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 (which was likely to be originated from bats) [2]. It is also found to be significantly less severe than the other two coronaviruses; Severe Acute Respiratory Syndrome (SARS-COV) and Middle East Respiratory Syndrome (MERS-COV) that caused an outbreak in 2002 and 2008 respectively [3]. The most important routes of human to human transmission of COVID-19 are respiratory droplets and contact transmission [4]. After the incubation period which is generally 2–14 days, the mild symptoms may persist from high degree fever, cough and shortness of breath to being severely ill and subsequently death [4].

As scientists all over the world are busy trying to develop a cure and vaccine, all hands must be put together to support and comply with the standard recommendations that can lower the transmissions of the disease. This is why, the following measures must be taken; social distancing, self-isolation, use of personal protective equipment (such as face mask, hand globes, overall gown, etc.), regular hand washing using soap or sanitizer, avoid having contact with person showing the symptoms and report any suspected case. Moreover, relevant authorities must engage in widely public orientation exercise for sensitization and enlightenment, banning of social (or religious) gathering and local (or international) trip, contact tracing and isolation of infected individuals, providing sanitizers at public domains like markets and car parks, fumigating exercise, and to the large extent imposing lockdown.

The scourge does not only cause apocalyptic proportion in terms of infection, morbidity and fatality, but also socio-economic consequences. To control the above mentioned problems, there is need to have better understanding on the transmission dynamics of the disease. This could be achieved by developing mathematical model that optimizes the possible control measures.

Optimal control is considered as an effective mathematical tool used to optimize the control problems arising in different field including epidemiology, aeronautic engineering, economics, finance, robotics, etc [5]. Mathematical model offers an insight in to the transmission and control of infectious disease [6–13]. Zhao et al. [14] developed a Susceptible, Un-quarantined infected, Quarantined infected, Confirmed infected (SUQC) model to characterize the dynamics of COVID-19 and explicitly parameterize the intervention effects of control measures. Song et al. [15] established deterministic mathematical model (SEIHR) to suit Korean outbreak, in which he estimated the reproduction number and the effect of preventive measures.

Tahir et al. [16] developed a mathematical model (for MERS) in form of nonlinear system of differential equations, in which he considered a camel to be the source of the infection. The virus is then spread to human population, then human to human transmission, then human to clinic center and then 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.

Yang et al. [17] proposed a mathematical model to investigate the current outbreak of the coronavirus disease (COVID-19) in Wuhan, China. The model described the multiple transmission pathways in the infection dynamics, and emphasized the role of environmental reservoir in the transmission and spread of the disease. However, the model employed non-constant transmission rates which change with the epidemiological status and environmental conditions and which reflect the impact of the ongoing disease control measures.

Chen et al. [18] modeled (based on SEIR) the outbreak in Wuhan with individual reaction and governmental action (holiday extension, city lockdown, hospitalization and quarantine) in which they estimated the preliminary magnitude of different effect of individual reaction and governmental action. Sunhwa et al. [19] developed a Bats-Hosts-Reservoir-People (BHRP) transmission network model for the potential transmission from the infection source (probably bats) to the human, which focuses on calculating R₀. 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. For the model to be more realistic, it has to be time dependent [21–26]. Here, we modified the work of Elhia et al. [20], by incorporating control functions with the aim of deriving optimal control that drastically minimizes the spread of the infection.

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 then the discussion follows.

Definition 1 (Optimal Control) [27]: A fairly general continuous time optimal control problem can be defined as follows:

Problem i: To find the control vector trajectory u:[t0,tf]∈ℝ→ℝn that minimizes the performance index:

(1) J(u)=φ(x(tf))+∫ t0tf L(x(t),u(t),t)dt.

Subject to:

(2) ẋ(t)=f(x(t),u(t),t),x(t0)=x0.

where x=(x1,x2,x3,…,xn)T, f=(f1,f2,f3,…,fn)T and φ:ℝn×ℝ→ℝ is a terminal cost function.

Problem ii: Find t_f and u(t) to minimize: J=∫ t0tf 1dt=tf-t0.

Subject to: ẋ(t)=f(x(t),u(t),t),x(t0)=x0.

