Analysis, modeling and optimal control of COVID-19 outbreak with three forms of infection in Democratic Republic of the Congo

Abstract

This paper deals with modeling and simulation of the novel coronavirus in which the infectious individuals are divided into three subgroups representing three forms of infection. The rigorous analysis of the mathematical model is provided. We provide also a rigorous derivation of the basic reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msub> </mml:mrow> </mml:math> . For <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msub> <mml:mo><</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> , we prove that the Disease Free Equilibium (DFE) is Globally Asymptotically Stable (GAS), thus COVID-19 extincts; whereas for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msub> <mml:mo>></mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> , we found the co-existing phenomena under some assumptions and parametric values. Elasticity indices for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msub> </mml:mrow> </mml:math> with respect to different parameters are calculated with baseline parameter values estimated. We also prove that a transcritical bifurcation occurs at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mrow><mml:mn>0</mml:mn></mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn></mml:mrow> </mml:math> . Taking into account the control strategies like screening, treatment and isolation (social distancing measures), we present the optimal control problem of minimizing the cost due to the application of these measures. By reducing the values of some parameters, such as death rates (representing a management effort for all categories of people) and recovered rates (representing the action of reduction in transmission, improved screening, treatment for individuals diagnosed positive to COVID-19 and the implementation of barrier measures limiting contamination for undiagnosed individuals), it appears that after <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>140</mml:mn> <mml:mo>-</mml:mo> <mml:mn>170</mml:mn></mml:mrow> </mml:math> days, the peak of the pandemic is reached and shows that by continuing with this strategy, COVID-19 could be eliminated in the population.