\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2019 (2019), No. 09, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2019 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2019/09\hfil Analysis of malaria transmission dynamics] {Global stability analysis of malaria transmission dynamics with vigilant compartment} \author[O. S. Obabiyi, S. Olaniyi \hfil EJDE-2019/09\hfilneg] {Olawale S. Obabiyi, Samson Olaniyi} \address{Olawale S. Obabiyi \newline Department of Mathematics, University of Ibadan, Nigeria} \email{obabiyios@yahoo.com} \address{Samson Olaniyi \newline Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology, PMB 4000, Ogbomoso, Nigeria} \email{solaniyi@lautech.edu.ng} \thanks{Submitted July 8, 2016. Published January 21, 2019.} \subjclass[2010]{92B05, 93A30, 93D20} \keywords{Global dynamics; malaria model; basic reproduction number; \hfill\break\indent Lyapunov function; vigilant human compartment} \begin{abstract} A deterministic compartmental model for the transmission dynamics of malaria incorporating vigilant human compartment is studied. The model is qualitatively analyzed to investigate its asymptotic behavior with respect to the equilibria. It is shown, using a linear Lyapunov function, that the disease-free equilibrium is globally asymptotically stable when the associated basic reproduction number is less than the unity. When the basic reproduction number is greater than the unity, under certain specifications on the model parameters, we prove the existence of a globally asymptotically stable endemic equilibrium with the aid of a suitable nonlinear Lyapunov function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction}\label{S1} Malaria is one of the most common infectious diseases that are posing great public health problem throughout the six World Health Organization regions today. It has been reported that an estimated 3.3 billion people across the globe are at risk of being infected with malaria while the burden is heaviest in the WHO African Region accounting for an estimated $90\%$ of all malaria deaths \cite{WorldHealthOrganization2014W}. The disease is caused by five species of parasites belonging to the genus \textit{Plasmodium}, namely \textit{P. falciparum}, \textit{P. vivax}, \textit{P. ovale}, \textit{P. malariae} and \textit{P. knowlesi}. Of these species, \textit{P. falciparum} and \textit{P. vivax} pose the greatest public health challenge \cite{WorldHealthOrganization2015W}. Malaria is spread between humans via the bite of female \textit{Anopheles} mosquitoes and it is characterized by symptoms which may include chills, illness, headaches, body aches, anemia, nausea and vomiting among others. However, the disease can be avoided and treated by adopting interventions such as vector control (which prevents mosquito from acquiring or passing on an infection through use of insecticide-treated mosquito nets (ITNs) or indoor residual spraying (IRS)); chemoprevention (which inhibits infections establishing themselves in humans); and case management (which includes prompt diagnosis and appropriate treatment) (see \cite{ LashariAlyHattafZamanJungLi2012,WorldHealthOrganization2015W}). Deterministic compartmental models describing the transmission of malaria between human and mosquito populations have been developed with attempts to facilitate the understanding of the mechanisms involved in the transmission dynamics of the disease (see, \cite{ForouzanniaGumel2015,NgarakanaBhunuMasochaMash2016} and the references therein). The impact of some of the intervention strategies mentioned earlier are investigated as control functions using time dependent models (non-autonomous systems) (see, for instance, \cite{AgustoMarcusOkosun2012,BlaynehCaoKwon2009,LashariAlyHattafZamanJungLi2012}). The choice of compartments used in mathematical models varies and largely depends on the observed features of the particular disease being modeled \cite{Hethcote2000}. Recently, an autonomous discrete-age-structured model proposed in \cite{ObabiyiOlaniyiIJAM2016} incorporated vigilant human compartment into the malaria transmission dynamics. This vigilant compartment comprises individuals who adhere to the intervention strategies with a view to preventing further spread of the disease in the population. The behavior of a dynamical system as its solution approaches a given equilibrium is an asymptotic stability property. In the literature of epidemic models, establishing global (unlike local) asymptotic stability results is usually a nontrivial and challenging mathematical problem \cite{ShuaiDriessche2013}. Methods such as Dulac's criterion with Poincar\'{e}-Bendixson theorem (see \cite{Vargas2011,XiaoRuan2007}); geometrical approach (see \cite{BuonomoLacitignola2008,LiMuldowney1996}); comparison theorem (see \cite{ForouzanniaGumel2015,MushayaTchuenche2011}); technique used in \cite{CastilloFengHuang2002}; and Lyapunov method (see, for instance, \cite{IboiOkuonghae2016,OlaniyiObabiyi2014,Korobeinikov2009,KorobeinikovWake2002}) can be used to study global stability properties of disease models. Lyapunov method, whose role cannot be overemphasized in this direction, requires the construction of a suitable Lyapunov function which is positive definite and whose value never increases along the solution paths of the system. In this paper, the method of Lyapunov function is sought to extend the analysis in \cite{ObabiyiOlaniyiIJAM2016} beyond only a small region near the disease-free and endemic equilibria of the system. This article is organized as follows. In section \ref{S2}, the description of the formulated model is given. The global asymptotic stability of the disease-free equilibrium and endemic equilibrium are explored in Sections \ref{S3} and \ref{S4} respectively. Also we provide concluding remarks. \section{Model description}\label{S2} We consider the normalized form of the malaria transmission dynamics obtained in \cite{ObabiyiOlaniyiIJAM2016} with the concept that humans may not have equal likelihood of being infected with malaria parasites. The human population at discrete-age $a_i$ (for $ i=0,1,2,\dots,L $ and $ a_L $ being the maximum age) and at time $t$ is subdivided into susceptible $ S_h(t,a_i) $; exposed $ E_h(t,a_i) $; infectious $ I_h(t,a_i) $; and vigilant $ V_h(t,a_i) $ individuals. On the other hand, the mosquito population is subdivided into susceptible $ S_m(t) $; exposed $ E_m(t) $; and infectious $ I_m(t) $ mosquitoes. It is assumed that susceptible humans are recruited into the population at a rate $\lambda_{h}(a_{i})$ whose fraction $\tau\lambda_h(a_i)$ are recruited vigilant. Susceptible humans acquire malaria through contact with infectious mosquitoes and become exposed humans at rate $b\beta_{h}(a_{i})\sigma$, where $b$ is the biting rate, $\beta_{h}(a_{i})$ is the probability that bite produces infection in