ABSTRACT
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.
ABSTRACT
Very recently, various mathematical models, for the dynamics of COVID-19 with main contribution of suspected-exposed-infected-recovered people have been proposed. Some models that account for the deceased, quarantined or social distancing functions were also presented. However, in any local space the real data reveals that the effects of lock-down and traveling are significant in decreasing and increasing the impact of this virus respectively. Here, discrete and continuum models for the dynamics of this virus are suggested. The continuum dynamical model is studied in detail. The present model deals with exposed, infected, recovered and deceased individuals (EIRD), which accounts for the health isolation and travelers (HIT) effects. Up to now no exact solutions of the parametric-dependent, nonlinear dynamical system NLDS were found. In this work, our objective is to find the exact solutions of a NLDS. To this issue, a novel approach is presented where a NLDS is recast to a linear dynamical system LDS. This is done by implementing the unified method (UM), with auxiliary equations, which are taken coupled linear ODE's (LDS). Numerical results of the exact solutions are evaluated, which can be applied to data in a local space (or anywhere) when the initial data for the IRD are known. Here, as an example, initial conditions for the components in the model equation of COVID-19, are taken from the real data in Egypt. The results of susceptible, infected, recovered and deceased people are computed. The comparison between the computed results and the real data shows an agreement up to a relative error 1 0 - 3 . On the other hand it is remarked that locking-down plays a dominant role in decreasing the number of infected people. The equilibrium states are determined and it is found that they are stable. This reveals a relevant result that the COVID-19 can be endemic in the case of a disturbance in the number of the exposed people. A disturbance in the form of an increase in the exposed number, leads to an increase in the number of infected people. This result is, globally, valid. Furthermore, initial states control is analyzed, where region of initial conditions for infected and exposed is determined. We developed a software tool to interact with the model and facilitate applying various data of different local spaces.