arXiv subjects

Rachidi B. Salako

Publications and source records attributed to Rachidi B. Salako.

17 recordsLinked to original sources

arXiv2024

Spatial profiles of a reaction-diffusion epidemic model with nonlinear incidence mechanism and varying total population

This paper considers a susceptible-infected-susceptible (SIS) epidemic reaction-diffusion model with no-flux boundary conditions and varying total population. The interaction of the susceptible and infected people is describe by the nonlinear transmission mechanism of the form $S^qI^p$, where $0 0$. In [39], we have studied a model with a constant total population. In the current paper, we extend our analysis to a model with a varying total population, incorporating birth and death rates. We investigate the asymptotic profiles of the endemic equilibrium when the dispersal rates of susceptible and/or infected individuals are small. Our work is motivated by disease control strategies that limit population movement. To illustrate the main findings, we conduct numerical simulations and provide a discussion of the theoretical results from the view of disease control. We will also compare the results for the models with constant or varying total population.

math.AP↗

arXiv2024

Dynamics of solutions to a multi-patch epidemic model with a saturation incidence mechanism

This study examines the behavior of solutions in a multi-patch epidemic model that includes a saturation incidence mechanism. When the fatality rate due to the disease is not null, our findings show that the solutions of the model tend to stabilize at disease-free equilibria. Conversely, when the disease-induced fatality rate is null, the dynamics of the model become more intricate. Notably, in this scenario, while the saturation effect reduces the basic reproduction number $\mathcal{R}_0$, it can also lead to a backward bifurcation of the endemic equilibria curve at $\mathcal{R}_0=1$. Provided certain fundamental assumptions are satisfied, we offer a detailed analysis of the global dynamics of solutions based on the value of $\mathcal{R}_0$. Additionally, we investigate the asymptotic profiles of endemic equilibria as population dispersal rates tend to zero. To support and illustrate our theoretical findings, we conduct numerical simulations.

q-bio.PE↗

arXiv2024

Asymptotic limits of the principal spectrum point of a nonlocal dispersal cooperative system and application to a two-stage structured population model

This work examines the limits of the principal spectrum point, $λ_p$, of a nonlocal dispersal cooperative system with respect to the dispersal rates. In particular, we provide precise information on the sign of $λ_p$ as one of the dispersal rates is : (i) small while the other dispersal rate is arbitrary, and (ii) large while the other is either also large or fixed. We then apply our results to study the effects of dispersal rates on a two-stage structured nonlocal dispersal population model whose linearized system at the trivial solution results in a nonlocal dispersal cooperative system. The asymptotic profiles of the steady-state solutions with respect to the dispersal rates of the two-stage nonlocal dispersal population model are also obtained. Some biological interpretations of our results are discussed.

math.AP↗

arXiv2023

Multiplicity of endemic equilibria for a diffusive SIS epidemic model with mass-action

We study a diffusive SIS epidemic model with the mass-action transmission mechanism and show, under appropriate assumptions on the parameters, the existence of multiple endemic equilibria (EE). Our results answer some open questions on previous studies related to disease extinction or persistence when $\mathcal{R}_0<1$ and the multiplicity of EE solutions when $\mathcal{R}_0>1$. Interestingly, even with such a simple nonlinearity induced by the mass-action, we show that the diffusive epidemic model may have an S-shaped or backward bifurcation curve of EE solutions.

math.AP↗

arXiv2023

Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments

This work examines the dynamics of solutions of a two-strain SIS epidemic model in patchy environments. The basic reproduction number $\mathcal{R}_0$ is introduced, and sufficient conditions are provided to guarantee the global stability of the disease-free equilibrium (DFE). In particular, the DFE is globally stable when either: (i) $\mathcal{R}_0\le \frac{1}{k}$, where $k\ge 2$ is the total number of patches, or (ii) $\mathcal{R}_0<1$ and the dispersal rate of the susceptible population is large. Moreover, the questions of competition-exclusion and coexistence of the strains are investigated when the single-strain reproduction numbers are greater than one. In this direction, under some appropriate hypotheses, it is shown that the strain whose basic reproduction number and local reproduction function are the largest always drives the other strain to extinction in the long run. Furthermore, the asymptotic dynamics of the solutions are presented when either both strain's local reproduction functions are spatially homogeneous or the population dispersal rate is uniform. In the latter case, the invasion numbers are introduced and the existence of coexistence endemic equilibrium (EE) is proved when these invasion numbers are greater than one. Numerical simulations are provided to complement the theoretical results.