human and $\sigma$ is the contact rate of mosquito per human per unit time. The per capita rate of progression of exposed individuals is given by $\alpha(a_i)$ whose fraction $\theta$ can become vigilant upon treating malaria infection (e.g., \textit{P}. vivax) usually at the dormant liver stage \cite{WorldHealthOrganization2015W} while the remaining fraction $(1-\theta)$ progresses to the infectious compartment following the development of the disease symptoms. Infectious humans become vigilant at per capita recovery rate $\gamma(a_i)$. It is further assumed that individuals in the vigilant compartment firmly adhere to the intervention strategies and cannot be re-infected. \begin{figure}[htb] \begin{center} \includegraphics[height=9cm, width=8cm]{fig1} % Newdiagram \end{center} \caption{Schematic diagram of the malaria transmission dynamics regardless of the discrete-age $ a_i $.} \label{Fig1} \end{figure} The mosquito population is increased at recruitment rate $\lambda_m$ assumed to be susceptible. Following effective contact with infectious humans, susceptible mosquitoes acquire infection and become exposed at rate $b\beta_m$, where $\beta_m$ is the probability that bite produces infection in the mosquito. Exposed mosquitoes progress to become infectious at per capita rate $\alpha_m$. The per capita natural death rates of humans and mosquitoes are, respectively, given by $\mu_h(a_i)$ and $\mu_m$. The diagrammatic representation of the foregoing assumptions can be seen in Figure \ref{Fig1} and the corresponding model is governed by the following system of ordinary differential equations: \begin{equation} \begin{gathered} \frac{dS_{h}(t,a_{i})}{dt}=(1-\tau)\lambda_{h}(a_{i}) -\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m -\mu_{h}(a_{i})S_{h}(t,a_{i}), \\ \frac{dE_{h}(t,a_{i})}{dt}=\sum^{L}_{i=0}b\beta_{h}(a_{i}) \sigma S_{h}(t,a_{i})I_m-(\alpha_{h}(a_{i})+\mu_{h}(a_{i}))E_{h}(t,a_{i}), \\ \frac{dI_{h}(t,a_{i})}{dt}=\sum^{L}_{i=0}(1-\theta)\alpha_{h}(a_{i})E_{h}(t,a_{i}) -(\gamma(a_{i})+\mu_{h}(a_{i})I_{h}(t,a_{i}), \\ \frac{dV_{h}(t,a_{i})}{dt}=\tau\lambda_{h}(a_{i}) +\theta\alpha_{h}(a_{i})E_{h}(t,a_{i})+\gamma(a_{i}) I_{h}(t,a_{i}) -\mu_{h}(a_{i})V_{h}(t,a_{i}), \\ \frac{dS_m}{dt}=\lambda_m-b\beta_mS_m(t)I_{h}(t,a_{i})-\mu_mS_m(t), \\ \frac{dE_m}{dt}=b\beta_mS_m(t)I_{h}(t,a_{i})-(\alpha_m+\mu_m)E_m(t),\\ \frac{dI_m}{dt}=\alpha_mE_m(t)-\mu_mI_m(t) \end{gathered} \label{eq.mdl.1} \end{equation} The parameters and variables of the formulated model \eqref{eq.mdl.1} are nonnegative since the model monitors human and mosquito populations. Further, it is supposed that the recruitment terms for human and mosquito populations are balanced by the natural deaths $\mu_h(a_i)$ and $\mu_m$ respectively. So that system \eqref{eq.mdl.1} can be analyzed in a positively invariant region $\mathfrak{D}=\mathfrak{D}_{h}\times\mathfrak{D}_m\subset\mathbb{R}_{+}^{4}\times \mathbb{R}_{+}^{3}$ with \begin{gather*} \mathfrak{D}_{h}=\big\{(S_{h},E_{h},I_{h},V_{h})\in \mathbb{R}_{+}^{4}:S_{h}+E_{h} +I_{h}+V_{h}= 1\big\}, \\ \mathfrak{D}_m=\big\{(S_m,E_m,I_m)\in \mathbb{R}_{+}^{3}:S_m+E_m+I_m = 1\big\}. \end{gather*} A key notion in the analysis of infectious disease models is the basic reproduction number $\mathcal{R}_0$, an epidemiological threshold that determines whether disease dies out or persists in the population. Following \cite{DriesscheWatmough2002}, $\mathcal{R}_0$ for system \eqref{eq.mdl.1} is given by \begin{equation} \mathcal{R}_0 =\Big(\sum^{L}_{i=0}\frac{b^2\beta_{h}(a_{i})\sigma\alpha_{h}(a_{i}) \beta_m\alpha_m(1-\theta)(1-\tau)}{(\alpha_{h}(a_{i}) +\mu_{h}(a_{i}))(\gamma(a_{i})+\mu_{h}(a_{i}))(\alpha_m +\mu_m)\mu_m}\Big)^{1/2} \label{cond.l.1} \end{equation} The basic reproduction number \eqref{cond.l.1} represents the average number of secondary cases (humans/mosquitoes) generated by one infectious case (mosquito/human) during the period of infectiousness in a completely susceptible (humans/mosquitoes) population. \section{Global stability of disease-free equilibrium}\label{S3} The steady-state solution of the model \eqref{eq.mdl.1}, the disease-free equilibrium, is given by \begin{equation}\label{solutions.de.P} \mathcal{E}_0=(1-\tau, 0, 0, \tau, 1, 0, 0) \end{equation} The following result establishes the global asymptotic behavior of system \eqref{eq.mdl.1} around \eqref{solutions.de.P} determined by the basic reproduction number \eqref{cond.l.1}. \begin{theorem}\label{globaldfe} The disease-free equilibrium \eqref{solutions.de.P} of model \eqref{eq.mdl.1} is globally asymptotically stable in $\mathfrak{D}$ whenever $\mathcal{R}_0\leq1$. \end{theorem} \begin{proof} Consider the linear Lyapunov function of the form \begin{equation}\label{linear.lyp.fxn} \mathfrak{F}=kE_{h}+\frac{I_{h}}{\gamma(a_{i})+\mu_{h}(a_{i})} +\frac{E_m}{b\beta_m} +\Big(\frac{\alpha_m+\mu_m}{b\beta_m\alpha_m}\Big)I_m, \end{equation} where \[ k=\frac{\alpha_{h}(a_{i})(1-\theta)}{(\alpha_{h}(a_{i}) +\mu_{h}(a_{i}))(\gamma(a_{i})+\mu_{h}(a_{i}))}. \] The time derivative of \eqref{linear.lyp.fxn} along the solutions of the system \eqref{eq.mdl.1} is \begin{equation}\label{lyapunov.der.dot} \begin{aligned} \mathfrak{\dot{F}} &=\sum^{L}_{i=0}\Big(\frac{\alpha_{h}(a_{i})(1-\theta)}{(\alpha_{h}(a_{i}) +\mu_{h}(a_{i}))(\gamma(a_{i})+\mu_{h}(a_{i}))}\Big)\\ &\quad \times[b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m-(\alpha_{h}(a_{i}) +\mu_{h}(a_{i}))E_{h}(t,a_{i})]\\ &\quad +\sum^{L}_{i=0}\Big(\frac{1}{\gamma(a_{i})+\mu_{h}(a_{i})}\Big)\\ &\quad \times[(1-\theta)\alpha_{h}(a_{i})E_{h}(t,a_{i})-(\gamma(a_{i}) +\mu_{h}(a_{i}))I_{h}(t,a_{i})]\\ &\quad +\big(\frac{1}{b\beta_m}\big)[b\beta_mS_mI_{h}(t,a_{i})-(\alpha_m +\mu_m)E_m\big]\\ &\quad +\big(\frac{\alpha_m+\mu_m}{b\beta_m\alpha_m}\big) [\alpha_mE_m-\mu_mI_m] \end{aligned} \end{equation} Algebraic expansion of \eqref{lyapunov.der.dot} and further simplification yield \begin{align*} \mathfrak{\dot{F}} &=\sum^{L}_{i=0}\Big(\frac{b\beta_{h}(a_{i})\sigma\alpha_{h}(a_{i})(1-\theta)} {(\alpha_{h}(a_{i})+\mu_{h}(a_{i}))(\gamma(a_{i})+\mu_{h}(a_{i}))}\Big)S_{h} (t,a_{i})I_m\\ &\quad -\sum^{L}_{i=0}\Big(\frac{\alpha_{h}(a_{i})(1-\theta)} {(\gamma(a_{i})+\mu_{h}(a_{i}))}\Big)E_{h}(t,a_{i}) -I_{h}(t,a_{i})+S_mI_{h}(t,a_{i})\\ &\quad +\sum^{L}_{i=0}\Big(\frac{\alpha_{h}(a_{i})(1-\theta)} {(\gamma(a_{i})+\mu_{h}(a_{i}))}\Big)E_{h}(t,a_{i}) -\Big(\frac{(\alpha_m+\mu_m)\mu_m}{b\beta_m\alpha_m}\Big)I_m\\ &\leq \sum^{L}_{i=0}\Big(\frac{b\beta_{h}(a_{i})\sigma\alpha_{h}(a_{i}) (1-\theta)(1-\tau)}{(\alpha_{h}(a_{i})+\mu_{h}(a_{i})) (\gamma(a_{i})+\mu_{h}(a_{i}))}\Big)I_m -\Big(\frac{(\alpha_m+\mu_m)\mu_m}{b\beta_m\alpha_m}\Big)I_m\\ &= \Big(\sum^{L}_{i=0}\frac{b\beta_{h}(a_{i})\sigma\alpha_{h}(a_{i}) (1-\theta)(1-\tau)}{(\alpha_{h}(a_{i})+\mu_{h}(a_{i}))(\gamma(a_{i}) +\mu_{h}(a_{i}))}-\frac{(\alpha_m+\mu_m)\mu_m}{b\beta_m\alpha_m}\Big) I_m\\ &= \Big(\frac{(\alpha_m+\mu_m)\mu_m}{b\beta_m\alpha_m}\Big) (\mathcal{R}^2_0-1)I_m \end{align*} It follows that $\mathfrak{\dot{F}}\leq 0$ whenever $\mathcal{R}_0\leq 1$ with $\mathfrak{\dot{F}}= 0$ if and only if $I_m=0.