West Nile Virus Model Research Articles

Existence and Uniqueness Solution for a Mathematical Model with Mittag-Leffler Kernel

In this work, we analyse the fractional order West Nile Virus model involving the Atangana-Baleanu derivatives. Existence and uniqueness solutions were obtained by the fixed-point theorem. Another impressive aspect of the work is illustrated by simulations of different fractional orders by calculating the numerical solutions of the mathematical model.

Sharp asymptotic profile of the solution to a West Nile virus model with free boundary

AbstractWe consider the long-time behaviour of a West Nile virus (WNv) model consisting of a reaction–diffusion system with free boundaries. Such a model describes the spreading of WNv with the free boundary representing the expanding front of the infected region, which is a time-dependent interval $[g(t), h(t)]$ in the model (Lin and Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary. J. Math. Biol. 75, 1381–1409, 2017). The asymptotic spreading speed of the front has been determined in Wang et al. (Spreading speed for a West Nile virus model with free boundary. J. Math. Biol. 79, 433–466, 2019) by making use of the associated semi-wave solution, namely $\lim _{t\to \infty } h(t)/t=\lim _{t\to \infty }[\!-g(t)/t]=c_\nu$ , with $c_\nu$ the speed of the semi-wave solution. In this paper, by employing new techniques, we significantly improve the estimate in Wang et al. (Spreading speed for a West Nile virus model with free boundary. J. Math. Biol. 79, 433–466, 2019): we show that $h(t)-c_\nu t$ and $g(t)+c_\nu t$ converge to some constants as $t\to \infty$ , and the solution of the model converges to the semi-wave solution. The results also apply to a wide class of analogous Ross–MacDonold epidemic models.

A West Nile virus nonlocal model with free boundaries and seasonal succession.

The paper deals with a West Nile virus (WNv) model, in which the nonlocal diffusion characterizes the long-range movement of birds and mosquitoes, the free boundaries describe their spreading fronts, and the seasonal succession accounts for the effect of the warm and cold seasons. The well-posedness of the mathematical model is established, and its long-term dynamical behaviours, which depend upon the generalized eigenvalues of the corresponding linearized differential operator, are investigated. For both spatially independent and nonlocal WNv models with seasonal successions, the generalized eigenvalues are studied and applied to determine whether the spreading or vanishing occurs. Our results extend those for the case with nonlocal diffusion but no free boundary and those for the case with free boundary but local diffusion, respectively. The generalized eigenvalues reveal that there exists positive correlation between the duration of the warm season and the risk of infection. Moreover, the initial infection length, the initial infection scale and the spreading ability to new areas all play important roles for the long time behaviors of the time dependent solutions.

Dynamics of West Nile virus driven by seasonal fluctuations in a spatially variable habitat

<p style='text-indent:20px;'>Seasonality is the common phenomena in ecological evolution. Firstly, we formulate and explore a simplified West Nile virus model, which describes the transmission of West Nile virus driven by seasonal fluctuations in a spatially variable habitat where the spatial movement of the infectious mosquitoes and infectious birds are described by Laplace diffusion. The range of the infected area is assumed to be a moving interval <inline-formula><tex-math id="M5">\begin{document}$ [0, h(t)] \subset \mathbb{R} $\end{document}</tex-math></inline-formula>, with its right end point representing the spreading fronts of the disease. We will mainly investigate the impact of spatial heterogeneity of environment and temporal periodicity on the persistence and extinction of West Nile virus. The basic reproduction number <inline-formula><tex-math id="M6">\begin{document}$ R_0^D $\end{document}</tex-math></inline-formula> in a fixed region and the spatial-temporal risk index <inline-formula><tex-math id="M7">\begin{document}$ R_0^F(t) $\end{document}</tex-math></inline-formula> for the free boundary problem, which depends on spatial heterogeneity, temporal periodicity and spatial diffusion, are defined by the associated linearized eigenvalue problems. Sufficient conditions for the spreading and vanishing of West Nile virus are presented for the spatial dynamics of the virus. At last, we explore the long time dynamical behavior of the solution to free boundary problem when the spreading occurs.</p>