This special type of optimal control problem is called the minimum time problem.

Definition 2 (Hamiltonian): A time varying Lagrange’s multiplier function λ:[t0,tf]→ℝ, also known as the co-state define Hamiltonian function H as:

(3) H(x(t),u(t),λ(t),t)=L(x(t),u(t),t)+λ(t)Tf(x(t),u(t),t),

such that

(4) J(u)=φ(x(tf))+∫ t0tf {H(x(t),u(t),λ(t),t)-λT(t)x}dt.

Theorem 1 (Pontryagin Maximum Principle): If u*(t),x*(t)(t∈[t0,tf]) is a solution of the optimal control problem Eqs. (1) and (2) then there exists a non-zero absolutely continuous function λ(t) such that λ(t),x*(t),u*(t) satisfy the system

(5) dxdt=∂H∂λ,dλdt=-∂H∂x,

such that, for almost all t∈[t0,tf] the function in Eq. (3) attains its maximum: H(λ(t),x*(t),u*(t))=M(λ,x),

(6) M(λ(tf),x*(tf))= sup{H(λ,x,u):u∈U}.

and such that at terminal time t_f the conditions M(λ(tf),x*(tf))=0,λ0(tf)≤,aresatisfied.

If the functions λ(t),x(t),u(t) satisfy the relation Eqs. (5) and (6) (i.e., x(t),u(t) are Portryagin extremals), then the condition M(t)=M(λ(t),x(t))=const.,λ0(t)=constholds.

Remark 1: Becerra states that for a minimum, it is necessary for the stationary (optimality) condition to give:

(7) ∂HT∂u=0.

We know that most people are susceptible to COVID-19 and the patients in the incubation period can infect healthy people. We denote the population of susceptible people with S, the patients in the incubation period and the patients that are yet to be diagnosed by I, patients in the hospital by H, removed people by R, respectively. Here the infectivity of the patients in the incubation period and the patients that are yet to be diagnosed are assumed to be the same.

After the outbreak of COVID-19, susceptible people are advised to lock themselves down at home, and all close contacts of infected individuals tracked are quarantined. Therefore, we divide the population of susceptible people into; susceptible people (S₁), the quarantined susceptible people (by close contacts tracked measure) (S₂) and general isolated susceptible people (due to community lockdown) (S₃). Infected people population is divided into general infected people, including the patients in the incubation period and the infected people that are yet to be diagnosed (I₁) and infected people that are quarantined (I₂). Here we assume that all susceptible people isolated at home cannot be infected and all infected people isolated at home cannot infect healthy people. Thus, we establish the transmission dynamics of the disease as in Fig. 1.

The transmission dynamics can be described by the nonlinear system of first order differential equations as follows:

(8) dS1dt=α1S2-βS1I1N-S2-S3-I2-H-q1-μS1+α2S3,

(9) dS2dt=q1-α1S2,

where, q1= min{S1,γb(I1+I2)S2+I2S2}andq2= min{I1,γb(I1+I2)S2+I2I2}. S1(0)≥0,S2(0)≥0,S3(0)≥0,I1(0)≥0,I2(0)≥0,H(0)≥0,R(0)≥0.

When patients go to hospital and are diagnosed (I₁ +I₂), by the close contacts tracked measure, susceptible people (q₁) and infected people (q₂) are quarantined by the proposition b. Here the number of quarantined susceptible people (q₁) and quarantined infected people (q₂) are less than the number of susceptible people (S₁) and infected people (I₁) respectively. Then q1= min{S1,γb(I1+I2)S2+I2S2} and q2= min{I1,γb(I1+I2)S2+I2I2}.

After the isolation of 14 days (1α1), the quarantined susceptible individuals become susceptible. When quarantined infected people have symptoms, they are hospitalized and diagnosed (I₂). After the time of treatment (1δ), they are removed from the hospital. The communities are isolated and healthy people are also advised to isolate themselves at home unless they have something urgent to deal with. Then μS1 and α2S3 denote the weak movements of population from susceptible to isolated susceptible and from isolated susceptible to susceptible, respectively.

Here the detail formulation and analysis of the optimal control problem with respect to the model Eqs. (8)–(14) is given.