On a global continuum model for COVID-19 virus in the presence of vaccine and induced immunity. Stability and initial states control

The dynamics of COVID-19 virus were investigated in the literature via mathematical models. These models take into account the action of the suspected-exposed-infected-recovered people (SEIR). Also, among them, those which account for quarantined, social distancing functions or health isolation, were presented. In the absence of effective vaccines or therapies, prevention and treatment strategies for COVID-19 infections can not issue to non-epidemic state. Over the world, vaccination against the virus is set on. This motivated us to develop a model for inspecting if this treatment will issue to non endemic state. To this end, a global continuum model for the dynamics of this virus in the presence of vaccine and stimulated immunity is constructed. The present model deals with EIR - deceased individuals (EIRD) together with action of the health isolation and travelers (HIT). Which is described by nonlinear dynamical system (NLDS). Our aim here is to reduce the problem of solving this system to the case of solving LDS. This is carried by introducing the unified method (UM) via an approach present by the authors. By the UM, the solutions of a NLDS are recast to solutions of LDS via auxiliary equations. Numerical results of the exact solutions are evaluated, with initial data for the EIRD together with the number of vaccinated people. Real data are taken from Egypt (can be from elsewhere) at the end of the first wave, and they are considered as the initial conditions. These results are compared with a previous work by the authors in the absence of vaccination. The results of exposed, infected, recovered and deceased people are computed. It is found that the number of infected people decays to zero asymptotically, while, the number of infected people decays to an asymptotic value. This is in contrast to the results found previously in the case of absence of vaccination, where, these numbers grow monotonically. This is completely new. It is shown that locking-down has a remarkable effect in diminishing the number of infected people. The region of initial conditions for I-E people, that guarantee non-epidemic, non-endemic states, is determined via initial states control analysis. A software tool, based on this model, for simplifying the utilization of various data of different countries is developed. It is worth to mention that, the exact solutions of nonlinear dynamical equations, found here, are novel.