Threshold dynamics of a nonlocal diffusion West Nile virus model with spatial heterogeneity

<abstract><p>In this study, we investigated the threshold dynamics of a spatially heterogeneous nonlocal diffusion West Nile virus model. By employing semigroup theory and continuous Fréchet-differentiable, we established the well-posedness of the solution. The expression for the basic reproduction number derived using the next-generation matrix method. The authors demonstrated the threshold dynamics of the system by constructing a Lyapunov function and applying the comparison principle. Finally, numerical simulations were used to validate the theorem results. It can be suggested that to control disease development rapidly, measures should be taken to reduce the spread of mosquitoes and birds.</p></abstract>

Threshold dynamics of a nonlocal and time-delayed West Nile virus model with seasonality

In this paper, we study a nonlocal periodic reaction–diffusion system modeling the West Nile virus transmission between mosquitoes and birds. For the subsystem of mosquito growth, we establish a global stability result on the extinction and persistence in terms of mosquito reproduction number RM, which is defined as the cone spectral radius of a monotone homogeneous map. For the full model, we introduce the basic reproduction number R0 and show that the disease transmission dynamics is determined by the sign of R0−1 under the assumption that RM>1. Moreover, we obtain the global attractivity of the positive constant steady state in the case where all the coefficients are constants. We also conduct numerical simulations to reveal the effects of diffusion, heterogeneity and periodic delay on R0.

Dynamics of a periodic West Nile virus model with mosquito demographics

<p style='text-indent:20px;'>In this paper, we propose a time-delayed model of West Nile virus with periodic extrinsic incubation period (EIP) and mosquito demographics including stage-structure, pair formation and intraspecific competition. We define two quantities <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_{\rm min} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_{\rm max} $\end{document}</tex-math></inline-formula> for mosquito population and the basic reproduction number <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> for our model. It is shown that the threshold dynamics are determined by these three parameters: (ⅰ) if <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_{\rm max}\leq 1 $\end{document}</tex-math></inline-formula>, the mosquito population will not survive; (ⅱ) if <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{R}_{\rm min}>1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \mathcal{R}_0<1 $\end{document}</tex-math></inline-formula>, then WNv disease will go extinct; (ⅲ) if <inline-formula><tex-math id="M7">\begin{document}$ \mathcal{R}_{\rm min}>1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \mathcal{R}_0>1 $\end{document}</tex-math></inline-formula>, then the disease will persist. Numerically, we simulate the long-term behaviors of solutions and reveal the influences of key model parameters on the disease transmission. A new finding is that <inline-formula><tex-math id="M9">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> is non-monotone with respect to the fraction of the aquatic mosquitoes maturing into adult male mosquitoes, which can help us implement more effective control strategies. Besides we observe that using the time-averaged EIP has the possibility of underestimating the infection risk.</p>

Spreading speed for some cooperative systems with nonlocal diffusion and free boundaries, part 1: Semi-wave and a threshold condition

We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. Such a system covers various models arising from mathematical biology, in particular a West Nile virus model and an epidemic model considered recently in [16] and [44], respectively, where a “spreading-vanishing” dichotomy is known to govern the long time dynamical behaviour, but the question on spreading speed was left open. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and travelling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave. This is Part 1 of a two part series. In Part 2, for some typical classes of kernel functions, we will obtain sharp estimates of the spreading rate for both the finite speed case, and the infinite speed case.

Spatial transmission and risk assessment of West Nile virus on a growing domain

This paper is concerned with a West Nile virus (WNv) model on a growing domain, which accounts for habitat expansion of mosquitoes because of climate warming. We aim to understand the relationship of the growing rate and the transmission risk of WNv. The basic reproduction number, which is related to the growing rate and diffusion rate, is introduced through spectral theory. The conditions to determine whether the virus vanishes or spreads are deduced. The obtained results reveal that domain growth leads to increased risk of infection and is detrimental to the control and prevention of WNv. To verify the feasibility of our analytical results on the long‐time behavior of WNv, some numerical simulations are given.

