Mathematical modeling and optimal control analysis for the transmission risk of COVID-19

Abstract

Coronavirus disease (COVID-19) is an infectious disease caused by SARS-CoV-2, a betacoronavirus.It is a pandemic disease affecting many countries. The present study was aims toinvestigate in the first phase the transmission risk of COVID-19 with optimal control. In thesecond case, to formulate and analyze a mathematical model of COVID-19 transmission dynamicsand apply it as case study in Ethiopia. Accordingly, we apply optimal control theory to a novelcoronavirus(COVID-19) transmission model given by a system of nonlinear ordinary differentialequations. An expression for the basic reproduction number is derived in terms of controlvariables. Then the sensitivity of basic reproduction number with respect to model parametersis also analyzed. Optimal control strategies are obtained by minimizing the number of exposedand infected population considering the cost of implementation. The existence of optimal controlsand characterization is established using Pontryagin’s Minimum Principle. Numerical simulationresults demonstrated good agreement with our analytical results. Finally, the findings of this studyshows that comprehensive impacts of prevention, intensive medical care and surface disinfectionstrategies outperform in reducing the disease epidemic with optimum implementation cost. In thesecond phase, we proposed a nonlinear deterministic mathematical model for the transmissiondynamics of COVID-19. Then, we analyzed the system properties such as boundedness of thesolutions, existence of disease-free and endemic equilibria, local and global stability of equilibriumpoints. Besides, we computed the basic reproduction number R0 and studied its normalizedsensitivity for model parameters to identify the most influencing parameter. The local stabilityof the disease-free equilibrium point is also verified via the help of the Jacobian matrix andRouth Hurwitz criteria. Moreover, the global stability of the disease-free equilibrium point isproved by using the approach of Castillo-Chavez and Song. We also proved the existence of theforward bifurcation using the center manifold theory. Then the model is fitted with COVID-19infected cases reported from March 13, 2020, to July 31, 2021, in Ethiopia. The values of modelparameters are then estimated from the data reported using the least square method togetherwith the fminsearch function in the MATLAB optimization toolbox. Finally, different simulationcases were performed using PYTHON software to compare with analytical results. The simulationresults suggest that the spread of COVID-19 can be managed via minimizing the contact rate ofinfected and increasing the quarantine of exposed individuals..