On a new SEIRDE o I o epidemic model eventually initiated from outside with delayed re-susceptibility and vaccination and treatment feedback controls

Abstract

Abstract This paper studies a new extended SEIR (susceptible-exposed-infectious recovered) epidemic model which incorporates the contribution of infective contagions to the resident population from infective exposed E o and infectious I o outsiders as well as eventual delayed re-susceptibility by partial loss of immunity. It is referred to as a S E I R D E o I o since the dead individuals caused by the disease are defined as a new subpopulation. The potential contribution of the disease propagation of non-resident external infected travellers is considered as well as a re-susceptibility which increases the susceptible numbers and a parallel loss of immunity caused by a potential delayed re-infection. The model is also studied under eventual vaccination and treatment controls each one of them including two additive terms including proportional feedback of the susceptible and infectious, respectively, as well as control actions being independent of the population numbers. The disease-free and endemic equilibrium points are characterized, as well as their dependence on the control gains, and their stability properties are also studied. It is found that they are unique and only one of the two is a global attractor depending on the model parameter values. Typically, the value of the basic reproduction number is crucial to characterize the stability or, alternatively, that of the coefficient transmission rate provided that the remaining parameters are prefixed. The controls are also useful to increase the admissible minimum threshold of the infection force compatible with the stability of the disease-free equilibrium point. Simulation results are given, some of them related to parameterizations of usefulness related to the recent COVID-19, while others to a varicella case study.