Analysis of a West Nile virus model with nonlocal diffusion and free boundaries**Research partially supported by the Australian Research Council.

We consider a West Nile virus model with nonlocal diffusion and free boundaries, in the form of a cooperative evolution system that can be viewed as a nonlocal version of the free boundary model of Lin and Zhu (2017 J. Math. Biol. 75 1381–1409). The model is a representative of a class of ‘vector-host’ models. We prove that this nonlocal model is well-posed, and its long-time dynamical behaviour is characterised by a spreading-vanishing dichotomy. We also find the criteria that completely determine when spreading and vanishing can happen, revealing some significant differences from the model in Lin and Zhu (2017 J. Math. Biol. 75 1381–1409). It is expected that the nonlocal model here may exhibit accelerated spreading (see remark part (c)), a feature contrasting sharply to the corresponding local diffusion model, which has been shown by Wang et al (2019 J. Math. Biol. 79 433–466) to have finite spreading speed whenever spreading happens. Many techniques developed here are applicable to more general cooperative systems with nonlocal diffusion.

A Periodic West Nile Virus Transmission Model with Stage-Structured Host Population

In this paper, we study an avian (host) stage-structured West Nile virus model, which incorporates seasonality as well as stage-specific mosquito biting rates. We first introduce the basic reproduction number R0 for this model and then show that the disease-free periodic solution is globally asymptotically stable when R0<1, while there exists at least one positive periodic solution and that the disease is uniformly persistent if R0>1. In the case where all coefficients are constants, for a special case, we obtain the global stability of the disease-free equilibrium, the uniqueness of the endemic equilibrium, and the permanence of the disease in terms of the basic reproduction number R0. Numerical simulations are carried out to verify the analytic result. Some sensitivity analysis of R0 is performed. Our finding shows that an increase in juvenile exposure will lead to more severe transmission. Moreover, we find that the ignorance of the seasonality may result in underestimation of the basic reproduction number R0.

Monotone dynamics and global behaviors of a West Nile virus model with mosquito demographics.

In this paper a mathematical model is formulated to study transmission dynamics of West Nile virus (WNv), which incorporates mosquito demographics including pair formation, metamorphic stages and intraspecific competition. The global behaviors of the model are obtained from a geometric approach and theory of monotone dynamics, even though bistability is present due to backward bifurcation. It turns out that the model can be investigated through two auxiliary subsystem, which are cooperative and K-competitive, respectively. Together with implement of compound matrices and Poincaré-Bendixson theorem, a thorough classification of dynamics of the full model is characterized by mosquito reproduction number [Formula: see text], WNv reproduction number [Formula: see text] and a bistability subthreshold [Formula: see text]. The theoretical results show that if [Formula: see text] is not greater than 1, mosquitoes will not survive, and the WNv will die out; if [Formula: see text] is greater than 1, then mosquitoes will persist, and disease may prevail or vanish depending on basin of attraction of the local attractors which are singletons. Our method in this paper can be applied to other mosquito-borne diseases such as malaria, dengue fever which have a similar monotonicity.

A West Nile Virus Model with Vertical Transmission and Periodic Time Delays

Seasonal change has played a critical role in the evolution dynamics of West Nile virus transmission. In this paper, we formulate and analyze a novel delay differential equation model, which incorporates seasonality, the vertical transmission of the virus, the temperature-dependent maturation delay and the temperature-dependent extrinsic incubation period in mosquitoes. We first introduce the basic reproduction ratio $$R_0$$ for this model and then show that the disease is uniformly persistent if $$R_0>1$$. It is also shown that the disease-free periodic solution is attractive if $$R_0<1$$, provided that there is only a small invasion. In the case where all coefficients are constants and the disease-induced death rate of birds is zero, we establish a threshold result on the global attractivity in terms of $$R_0$$. Numerically, we study the West Nile virus transmission in Orange County, California, USA.

