Dynamics and optimal control of an SIVR immuno-epidemiological model with standard incidence

Abstract

Based on the immuno-epidemiological model concept, we propose a susceptible–infected–vaccinated–recovered epidemic model with between-host transmission and within-host infection, where disease transmission between hosts is described by a standard incidence rate and the within-host infection process is governed by a bilinear incidence rate. The basic reproduction number $R_0\left(\psi \right)$ in the between-host model strongly depends on the within-host infection process. If $R_0\left(\psi \right)<1$, the disease-free steady state $E^0$ of the between-host epidemic model is locally stable, and if $R_0\left(\psi \right)>1$, the endemic steady state $E^{⁎}$ of the between-host epidemic model is locally stable. If $R_0\left(0\right)<1$, the disease-free steady state $E^0$ of the between-host epidemic model is globally stable. Furthermore, to better understand the roles of within-host treatment and between-host control in disease transmission, we formulated and studied an optimal control problem for the immuno-epidemiological model involving treatment and vaccination. Numerical simulations were conducted to demonstrate the effectiveness of the control strategies in various infection processes. The results showed that the duration of within-host treatment must be longer than the duration of vaccination to better control the spread of the disease.