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In this paper, we formulate a mathematical model of vector-borne disease dynamics. The model is constructed by considering two models : a baseline model of vector population dynamics due to Lutambi et al. that takes into account the development of the aquatic stages and the female mosquitoes gonotrophic cycle and an SI-SIR model describing the interaction between mosquitoes and human hosts. We briefly study the baseline model of vectors dynamics and, for the transmission model, we explicitly compute the equilibrium points, and by using the method of Van den Driesshe and J. Watmough, we derive the basic reproduction number ℛ0. Otherwise, thanks to Lyapunov’s principle, Routh-Hurwitz criteria and a favorable result due to Vidyasagar, we establish the local and global stability results of the equilibrium points. Furthermore, we establish an interesting relationship between the mosquito reproduction number ℛv and the basic reproduction number ℛ0. It then follows that aquatic stages and behavior of adult mosquitoes have a significant impact on disease transmission dynamics. Finally, some numerical simulations are carried out to support the theoretical findings of the study.

In this article, we propose three mathematical models of mosquito dynamics. The first model describes the dynamics of wild mosquitoes in combination with mechanical control and larvicide. For this model, we derive a threshold parameter $ \mathcal {N} $ N , called the basic number of offspring, and show that the trivial equilibrium is globally asymptotically stable when $ \mathcal {N}\leq ~1 $ N ≤ 1 , while if $ \mathcal {N} \gt 1 $ N > 1 , the non-trivial equilibrium is globally asymptotically stable. In the second model, we evaluated a constant release of Wolbachia-infected males in the wild mosquito population. Our analysis identified a critical threshold for releasing Wolbachia-infected male mosquitoes $ M_{i}^{\rm crit} $ M i crit to reduce or eliminate the wild mosquito population. In the third model, we assessed wild mosquito dynamics by periodically releasing Wolbachia-infected male mosquitoes. The analysis of this model produced two thresholds, $ T^{*} $ T ∗ the threshold on the release period and $ \Lambda ^{*} $ Λ ∗ the threshold on the release quantity. The existence of trivial periodic solutions is obtained, and the corresponding local and global stability conditions are proved by Floquet's theory and Lyapunov's stability theorem, respectively. We also prove the existence conditions for non-trivial periodic solutions and their local stability. Finally, we perform some numerical simulations to illustrate our theoretically pertinent results. [ABSTRACT FROM AUTHOR]

In this paper, we formulate a temperature-dependent model for malaria transmission dynamics which includes immature stages of mosquitoes. The model is constructed by using ordinary differential equations with some parameters which are periodic functions. Two thresholds dynamics associated to the model have been derived: the vector reproduction ratio ℛv and the basic reproduction ratio ℛ0. Through a rigorous analysis via theories and methods of dynamical systems, we prove that the global behavior of the model depends strongly on these two parameters. More precisely, we show that if ℛv is greater than one and ℛ0 is less than one then, the disease-free periodic equilibrium is globally attractive. If ℛv is greater than one and ℛ0 is greater than one, the disease remains persistent and the system admits at least one positive periodic solution. Finally, using the reported monthly mean temperature for Burkina Faso, numerical simulations are carried out to illustrate our mathematical results.

In this paper, we present a mathematical model of malaria transmission dynamics with age structure for the vector population and a periodic biting rate of female anopheles mosquitoes. The human population is divided into two major categories: the most vulnerable called non-immune and the least vulnerable called semi-immune. By applying the theory of uniform persistence and the Floquet theory with comparison principle, we analyse the stability of the disease-free equilibrium and the behaviour of the model when the basic reproduction ratio $ \mathcal {R}_0 $ is greater than one or less than one. At last, numerical simulations are carried out to illustrate our mathematical results.

In this paper, we formulate a mathematical model of malaria transmission dynamics with impulsive and periodic release of Wolbachia-infected male mosquitoes. Our objective is to evaluate the impact of periodic release of Wolbachia-infected male mosquitoes on malaria transmission. Subsequently, to find out how often and in what quantities Wolbachia-infected male mosquitoes need to be released to effectively reduce malaria transmission. By using the comparison principle, Floquet theory and some of the analytical methods, we obtain the basic reproduction number. Then, we demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity, and we prove the uniform persistence of the disease if the reproduction number is greater than unity. Finally, we perform numerical simulations to illustrate our theoretically pertinent 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 have formulated and analyzed a mathematical model describing the dynamics of the phytoplankton producing toxin and the fish population by using an ordinary differential equations system. The phytoplankton population is divided into two groups, namely infected phytoplankton and susceptible phytoplankton. We aim to analyze the effect of the toxic substance on the fish population. The equilibria stability of the model has been studied locally and globally around the basic reproduction ratio R0. The mathematical analysis of the model shows that the equilibrium without disease is globally asymptotically stable if R0 ≤ 1 and the endemic equilibrium is globally asymptotically stable if R0 > 1. Numerical simulations are carried out to illustrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we propose a nonlinear reaction-diffusion system describing the interaction between toxin-producing phytoplankton and fish population. We analyze the effect of self- and cross-diffusion on the dynamics of the system. The existence, uniqueness and uniform boundedness of solutions are established in the positive octant. The system is analyzed for various interesting dynamical behaviors which include boundedness, persistence, local stability, global stability around each equilibria based on some conditions on self- and cross-diffusion coefficients. The analytical findings are verified by numerical simulation. [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]