Fractional optimal control in transmission dynamics of West Nile virus model with state and control time delay: a numerical approach

In this paper, an optimal control for a novel fractional West Nile virus model with time delay is presented. The proposed model is governed by a system of fractional delay differential equations, where the fractional derivative is defined in the Grünwald–Letnikov sense. Stability analysis of fixed points is studied. Corresponding fractional optimal control problem, with time delays in both state and control variables, is formulated and studied. Two simple numerical methods are used to study the nonlinear fractional delay optimal control problem. The methods are standard finite difference method and nonstandard finite difference method. Comparative studies are implemented, it is found that the nonstandard finite difference method is better than the standard finite difference method.

Spreading speed for a West Nile virus model with free boundary.

The purpose of this paper is to determine the precise asymptotic spreading speed of the virus for a West Nile virus model with free boundary, introduced recently in Lin and Zhu (J Math Biol 75:1381-1409, 2017), based on a model of Lewis et al. (Bull Math Biol 68:3-23, 2006). We show that this speed is uniquely defined by a semiwave solution associated with the West Nile virus model. To find such a semiwave solution, we firstly consider a general cooperative system over the half-line [Formula: see text], and prove the existence of a monotone solution by an upper and lower solution approach; we then establish the existence and uniqueness of the desired semiwave solution by applying this method together with some other techniques including the sliding method. Our result indicates that the asymptotic spreading speed of the West Nile virus model with free boundary is strictly less than that of the corresponding model in Lewis et al. (2006).

Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment.

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.

Global dynamics of a West Nile virus model in a spatially variable habitat

In this paper, we investigate a mathematical model that describes the transmission dynamics of West Nile virus (WNV) associated with the mosquito–bird population in a continuous bounded habitat. Our model is given by a spatial reaction–diffusion system with the zero-flux condition on the boundary, which is motivated by the models in the previous works of Wonham et al. (2004) and Lewis et al. (2006). By using the comparison theorem and the theory of uniform persistence, we show that the global dynamics of the model can be determined by two indices, mosquito reproduction number and infection invasion threshold.

Asymptotic periodicity in a diffusive West Nile virus model in a heterogeneous environment

This paper is concerned with a diffusive West Nile virus model (WNv) in a heterogeneous environment. The basic reproduction number [Formula: see text] for spatially homogeneous model is first introduced. We then define a threshold parameter [Formula: see text] for the corresponding diffusive WNv model in a heterogeneous environment. It is shown that if [Formula: see text], the model admits at least one nontrivial T-periodic solution, whereas if [Formula: see text], the model has no nontrivial T-periodic solution. By means of monotone iterative schemes, the true solution can be obtained and the asymptotic behavior of periodic solutions is presented. The paper is closed with some numerical simulations to illustrate our theoretical results.

Comparative studies for the fractional optimal control in transmission dynamics of West Nile virus

In this paper, optimal control for a novel West Nile virus (WNV) model of fractional order derivative is presented. The proposed model is governed by a system of fractional differential equations (FDEs), where the fractional derivative is defined in the Caputo sense. An optimal control problem is formulated and studied theoretically using the Pontryagin maximum principle. Two numerical methods are used to study the fractional-order optimal control problem. The methods are, the iterative optimal control method (OCM) and the generalized Euler method (GEM). Positivity, boundedness and convergence of the IOCM are studied. Comparative studies between the proposed methods are implemented, it is found that the IOCM is better than the GEM.

Spreading and vanishing in a West Nile virus model with expanding fronts

In this paper, we study a simplified version of a West Nile virus model discussed by Lewis et al. [28], which was considered as a first approximation for the spatial spread of WNv. The basic reproduction number $R_0$ for the non-spatial epidemic model is defined and a threshold parameter $R_0 ^D$ for the corresponding problem with null Dirichlet boundary condition is introduced. We consider a free boundary problem with coupled system, which describes the diffusion of birds by a PDE and the movement of mosquitoes by a ODE. The risk index $R_0^F (t)$ associated with the disease in spatial setting is represented. Sufficient conditions for the WNv to eradicate or to spread are given. The asymptotic behavior of the solution to system when the spreading occurs are considered. It is shown that the initial number of infected populations, the diffusion rate of birds and the length of initial habitat exhibit important impacts on the vanishing or spreading of the virus. Numerical simulations are presented to illustrate the analytical results.