parameter between the extinction and the persistence of the disease. Thus, by applying the theorem of comparison and the theory of uniform persistence, we prove that if the basic reproduction ratio is less than 1, then the disease-free equilibrium is globally asymptotically stable and if it is greater than 1, then there exists at least one positive periodic solution. Finally, numerical simulations are carried out to illustrate our analytical results. [ABSTRACT FROM AUTHOR]

In this paper we present a mathematical model of malaria transmission. The model is an autonomous system, constructed by considering two models: a model of vector population and a model of virus transmission. The threshold dynamics of each model is determined and a relation between them established. Furthermore, the Lyapunov principle is applied to study the stability of equilibrium points. The common basic reproduction number has been determined using the next generation matrix and its implication for malaria management analyzed. Hence, we show that if the threshold dynamics quantities are less than unity, the mosquitoes population disappears leading to malaria disappearance; but if they are greater than unity, mosquitoes population persists and malaria also.Finally, numerical simulations are carried out to support our mathematical results. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model of malaria transmission which takes into account the four distinct mosquito metamorphic stages is presented. The model is formulated thanks to the coupling of two sub-models, namely the model of mosquito population and the model of malaria parasite transmission due to the interaction between mosquitoes and humans. Moreover, considering that climate factors have a great impact on the mosquito life cycle and parasite survival in mosquitoes, we incorporate seasonality in the model by considering some parameters which are periodic functions. Through a rigorous analysis via theories and methods of dynamical systems, we prove that the global behavior of the model depends strongly on two biological insightful quantities : the vector reproduction ratio R v and the basic reproduction ratio R 0. Indeed, if R v < 1 , mosquitoes and malaria die out; if R v > 1 and R 0 < 1 , the disease-free periodic equilibrium is globally attractive; and if R v > 1 and R 0 > 1 the disease remains persistent. Finally, using the reported monthly mean temperature of Burkina Faso, we perform some numerical simulations to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]