Existence and Uniqueness of Solution for an Infection-Age Structured Mathematical Model of Ebola Virus Disease Dynamics

Abstract

This study examines an infection-age-structured epidemic model for Ebola Virus Disease (EVD) dynamics. We incorporate the effects of multiple quarantine stages and dead bodies of Ebola victims into a hybrid system comprising ordinary differential equations (ODEs), partial differential equations (PDEs), and integro-differential equations to investigate the transmission dynamics of EVD. We prove the existence and uniqueness of solutions using Volterra integral equations and classical Gronwall’s inequality $B(t) \leq |\beta|_{L} e^{t |\beta|_{L}} |\psi|_{L}$. We also show that the solutions of partial differential equations and integro-differential equations depend continuously on the initial infection age of EVD distributions. We show that the solutions of ordinary differential equations exist and are unique. The effective reproduction number $(R_{eff})$ was obtained with a threshold value of $R_{P} + R_{Q} + R_{G}$ and is used to analyze the local and global stability of disease-free equilibrium. The local stability of the disease-free equilibrium was linearized to obtain $\left| G_1(a_1) \right| \leq G_1(0) < 1$, where $ G_1(a_1)$ is a function of $R_{eff}$. A suitable Lyapunov function was constructed and used to analyze the global stability of the disease-free equilibrium. The result obtained was $q(t, \tau)\geq 0$, where $q(t, \tau)$ is the density function of infection-age for all $t> \tau $. This implies that disease-free equilibrium is locally and globally asymptotically stable if $R_{eff}< 1$ and $R_{eff} \leq 1$ respectively. The local and global stability of endemic equilibrium were analyzed using LaSalle’s Invariance theory and Lyapunov function.