$ Further, one sees that $\left(S_{h}(t,a_{i}),E_{h}(t,a_{i}),I_{h}(t,a_{i}),V_{h}(t,a_{i}), S_m(t),E_m(t)\right)$ tends to $\left((1-\tau),0,0,0,1,0\right)$ as $t\to \infty $ since $I_m\to 0$ as $t\to \infty $. By LaSalle's invariance principle \cite{LaSalle1976}, one concludes that every solution of the model \eqref{eq.mdl.1} in $\mathfrak{D}$ approaches the disease-free equilibrium \eqref{solutions.de.P} as $t\to\infty$. Hence $\mathcal{E}_0$ is globally asymptotically stable. \end{proof} The epidemiological implication of Theorem \ref{globaldfe} shows that malaria can be controlled or eliminated from the community if the associated basic reproduction number of the model \eqref{eq.mdl.1} is less than the unity. \section{Global stability of endemic equilibrium}\label{S4} The disease-present (endemic) equilibrium of the model \eqref{eq.mdl.1} is referred to the steady-state solution where at least one of the infected compartments is nonzero. Let the arbitrary endemic equilibrium of the model \eqref{eq.mdl.1} be represented by $$ \mathcal{E}_{1}=(S^{**}_h(a_i),E^{**}_h(a_i),I^{**}_h(a_i), V^{**}_h(a_i),S^{**}_m,E^{**}_m,I^{**}_m), $$ considering the fact that the method of Lyapunov function requires no knowledge of solutions in establishing the global stability \cite{Vargas2011}. However, see \cite{ObabiyiOlaniyiIJAM2016}, for possible existence of the endemic equilibrium $\mathcal{E}_{1}$ of the model \eqref{eq.mdl.1} at $\mathcal{R}_0>1$. Here, the global asymptotic stability of the endemic equilibrium $\mathcal{E}_{1}$ is explored for a special case of the model \eqref{eq.mdl.1} where $\tau=\theta=0$. Let $$ \mathfrak{D}_0 = \big\{ (S_h,E_h,I_h,V_h,S_m,E_m,I_m)\in\mathfrak{D}: E_h=I_h=V_h=E_m=I_m=0\big\} $$ be the stable manifold of the disease-free equilibrium $\mathcal{E}_0$. The following result is claimed. \begin{theorem}\label{globalEe} The endemic equilibrium of the model \eqref{eq.mdl.1} is globally asymptotically stable in $\mathfrak{D}\backslash\mathfrak{D}_0$ whenever $\mathcal{R}_0| _{\tau=\theta=0}> 1$ \end{theorem} \begin{proof} Consider the Goh-Volterra nonlinear Lyapunov function \begin{equation}\label{lyapunov.non.linear} \begin{aligned} \mathcal{V}&=S_{h}(t,a_{i})-S^{*}_{h}(a_{i})-S^{*}_{h}(a_{i}) \ln\Big(\frac{S_{h}(t,a_{i})}{S^{*}_{h}(a_{i})}\Big)\\ &\quad +E_{h}(t,a_{i})-E^{*}_{h}(a_{i})-E^{*}_{h}(t,a_{i}) \ln\Big(\frac{E_{h}(t,a_{i})}{E^{*}_{h}(a_{i})}\Big)\\ &\quad +\sum^{L}_{i=0}\frac{\alpha_{h}(a_{i}) +\mu_{h}(a_{i})}{\alpha_{h}(a_{i})} \Big[I_{h}(t,a_{i})-I^{*}_{h}(a_{i})-I^{*}_{h}(a_{i}) \ln\Big(\frac{I_{h}(t,a_{i})}{I^{*}_{h}(a_{i})}\Big)\Big]\\ &\quad +S_m-S^{*}_m-S^{*}_m\ln\Big(\frac{S_m}{S^{*}_m}\Big) +E_m-E^{*}_m-E^{*}_m\ln\big(\frac{E_m}{E^{*}_m}\big)\\ &\quad +\frac{\alpha_m+\mu_m}{\alpha_m} \Big[I_m-I^{*}_m-I^{*}_m\ln\Big(\frac{I_m}{I^{*}_m}\Big)\Big], \end{aligned} \end{equation} The time derivative of \eqref{lyapunov.non.linear} along the solution of \eqref{eq.mdl.1} gives \begin{equation} \label{nonlyap.der.1} \begin{aligned} \mathcal{\dot{V}} &=\dot{S}_{h}(t,a_{i})-\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})}\dot{S}_{h} (t,a_{i})+\dot{E}_{h}(t,a_{i})-\frac{E^{*}_{h}(a_{i})}{E_{h}(t,a_{i})} \dot{E}_{h}(t,a_{i}) \\ &\quad +\sum^{L}_{i=0}\frac{\alpha_{h}(a_{i})+\mu_{h}(a_{i})}{\alpha_{h}(a_{i})} \Big(\dot{I}_{h}(t,a_{i})-\frac{I^{*}_{h}(a_{i})}{I_{h}(t,a_{i})} \dot{I}_{h}(t,a_{i})\Big) \\ &\quad +\dot{S}_m-\frac{S^{*}_m}{S_m}\dot{S}_m+\dot{E}_m -\frac{E^{*}_m}{E_m}\dot{E}_m+\frac{\alpha_m+\mu_m}{\alpha_m} \big(\dot{I}_m-\frac{I^{*}_m}{I_m}\dot{I}_m\big). \end{aligned} \end{equation} Substituting the appropriate equations of the model \eqref{eq.mdl.1} into \eqref{nonlyap.der.1} gives \begin{align*} \mathcal{\dot{V}} &=(1-\tau)\mu_{h}(a_{i})-\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m -\mu_{h}(a_{i})S_{h}(ta_{i})\\ &\quad -\sum^{L}_{i=0}\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})} \left((1-\tau)\mu_{h}(a_{i})-b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m-\mu_{h} (a_{i})S_{h}(t,a_{i})\right)\\ &\quad +\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m -(\alpha_{h}(a_{i})+\mu_{h}(a_{i}))E_{h}(t,a_{i})\\ &\quad -\sum^{L}_{i=0}\frac{E^{*}_{h}(a_{i})}{E_{h}(t,a_{i})} \left(b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m -\left[\alpha_{h}(a_{i})+\mu_{h}(a_{i})\right]E_{h}(t,a_{i})\right)\\ &\quad +\sum^{L}_{i=0}\frac{\alpha_{h}(a_{i})+\mu_{h}(a_{i})}{\alpha_{h}(a_{i})} \left[(1-\theta)\alpha_{h}(a_{i})E_{h}(t,a_{i}) -(\gamma(a_{i})+\mu_{h}(a_{i}))I_{h}(t,a_{i})\right]\\ &\quad -\sum^{L}_{i=0}\frac{\alpha_{h}(a_{i})+\mu_{h}(a_{i})}{\alpha_{h}(a_{i})} \Big(\frac{I^{*}_{h}(a_{i})}{I_{h}(t,a_{i})}\Big)\\ &\quad \times\left[(1-\theta)\alpha_{h}(a_{i})E_{h}(t,a_{i}) -(\gamma(a_{i})+\mu_{h}(a_{i}))I_{h}(t,a_{i})\right]\\ &\quad +\mu_m-b\beta_{h}(a_{i})S_mI_{h}(t,a_{i})-\mu_mS_m -\frac{S^{*}_m}{S_m}\left(\mu_m-b\beta_{h}(a_{i})S_mI_{h}(t,a_{i})-\mu_mS_m\right)\\ &\quad +b\beta_mS_mI_{h}(t,a_{i})-\left[\alpha_m+\mu_m\right]E_m -\frac{E^{*}_m}{E_m}\left(b\beta_mS_mI_{h}-\left[\alpha_m+\mu_m\right]E_m\right)\\ &\quad +\frac{\alpha_m+\mu_m}{\alpha_m}\Big[\alpha_mE_m-\mu_mI_m -\frac{I^{*}_m}{I_m}\left(\alpha_mE_m-\mu_mI_m\right)\Big]. \end{align*} Further simplification yields \begin{align} \mathcal{\dot{V}} &=\sum^{L}_{i=0}\mu_{h}(a_{i})\Big(1-\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})}\Big) -\mu_{h}(a_{i})S_{h}(t,a_{i})\Big(1-\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})}\Big) \nonumber\\ &\quad +\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I_m -\frac{E^{*}_{h}(a_{i})b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m}{E_{h}(t,a_{i})} \nonumber\\ &\quad +\sum^{L}_{i=0}(\alpha_{h}(a_{i})+\mu_{h}(a_{i}))E^{*}_{h}(a_{i}) -\frac{(\alpha_{h}(a_{i})+\mu_{h}(a_{i}))(\gamma(a_{i}) +\mu_{h}(a_{i}))I_{h}(t,a_{i})}{\alpha_{h}(a_{i})} \nonumber\\ &\quad -\sum^{L}_{i=0}\frac{(\alpha_{h}(a_{i}) +\mu_{h}(a_{i}))I^{*}_{h}(a_{i})E_{h}(t,a_{i})}{I_{h}(t,a_{i})} +\mu_m\Big(1-\frac{S^{*}_m}{S_m}\Big) \label{fur.ther.simp} \\ &\quad +\frac{(\alpha_{h}(a_{i})+\mu_{h}(a_{i}))(\gamma(a_{i}) +\mu_{h}(a_{i}))I^{*}_{h}(a_{i})}{\alpha_{h}(a_{i})} -\mu_mS_m\Big(1-\frac{S^{*}_m}{S_m}\Big) \nonumber\\ &\quad +b\beta_mS^{*}_mI_{h}(t,a_{i}) -\frac{E^{*}_mb\beta_mS_mI_{h}(t,a_{i})}{E_m}+(\alpha_m+\mu_m)E^{*}_m -\frac{(\alpha_m+\mu_m)\mu_mI_m}{\alpha_m} \nonumber\\ &\quad -\frac{(\alpha_m+\mu_m)I^{*}_mE_m}{I_m} +\frac{(\alpha_m+\mu_m)\mu_mI_m}{\alpha_m}. \nonumber \end{align} One sees that the following equilibrium relations hold from model \eqref{eq.mdl.1}, \begin{equation}\label{eq.rla.tn} \begin{gathered} \mu_{h}(a_{i})=\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m +\mu_{h}(a_{i})S^{*}_{h}(a_{i}),\\ \alpha_{h}(a_{i})+\mu_{h}(a_{i})=\sum^{L}_{i=0} \frac{b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m}{E^{*}_{h}(a_{i})},\\ \gamma(a_{i})+\mu_{h}(a_{i})=\sum^{L}_{i=0}\frac{\alpha_{h} (a_{i})E^{*}_{h}(a_{i})}{I^{*}_{h}(a_{i})},\\ \mu_m=b\beta_mS^{*}_mI^{*}_{h}(t,a_{i})+\mu_mS^{*}_m,\\ \alpha_m+\mu_m=\frac{b\beta_mS^{*}_mI^{*}_{h}(t,a_{i})}{E^{*}_m},\quad \mu_m=\frac{\alpha_mE^{*}_m}{I^{*}_m} \end{gathered} \end{equation} Consequently, using \eqref{eq.rla.tn} in \eqref{fur.ther.simp} gives \begin{align} \mathcal{\dot{V}} &=\sum^{L}_{i=0}\mu_{h}(a_{i})S^{*}_{h}(a_{i}) \Big(2-\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})}- \frac{S_{h}(t,a_{i})}{S^{*}_{h}(a_{i})}\Big) \nonumber\\ &\quad +\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m -\frac{b\beta_{h}(a_{i})\sigma (S^{*}_{h}(a_{i}))^{2}I^{*}_m}{S_{h}(t,a_{i})} \nonumber\\ &\quad +\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I_m -\frac{E^{*}_{h}(a_{i})b\beta_{h}(a_{i})\sigma S_{h}(t,a_{i})I_m}{E_{h}(t,a_{i})} \nonumber\\ &\quad +\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m -\frac{b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I_{h}(t,a_{i}) I^{*}_m}{I^{*}_{h}(a_{i})} \nonumber\\ &\quad -\sum^{L}_{i=0}\frac{b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_{h}(a_{i}) E_{h}(t,a_{i})I^{*}_m}{E^{*}_{h}(a_{i})I_{h}(t,a_{i})} +b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m \nonumber\\ &\quad +\mu_mS^{*}_m\Big(2-\frac{S^{*}_m}{S_m}-\frac{S_m}{S^{*}_m}\Big) +b\beta_mS^{*}_mI^{*}_{h}-\frac{b\beta_m(S^{*}_m)^{2} I^{*}_{h}}{S_m} \nonumber\\ &\quad +b\beta_mS^{*}_mI_{h}(t,a_{i})-\frac{E^{*}_mb\beta_mS_mI_{h}(t,a_{i})} {E_m}+b\beta_mS^{*}_mI^{*}_{h}(a_{i}) \nonumber\\ &\quad -\frac{b\beta_mS^{*}_mI_mI^{*}_{h}(a_{i})}{I^{*}_m} -\frac{b\beta_mS^{*}_mI^{*}_{h}E_mI^{*}_{h}}{E^{*}_mI_m} +b\beta_mS^{*}_mI^{*}_{h}(a_{i}). \label{eq.rla.tn2} \end{align} Adding and subtracting $b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m$, \[ \sum^{L}_{i=0}\frac{b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I_{h}(t,a_{i}) (I^{*}_m)^{2}}{I^{*}_{h}(a_{i})I_m}, \] $b\beta_mS^{*}_mI^{*}_{h}(a_{i})$ and \[ \frac{b\beta_mS^{*}_mI_m (I^{*}_{h}(a_{i}))^{2}}{I^{*}_mI_{h}(t,a_{i})} \] in \eqref{eq.rla.tn2} systematically, gives \begin{align*} \mathcal{\dot{V}} &=\sum^{L}_{i=0}\mu_{h}(a_{i})S^{*}_{h}(a_{i}) \Big(2-\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})} -\frac{S_{h}(t,a_{i})}{S^{*}_{h}(a_{i})}\Big) +\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m\\ &\quad \times\Big[4-\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})} -\frac{E^{*}_{h}(a_{i})S_{h}(t,a_{i})I_m}{E_{h}(t,a_{i})S^{*}_{h}(a_{i})I^{*}_m} -\frac{I^{*}_{h}(a_{i})E_{h}(t,a_{i})}{I_{h}(t,a_{i})E^{*}_{h}(a_{i})}- \frac{I_{h}(t,a_{i})I^{*}_m}{I^{*}_{h}(a_{i})I_m}\Big]\\ &\quad +\sum^{L}_{i=0}b\beta_{h}(a_{I})\sigma S^{*}_{h}(a_{I})I_m -\frac{b\beta_{h}(a_{I})\sigma S^{*}_{h}(a_{I})I_{h} (t,a_{I})I^{*}_m}{I^{*}_{h}(t,a_{I})}\\ &\quad +\sum^{L}_{i=0}\frac{b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I_{h} (t,a_{i})(I^{*}_m)^{2}}{I^{*}_{h}(a_{i})I_m} -b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m\\ &\quad + \mu_mS^{*}_m\left(2-\frac{S^{*}_m}{S_m} -\frac{S_m}{S^{*}_m}\right)+b\beta_mS^{*}_mI^{*}_{h}(a_{i})\\ &\quad \times\left[4-\frac{S^{*}_m}{S_m} -\frac{E^{*}_mS_mI_{h}(t,a_{i})}{E_mS^{*}_mI^{*}_{h}(a_{i})} -\frac{I^{*}_mE_m}{I_mE^{*}_m} -\frac{I_mI^{*}_{h}(a_{i})}{I^{*}_mI_{h}(t,a_{i})}\right]\\ &\quad + b\beta_mS^{*}_mI_{h}(t,a_{i}) -\frac{b\beta_mS^{*}_mI_mI^{*}_{h}(a_{i})}{I^{*}_m} +\frac{b\beta_mS^{*}_mI_m(I^{*}_{h}(a_{i}))^{2}}{I^{*}_mI_{h}(t,a_{i})}\\ &\quad -b\beta_mS^{*}_mI^{*}_{h}(a_{i}) \end{align*} Further algebraic simplification yields \begin{equation}\label{lyp.fin.al} \begin{aligned} \mathcal{\dot{V}} &=-\mathcal{V}_{1}-\mathcal{V}_{2}-\sum^{L}_{i=0}b\beta_{h}(a_{i}) \sigma S^{*}_{h}(a_{i})I^{*}_m\Big(\frac{I_m}{I_{h}(t,a_{i})} -\frac{I^{*}_m}{I^{*}_{h}(a_{i})}\Big)\Big(1-\frac{I_m}{I^{*}_m}\Big)\\ &\quad -\mathcal{V}_{3}-\mathcal{V}_{4}-\sum^{L}_{i=0}b\beta_mS^{*}_mI^{*}_{h}(a_{i}) \Big(\frac{I_{h}(t,a_{i})}{I_m}- \frac{I^{*}_{h}(a_{i})}{I^{*}_m}\Big) \Big(1-\frac{I_{h}(t,a_{i})}{I^{*}_{h}(a_{i})}\Big), \end{aligned} \end{equation} where \begin{gather*} \mathcal{V}_{1}=\sum^{L}_{i=0}\mu_{h}(a_{i})S^{*}_{h}(a_{i}) \Big(\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})} +\frac{S_{h}(t,a_{i})}{S^{*}_{h}(a_{i})}-2\Big),\\ \begin{aligned} \mathcal{V}_{2} &=\sum^{L}_{i=0}b\beta_{h}(a_{i})\sigma S^{*}_{h}(a_{i})I^{*}_m \Big[\frac{S^{*}_{h}(a_{i})}{S_{h}(t,a_{i})} +\frac{E^{*}_{h}(a_{i})S_{h}(t,a_{i})I_m}{E_{h}(t,a_{i})S^{*}_{h}(a_{i})I^{*}_m}\\ &\quad +\frac{I^{*}_{h}(a_{i})E_{h}(t,a_{i})}{I_{h}(t,a_{i})E^{*}_{h}(a_{i})} +\frac{I_{h}(t,a_{i})I^{*}_m}{I^{*}_{h}(a_{i})I_m}-4\Big], \end{aligned}\\ \mathcal{V}_{3}=\mu_mS^{*}_m\Big(\frac{S^{*}_m}{S_m}+\frac{S_m}{S^{*}_m}-2\Big),\\ \mathcal{V}_{4}=b\beta_mS^{*}_mI^{*}_{h}(a_{i}) \Big[\frac{S^{*}_m}{S_m}+\frac{E^{*}_mS_mI_{h}(t,a_{i})}{E_mS^{*}_mI^{*}_{h}(a_{i})} +\frac{I^{*}_mE_m}{I_mE^{*}_m} +\frac{I_mI^{*}_{h}(a_{i})}{I^{*}_mI_{h}(t,a_{i})}-4\Big]. \end{gather*} Since arithmetic mean is greater than or equal to the geometric mean (AM--GM inequality), one sees that $\mathcal{V}_{1}\geq0$, $\mathcal{V}_{2}\geq0$, $\mathcal{V}_{3}\geq0$, $\mathcal{V}_{4}\geq0$ and whenever the sign conditions \[ \Big(\frac{I_m}{I_{h}(t,a_{i})}-\frac{I^{*}_m}{I^{*}_{h}(a_{i})}\Big) \Big(1-\frac{I_m}{I^{*}_m}\Big)\geq0 \] with \[ \Big(\frac{I_{h}(t,a_{i})}{I_m}-\frac{I^{*}_{h}(a_{i})}{I^{*}_m}\Big) \Big(1-\frac{I_{h}(t,a_{i})}{I^{*}_{h}(a_{i})}\Big)\geq0 \] hold, it follows from \eqref{lyp.fin.al} that $\mathcal{\dot{V}}\leq 0$ with $\mathcal{\dot{V}}= 0$ if and only if $S_{h}(t,a_{i})=S^{**}_{h}(a_{i})$, $E_{h}(t,a_{i})=E^{**}_{h}(a_{i})$, $I_{h}(t,a_{i})=I^{**}_{h}(a_{i})$, $S_m=S^{**}_m$, $E_m=E^{**}_m$, $I_m=I^{**}_m$. This further implies that $V_{h}(t,a_{i})\to \gamma(a_{i}) I^{*}_{h}(a_{i})/\mu_{h}(a_{i})=V^{*}_{h}(a_{i})$ as $t\to \infty$ since $(S_{h},E_{h},I_{h},S_m,E_m,I_m)\to (S^{**}_{h}(a_{i}),E^{**}_{h}(a_{i}),I^{**}_{h} (a_{i}),S^{**}_m,E^{**}_m,I^{**}_m)$. Therefore, by LaSalle's principle \cite{LaSalle1976}, it follows that every solution of the model \eqref{eq.mdl.1} starting in $\mathfrak{D}\backslash\mathfrak{D}_0$ approaches the endemic equilibrium $\mathcal{E}_{1}$ as $t\to \infty$. \end{proof} The epidemiological implication of Theorem \ref{globalEe} is that malaria can persist in the population whenever the intervention strategies are not adhered to and the associated basic reproduction is greater than one. \subsection*{Conclusion}%\label{S5} In this article, a malaria transmission dynamics with vigilant compartment governed by system of differential equations has been theoretically analyzed. 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