math.AP↗

arXiv2021

Analysis of a time-delayed HIV/AIDS epidemic model with education campaigns

We consider a time-delayed HIV/AIDS epidemic model with education dissemination and study the asymptotic dynamics of solutions as well as the asymptotic behavior of the endemic equilibrium with respect to the amount of information disseminated about the disease. Under appropriate assumptions on the infection rates, we show that if the basic reproduction number is less than or equal to one, then the disease will be eradicated in the long run and any solution to the Cauchy problem converges to the unique disease-free equilibrium of the model. On the other hand, when the basic reproduction number is greater than one, we prove that the disease will be permanent but its impact on the population can be significantly minimized as the amount of education dissemination increases. In particular, under appropriate hypothesis on the model parameters, we establish that the size of the component of the infected population of the endemic equilibrium decreases linearly as a function of the amount of information disseminated. We also fit our model to a set of data on HIV/AIDS in order to estimate the infection, effective response, and information rates of the disease. We then use these estimates to present numerical simulations to illustrate our theoretical findings.

math.DS↗

arXiv2020

Long time behavior of random and nonautonomous Fisher-KPP equations. Part I. Stability of equilibria and spreading speeds

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(θ_tω)u(1-u),\quad x\in\mathbb{R} $$ where $ω\inΩ$, $(Ω, \mathcal{F},\mathbb{P})$ is a given probability space, $θ_t$ is an ergodic metric dynamical system on $Ω$, and $a(ω)>0$ for every $ω\inΩ$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),\quad x\in\mathbb{R}$$ where $a_0(t)$ is a positive locally Hölder continuous function. In this first part of the series, we investigate the stability of positive equilibria and the spreading speeds. Under some proper assumption on $a(ω)$, we show that the constant solution $u=1$ of (1) is asymptotically stable with respect to strictly positive perturbations and show that (1) has a deterministic spreading speed interval $[2\sqrt{\underline a}, 2\sqrt{\bar a}]$, where $\underline{a}$ and $\bar a$ are the least and the greatest means of $a(\cdot)$, respectively, and hence the spreading speed interval is linearly determinant. It is shown that the solution of (1) with the initial function which is bounded away from $0$ for $x\ll -1$ and is $0$ for $x\gg 1$ propagates at the speed $2\sqrt {\hat a}$, where $\hat a$ is the average of $a(\cdot)$. Under some assumption on $a_0(\cdot)$, we also show that the constant solution $u=1$ of (2) is asymptotically stably and (2) admits a bounded spreading speed interval. It is not assumed that $a(ω)$ and $a_0(t)$ are bounded above and below by some positive constants. The results obtained in this part are new and extend the existing results in literature on spreading speeds of Fisher-KPP equations. In the second part of the series, we will study the existence and stability of transition fronts of (1) and (2).

math.AP↗

arXiv2020

Entire Solutions of Diffusive Lotka-Volterra System

This work is concerned with the existence of entire solutions of the diffusive Lotka-Volterra competition system \begin{equation}\label{eq:abstract} \begin{cases} u_{t}= u_{xx} + u(1-u-av), & \qquad \ x\in\mathbb{R} \cr v_{t}= d v_{xx}+ rv(1-v-bu), & \qquad \ x\in\mathbb{R} \end{cases} \quad (1) \end{equation} where $d,r,a$, and $b$ are positive constants with $a\neq 1$ and $b\neq 1$. We prove the existence of some entire solutions $(u(t,x),v(t,x))$ of $(1)$ corresponding to $(Φ_{c}(ξ),0)$ at $t=-\infty$ (where $ξ=x-ct$ and $Φ_c$ is a traveling wave solution of the scalar Fisher-KPP defined by the first equation of $(1)$ when $a=0$). Moreover, we also describe the asymptotic behavior of these entire solutions as $t\to+\infty$. We prove existence of new entire solutions for both the weak and strong competition case. In the weak competition case, we prove the existence of a class of entire solutions that forms a 4-dimensional manifold.

math.AP↗

arXiv2019

Existence of traveling wave solutions of a deterministic vector-host epidemic model with direct transmission

We consider an epidemic model with direct transmission given by a system of nonlinear partial differential equations and study the existence of traveling wave solutions. When the basic reproductive number of the considered model is less than one, we show that there is no nontrivial traveling wave solution. On the other hand, when the basic reproductive number is greater than one, we prove that there is a minimum wave speed $c^*$ such that the system has a traveling wave solution with speed $c$ connecting both equilibrium points for any $c\ge c^*$. Moreover, under suitable assumption on the diffusion rates, we show that there is no traveling wave solution with speed less than $c^*$. We conclude with numerical simulations to illustrate our findings. The numerical experiments supports the validity of our theoretical results.

math.AP↗

arXiv2019

Dynamics of a parabolic-ODE competition system in heterogeneous environments

This work is concerned with the large time behavior of the solutions of a parabolic-ODE hybrid system, modeling the competition of two populations which are identical except their movement behaviors: one species moves by random dispersal while the other does not diffuse. We show that the non-diffusing population will always drive the diffusing one to extinction in environments with sinks. In contract, the non-diffusing and diffusing populations can coexist in environments without sinks.

math.AP↗

arXiv2019

Can chemotaxis speed up or slow down the spatial spreading in parabolic-elliptic Keller-Segel systems with logistic source?

