Abstract
As the corona virus (COVID-19) pandemic ravages socio-economic activities in addition to devastating infectious and fatal consequences, optimal control strategy is an effective measure that neutralizes the scourge to its lowest ebb. In this paper, we present a mathematical model for the dynamics of COVID-19, and then we added an optimal control function to the model in order to effectively control the outbreak. We incorporate three main control efforts (isolation, quarantine and hospitalization) into the model aimed at controlling the spread of the pandemic. These efforts are further subdivided into five functions; u1(t) (isolation of the susceptible communities), u2(t) (contact track measure by which susceptible individuals with contact history are quarantined), u3(t) (contact track measure by which infected individualsare quarantined), u4(t) (control effort of hospitalizing the infected I1) and u5(t) (control effort of hospitalizing the infected I2). We establish the existence of the optimal control and also its characterization by applying Pontryaging maximum principle. The disease free equilibrium solution (DFE) is found to be locally asymptotically stable and subsequently we used it to obtain the key parameter; basic reproduction number. We constructed Lyapunov function to which global stability of the solutions is established. Numerical simulations show how adopting the available control measures optimally, will drastically reduce the infectious populations.
Highlights
The novel coronavirus pneumonia which was officially named as Corona Virus Disease 2019 (COVID-19) by World Health Organization (WHO) was reported first in late December 2019, in Wuhan, China [1]
The source of the virus is not yet known, but genetic investigation revealed that COVID-19 virus has the same genetic characteristics with SARS-CoV2 [2]
Elhia et al [20] developed a mathematical model based on the epidemiology of COVID-19, incorporating the isolation of healthy people, confirmed cases and close contacts
Summary
The novel coronavirus pneumonia which was officially named as Corona Virus Disease 2019 (COVID-19) by World Health Organization (WHO) was reported first in late December 2019, in Wuhan, China [1]. The virus is spread to human population, human to human transmission, human to clinic center and human to care center They used Lyapunov function to investigate the global stability analyses of the equilibrium solutions and subsequently obtained the basic reproduction number or roughly, a key parameter describing transmission of the infection. Elhia et al [20] developed a mathematical model based on the epidemiology of COVID-19, incorporating the isolation of healthy people, confirmed cases and close contacts. Most of these models have a general shortcoming of not taking into consideration time dependent control strategies. The paper is arranged in the following order: Chapter 1 gives the introduction, Chapter 2 gives preliminary definitions and theorems, Chapter 3 is the model formulation, Chapter 4 discusses the formulation and analysis of optimal control, Chapter 5 presents local and global stability analyses of the solutions of the model and the derivation of the reproduction number and lastly Chapter 6 gives numerical simulation results and the discussion follows
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