Mathematical Analysis and Computer Simulations of a Generalized Model of Sars-CoV-2 Induced Covid- 19 Epidemiological Dynamics

A generalized non-linear deterministic mathematical model was constructed and analyzed depicting the epidemiological dynamics of SARS-CoV-2 virus induced COVID-19 disease in a domain of humans with varying susceptibilities in a non-isolated population. The model incorporates biophysically relevant characteristics such as compartmentalization of the target population into six categories: number of susceptible persons; number of persons infected with SARS-CoV-2 virus with a positive PCR test; number of persons exposed to SARS-CoV-2 virus, but who are either asymptomatic or do not have a positive PCR test; number of persons who are infected with SARS-CoV-2 virus and are hospitalized; number of persons who are recovered or recovering or in quarantine; and number of persons who are dying due to exposure, infection, and complications due to SARS-CoV-2 virus. The model rate constants, parameters, and stoichiometric coefficients are epidemiologically quantifiable and biophysically measurable. Explicit investigative computer simulations are presented depicting various plausible hypothetical regimes of the dynamics of SARS-CoV-2 induced COVID-19 under various population characteristics and values of R0 and Re. The criteria for existence, local stability, and global stability of the disease-free equilibrium E1 = [x1, 0, 0, 0, 0] are clearly exhibited in terms of biophysically measurable model parameters, using non-linear systems analysis and principles of linearized stability via the Hartman-Grobman theorem, LaSalle’s theory, Hopf-Andronov-Poincare bifurcation, and Lyapunov methods. In addition, the criteria for the persistence of SARS-CoV-2 in a population have been presented. It is shown that a Hopf-Andronov-Poincare bifurcation occurs at the Hopf point(E1,μ0) with μ0 = (k01+ k k 12 02 x + 1 k11) , and that the disease-free equilibrium E1 loses its asymptotic stability and is surrounded by periodic oscillations. Robust criteria have been derived under which the doomsday equilibrium E2 = [0, 0, 0, 0, 0] which represents total population annihilation, will manifest locally as a local attractor or as an unstable equilibrium point. The conditions under which SARS-CoV-2 virus induced COVID-19 will persist endemically in the population or exhibit endemic global stability have been stated and proved analytically using explicitly stated theorems.