The current paper is concerned with the spatial spreading speed and minimal wave speed of the following Keller-Segel chemoattraction system, \begin{equation}\label{abstract-eq1} \begin{cases} u_t=u_{xx}-χ(uv_x)_x +u(a-bu),\quad x\in\R\cr 0=v_{xx}- λv+μu,\quad x\in\R, \end{cases} \end{equation} where $χ$, $a$, $b$, $λ$, and $μ$ are positive constants. Assume $b>χμ$. Then if in addition, $\big(1+\frac{1}{2}\frac{(\sqrt{a}-\sqrtλ)_+}{(\sqrt{a}+\sqrt{\la})}\big)χμ{ \leq} b$ holds, it is proved that $c_0^*=2\sqrt a$ is the spreading speed of the solutions of \eqref{abstract-eq1} with nonnegative continuous initial function $u_0$ with nonempty compact support, that is, $$ \limsup_{|x|\ge ct, t\to\infty}u(t,x;u_0)=0\quad \forall\, c>c_0^* $$ and $$ \liminf_{|x|\le ct,t\to\infty} u(t,x;u_0)>0\quad \forall \, 0 2χμ$ and $λ\geq a$ holds, then $c_0^*=2\sqrt a$ is the minimal speed of the traveling wave solutions of \eqref{abstract-eq1} connecting $(0,0)$ and $(\frac{a}{b},\fracμλ\frac{a}{b})$, that is, for any $c\ge c_0^*$, \eqref{abstract-eq1} has a traveling wave solution connecting $(0,0)$ and $(\frac{a}{b},\fracμλ\frac{a}{b})$ with speed $c$, and \eqref{abstract-eq1} has no such traveling wave solutions with speed less than $c_0^*$. Note that $c_0^*=2\sqrt a$ is the spatial spreading speed as well as the minimal wave speed of the following Fisher-KPP equation, \begin{equation} \label{abstract-eq2} u_t=u_{xx}+u(a-bu),\quad x\in\R. \end{equation}

math.AP↗

arXiv2018

Long time behavior of random and nonautonomous Fisher-KPP equations. Part II. Transition fronts

In the current series of two papers, we study the long time behavior of the following random Fisher-KPP equation $$ u_t =u_{xx}+a(θ_tω)u(1-u),\quad x\in\R, \eqno(1) $$ where $ω\inΩ$, $(Ω, \mathcal{F},\mathbb{P})$ is a given probability space, $θ_t$ is an ergodic metric dynamical system on $Ω$, and $a(ω)>0$ for every $ω\inΩ$. We also study the long time behavior of the following nonautonomous Fisher-KPP equation, $$ u_t=u_{xx}+a_0(t)u(1-u),\quad x\in\R, \eqno(2) $$ where $a_0(t)$ is a positive locally Hölder continuous function. In the first part of the series, we studied the stability of positive equilibria and the spreading speeds of (1) and (2). In this second part of the series, we investigate the existence and stability of transition fronts of (1) and (2). We first study the transition fronts of (1). Under some proper assumption on $a(ω)$, we show the existence of random transition fronts of (1) with least mean speed greater than or equal to some constant $\underline{c}^*$ and the nonexistence of ranndom transition fronts of (1) with least mean speed less than $\underline{c}^*$. We prove the stability of random transition fronts of (1) with least mean speed greater than $\underline{c}^*$. Note that it is proved in the first part that $\underline{c}^*$ is the infimum of the spreading speeds of (1). We next study the existence and stability of transition fronts of (2). It is not assumed that $a(ω)$ and $a_0(t)$ are bounded above and below by some positive constants. Many existing results in literature on transition fronts of Fisher-KPP equations have been extended to the general cases considered in the current paper. The current paper also obtains several new results.

math.AP↗

arXiv2018

Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. I. Persistence and asymptotic spreading

