On a New Epidemic Model with Asymptomatic and Dead-Infective Subpopulations with Feedback Controls Useful for Ebola Disease

Abstract

This paper studies the nonnegativity and local and global stability properties of the solutions of a newly proposed SEIADR model which incorporates asymptomatic and dead-infective subpopulations into the standard SEIR model and, in parallel, it incorporates feedback vaccination plus a constant term on the susceptible and feedback antiviral treatment controls on the symptomatic infectious subpopulation. A third control action of impulsive type (or “culling”) consists of the periodic retirement of all or a fraction of the lying corpses which can become infective in certain diseases, for instance, the Ebola infection. The three controls are allowed to be eventually time varying and contain a total of four design control gains. The local stability analysis around both the disease-free and endemic equilibrium points is performed by the investigation of the eigenvalues of the corresponding Jacobian matrices. The global stability is formally discussed by using tools of qualitative theory of differential equations by using Gauss-Stokes and Bendixson theorems so that neither Lyapunov equation candidates nor the explicit solutions are used. It is proved that stability holds as a parallel property to positivity and that disease-free and the endemic equilibrium states cannot be simultaneously either stable or unstable. The periodic limit solution trajectories and equilibrium points are analyzed in a combined fashion in the sense that the endemic periodic solutions become, in particular, equilibrium points if the control gains converge to constant values and the control gain for culling the infective corpses is asymptotically zeroed.