Estimation and optimal control of the multi-scale dynamics of the Covid-19

Abstract

AbstractThis work aims at a better understanding and the optimal control of the spread of the new severe acute respiratory corona virus 2 (SARS-CoV-2). We first propose a multi-scale model giving insights on the virus population dynamics, the transmission process and the infection mechanism. We consider 10 compartments in the human population in order to take into accounts the effects of different specific mitigation policies: susceptible, infected, infectious, quarantined, hospitalized, treated, recovered, non-infectious dead, infectious dead, buried. The population of viruses is also partitioned into 10 compartments corresponding respectively to each of the first nine human population compartments and the free viruses available in the environment. Indeed, we have human to human virus transmission, human to environment virus transmission, environment to human virus transmission and self infection by susceptible individuals. We show the global stability of the disease free equilibrium if a given threshold 𝒯0 is less or equal to 1 and we provide how to compute the basic reproduction number ℛ0. A convergence index 𝒯1 is also defined in order to estimate the speed at which the disease extincts and an upper bound to the time of extinction is given. The existence of the endemic equilibrium is conditional and its description is provided. We evaluate the sensitivity of ℛ0, 𝒯0 and 𝒯1 to control parameters such as the maximal human density allowed per unit of surface, the rate of disinfection both for people and environment, the mobility probability, the wearing mask probability or efficiency, and the human to human contact rate which results from the previous one. Except the maximal human density allowed per unit of surface, all those parameters have significant effects on the qualitative dynamics of the disease. The most significant is the probability of wearing mask followed by the probability of mobility and the disinfection rate. According to a functional cost taking into consideration economic impacts of SARS-CoV-2, we determine and discuss optimal fighting strategies. The study is applied to real available data from Cameroon and an estimation of model parameters is done. After several simulations, social distancing and the disinfection frequency appear as the main elements of the optimal control strategy.