The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$\partial_tu=Δu-χ\nabla(u\nabla v)+u(a(x,t)-ub(x,t))\ ,\ 0=Δv-λv+μu \ \ (1)$$where $χ, λ, μ$ are positive constants, $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of the series, we investigate the persistence and asymptotic spreading. Under some explicit condition on the parameters, we show that (1) has a unique nonnegative time global classical solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ with $u(x,t_0;t_0,u_0)=u_0(x)$ for every $t_0\in R$ and every $u_0\in C^{b}_{\rm unif}(R^N)$, $u_0\geq 0$. Next we show the pointwise persistence phenomena in the sense that, for any solution $(u(x,t;t_0,u_0),v(x,t;t_0,u_0))$ of (1) with strictly positive initial function $u_0$, then$$0<\inf_{t_0\in R, t\geq 0}u(x,t+t_0;t_0,u_0)\le\sup_{t_0\in R, t\geq 0} u(x,t+t_0;t_0,u_0)<\infty$$and show the uniform persistence phenomena in the sense that there are $0 0$ such that$$m\le u(x,t+t_0;t_0,u_0)\le M\ \forall\,t\ge T(u_0),\ x\in R^N.$$We then discuss the spreading properties of solutions to (1) with compactly supported initial and prove that there are positive constants $0 c_+^*,$$and$$\liminf_{t\to\infty}\sup_{|x|\le ct}u(x,t+t_0;t_0,u_0)>0, \ \forall 0<c<c_-^*.$$We also discuss the spreading properties of solutions to (1) with front-like initial functions. In the second and third of the series, we will study the existence, uniqueness, and stability of strictly positive entire solutions and the existence of transition fronts, respectively.

math.AP↗

arXiv2018

Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. II. Existence, uniqueness, and stability of strictly positive entire solutions

This work is the second of the series of three papers devoted to the study of asymptotic dynamics in the chemotaxis system with space and time dependent logistic source,$$\partial_tu=Δu-χ\nabla\cdot(u\nabla v)+u(a(x,t)-ub(x,t)),\quad 0=Δv-λv+μu ,\ x\in\mathbb{R}^N, \ (1) $$where $N\ge1$ is a positive integer, $χ,λ,μ>0$, and the functions $a(x,t), b(x,t)$ are positive and bounded. In the first of the series, we studied the phenomena of pointwise and uniform persistence, and asymptotic spreading for solutions with compactly supported or front like initials. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions. In this direction, we prove that, if $0\leμχ 0$, where $(u_χ(x,t+t_0;t_0,u_0),v_χ(x,t+t_0;t_0,u_0))$ denotes the unique classical solution of (1) with $u_χ(x,t_0;t_0,u_0)=u_0(x)$, for every $0\leχ<b_\inf$.

math.AP↗

arXiv2017

Global classical solutions, stability of constant equilibria, and spreading speeds in attraction-repulsion chemotaxis systems with logistic source on $\mathbb{R}^{N}$

We consider the following chemotaxis systems $$\begin{cases}u_t=Δu-χ_1\nabla(u\nabla v_1)+χ_2\nabla(u\nabla v_2)+u(a-bu),\ \ x\in\mathbb R^N,t>0,\\0=(Δ-λ_1I)v_1+μ_1u,\ \ x\in\mathbb R^N,t>0,\\0=(Δ-λ_2I)v_2+μ_2u,\ \ \text{in}\ x\in\mathbb R^N,\ t>0,\\u(\cdot,0)=u_0,\ \ x\in\mathbb R^N,\end{cases}$$where $χ_i,\ λ_i,\ μ_i,\ i=1,2$ and $a,\ b$ are positive constant real numbers and $N$ is a positive integer. Under some conditions on the parameters, we prove the global existence and boundedness of classical solutions $(u(x,t;u_0),v_1(x,t;u_0),v_2(x,t;u_0))$ for nonnegative, bounded, and uniformly continuous initials $u_0(x)$. Next, we show that, for every strictly positive initial \,$u_0(x)$,$$\lim_{t\to\infty}\left[\|u(\cdot,t;u_0)-\frac{a}{b}\|_{\infty}+\|λ_1v_1(\cdot,t;u_0)-\frac{a}{b}μ_1\|_{\infty}+\|λ_2v_2(\cdot,t;u_0)-\frac{a}{b}μ_2\|_{\infty}\right]=0.$$ Finally, we explore the spreading properties of the global solutions and prove that there are two positive numbers $0 c^*_+(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)$. Furthermore we show that$$\lim_{(χ_1,χ_2)\to(0,0)}c^*_-(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=\lim_{(χ_1,χ_2)\to(0,0)}c^*_+(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=2\sqrt{a}.$$

math.AP↗

arXiv2017

Existence of Traveling wave solutions to parabolic-elliptic-elliptic chemotaxis systems with logistic source

We study traveling wave solutions of the following chemotaxis systems,$$\begin{cases}u_t=Δu-χ_1\nabla(u\nabla v_1)+χ_2\nabla(u\nabla v_2)+u(a-bu),\ x\in\mathbb{R}^N\\ 0=Δv_1-λ_1v_1+μ_1u,\ x\in\mathbb{R}^N,\\ 0=Δv_2-λ_2v_2+μ_2u,\ x\in\mathbb{R}^N,\end{cases}$$where $u(x,t), v_1(x,t)$ and $v_2(x,t)$ represent the population densities of a mobile species, a chemoattractant, and a chemo-repulsion, respectively. In an earlier work, we proved that there is a constant $K\geq0$ such that if $b+χ_2μ_2>χ_1μ_1+K$, then the steady solution $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ is asymptotically stable with respect to positive perturbations. In this paper, we prove that if $b+χ_2μ_2>χ_1μ_1+K$, then there exist a number $c^*(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)\geq 2\sqrt a$ such that for every $c\in ( c^*(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2) , \infty)$ and $ξ\in S^{N-1}$, the system has a traveling wave solution $(u,v_1,v_2)=(U(x\cdotξ-ct),V_1(x\cdotξ-ct),V_2(x\cdotξ-ct))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ and $(0,0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt a$. Moreover we prove that$$\lim_{(χ_1,χ_2)\to(0^+,0^+)}c^{*}(χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=\begin{cases}2\sqrt a\ \text{if}\ a\leq \min\{λ_1, λ_2\}\\ \frac{a+λ_1}{\sqrt{λ_1}}\ \text{if}\ λ_1\leq \min\{a, λ_2\}\\ \frac{a+λ_2}{\sqrt{λ_2}}\ \text{if}\ λ_2\leq \min\{a, λ_1\}\end{cases},\forall λ_1,λ_2,μ_1,μ_2>0,$$and$$\lim_{x\to\infty}\frac{U(x)}{e^{-\sqrt{a}μx}}=1,$$ where $μ$ solves $\sqrt a(μ+\frac{1}μ)=c$ in the interval $(0 , \min\{1,\sqrt{\frac{λ_1}{a}},\sqrt{\frac{λ_2}{a}}\})$.

math.AP↗

arXiv2016

Existence of Traveling wave solutions of parabolic-parabolic chemotaxis systems

The current paper is devoted to the study of traveling wave solutions of the following parabolic-parabolic chemotaxis systems, $$ \begin{cases} u_{t}= Δu-χ\nabla \cdot (u \nabla v) + u(a-bu),\quad x\in\mathbb{R}^N τv_t=Δv-v+u, \quad x\in\mathbb{R}^N, \end{cases} $$ where $u(x,t)$ represents the population density of a mobile species and $v(x,t)$ represents the population density of a chemoattractant, and $χ$ represents the chemotaxis sensitivity. We prove that for every $τ>0$, there is $0<χ_τ^*<\frac{b}{2}$ such that for every $0<χ<χ_τ^*$, there exist two positive numbers $2\sqrt a \le c^{*}(χ,τ)<c^{**}(χ,τ)$ satisfying that for every $ c\in [ c^{*}(χ,τ)\,\ c^{**}(χ,τ))$ and $ξ\in S^{N-1}$, the system has a traveling wave solution $(u(x,t),v(x,t))=(U(x\cdotξ-ct;τ),V(x\cdotξ-ct;τ))$ with speed $c$ connecting the constant solutions $(\frac{a}{b},\frac{a}{b})$ and $(0,0)$, and it does not have such traveling wave solutions of speed less than $2\sqrt a$. Moreover, $$ \lim_{χ\to 0^+}c^{**}(χ,τ)=\infty,$$ $$ \lim_{χ\to 0^+}c^{*}(χ,τ)=\begin{cases} 2\sqrt{a}\qquad \qquad \qquad\ \text{if} \quad 0<a\leq \frac{1+τa}{(1-τ)_+}\cr \frac{1+τa}{(1-τ)_{+}}+\frac{a(1-τ)_{+}}{1+τa}\quad \text{if} \quad a\geq \frac{1+τa}{(1-τ)_+}, \end{cases} $$ and $$ \lim_{x\to -\infty}\frac{U(x;τ)}{e^{-μx}}=1, $$ where $μ$ is the only solution of the equation $μ+\frac{a}μ=c$ in the interval $(0, \min\{\sqrt a, \sqrt{\frac{1+τa}{(1-τ)_+}}\})$. Furthermore, it hods that $\lim_{τ\to 0^+}χ_τ^*=\frac{b}{2}$.

math.AP↗