Mathematical model of COVID-19 in Nigeria with optimal control

The novel Coronavirus Disease 2019 (COVID-19) is a highly infectious disease caused by a new strain of severe acute respiratory syndrome of coronavirus 2 (SARS-CoV-2). In this work, we proposed a mathematical model of COVID-19. We carried out the qualitative analysis along with an epidemic indicator which is the basic reproduction number (R0) of this model, stability analysis of COVID-19 free equilibrium (CFE) and Endemic equilibrium (EE) using Lyaponuv function are considered. We extended the basic model into optimal control system by incorporating three control strategies. These are; use of face-mask and hand sanitizer along with social distancing; treatment of COVID-19 patients and active screening with testing and the third control is prevention against recurrence and reinfection of humans who have recovered from COVID-19. Daily data given by Nigeria Center for Disease Control (NCDC) in Nigeria is used for simulation of the proposed COVID-19 model to see the effects of the control measures. The biological interpretation of this findings is that, COVID-19 can be effectively managed or eliminated in Nigeria if the control measures implemented are capable of taking or sustaining the basic reproductive number R0 to a value below unity. If the three control strategies are well managed by the government namely; NCDC, Presidential Task Force (PTF) and Federal Ministry of Health (FMOH) or policymakers, then COVID-19 in Nigeria will be eradicated.


Introduction
A novel coronavirus disease (2019-nCoV) outbreak in Wuhan City, Hubei, China was announced on December 31, 2019 by World Health Organization (WHO) [1]. On February 11, 2020, an International Committee on Taxonomy of Viruses gave the disease a name called Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) [2]. Due to the high rate of global spread of this virus, World Health Organization declared it to be a global pandemic on March 11, 2020 [3]. As at September 7, 2020, by 10:26 h, the global confirmed cases of COVID-19 stood at 27,300, 888 (+25,452) with 893,130 (+455) that is, 3.27% deaths. United states of America (USA) had the highest number of confirmed cases of 6,460,421 (+171) with 193,253 (+3) deaths. This was followed by India which had the number of confirmed cases of 4,208,645 (+6,083) with 71,711 (+24) deaths. Brazil was the third country with the highest number of confirmed cases of COVID-19 with 4,137,606 confirmed cases and 126, 686 deaths [4]. Nigeria, Africa's most populous country with an estimated 200 million population [5], is one of the COVID-19 epicenters in Africa. On 27th February 2020, Nigeria recorded its first COVID-19 case when an Italian citizen employed in Nigeria was diagnosed with the disease when he had returned to Nigeria during his visit to Milan, Italy [6]. As of September 7, 2020, Nigeria was ranked 52nd with the highest number of confirmed cases of 55, 005 with 1,057 deaths [6]. Among other symptoms of COVID-19 are; sore throat, dry cough, fever, loss of taste, muscle pain, fatigue and cold [6]. On 24th January 2021, there was a rise in the number of confirmed case for coronavirus  in Nigeria, reaching a total of 142,579 confirmed cases. A total of 1,702 casualties and 118 thousand recoveries were recorded in the country as of the same date. As regards reported cases, Nigeria is the sixth top-ranked African country. The highest daily increase in cases in Nigeria since the beginning of the pandemic was reported on January 23. The Cumulative number of confirmed cases of COVID-19 in Nigeria was recorded from February 28, 2020 to February 10, 2021 [7]. Over the years, mathematical models have proved to be reliable and powerful tools used to implement control measures to prevent and minimize the impact of infectious diseases [8,9].
The need to develop a mathematical model to study the dynamics of the transmission and spread of diseases with control measures is important because of the rapid spread of COVID-19 in Nigeria. This study is motivated by the "on-going" community spread of the novel disease of COVID-19 in Nigeria with a view to flattening the infection curve. In order to achieve this goal, we develop a new deterministic mathematical model with five-compartments based on the epidemiological status of individuals. In order to determine the optimal level of time-dependent prevention and management steps that will substantially reduce the number of infectious humans in the population, the optimal control strategy is used in the proposed COVID-19 model.

COVID-19 model formulation
In this section, we use mathematical model to illustrate the total population of humans at time t denoted by N(t). The total population is subdivided into five compartments. That is, susceptible humans (S(t)), exposed humans (E(t)), humans infected with COVID-19 disease (I(t)), quarantine (Q(t)) and recovered humans from COVID-19 (R(t)). Therefore Λ is the recruitment rate of humans, ϕ c is the contact rate of COVID-19 transmission. Also, υ 1 and θ are the recovery rates of COVID-19 from I(t) and Q(t) respectively. μ is the natural death rate, δ is the COVID-19induced death rate. α is the detection rate from I(t) and υ 2 is the relapse rate of COVID-19. Therefore, susceptible humans develop COVID-19 infection at the rate β c = ϕ c I N . The formulation above with the assumptions and the movement of human from one stage to another at different rates can be shown in the COVID-19 flow chart model in Fig. 2.The flow chart of the COVID-19 model is represented by the following differential equations written below (2)

Analysis of COVID-19 model
In this section, we will carry out the quantitative study of COVID-19 which comprises the positivity and boundedness of the solutions, invariant region of the COVID-19 model and existence of equilibrium points for COVID-19 model. Others include, the basic reproduction number (R 0 ) of COVID-19 model, global stability of COVID-19 free and endemic equilibria.    Integrating both sides from t = 0 to t = t * we obtain

Positivity of the Solution
From the result, S(t * ) is greater than or equal to the sum of positive terms. By the same argument, we can also prove that

Boundedness of the solution
Proof. To prove boundedness of the solution, 0 < I(t) ⩽N(t). Adding all the equations in model (2), we obtain The solutions of the proposed model in (2) are bounded. Hence, from eq.(5), Λ − (μ + δ)N(t)⩽ dN(t) dt ⩽Λ − μN(t). and this can now be written as

Invariant region of COVID-19 model
Considering COVID-19 model (2) with positive initial conditions, all the state variables must be feasible in the region Ω. Therefore, the feasible region of the COVID-19 model (2) can be expressed as Ω⊂R 5 The following steps are considered to verify the positive invariance of Ω that is, the solution in Ω still remains in Ω∀t > 0. This follows that dN(t) dt ⩽Λ − μN(t). By standard comparison theorem [32], we obtain In particular, Hence, the region Ω is positive invariant. It is sufficient to consider the dynamics of the flow formulated by model (2) in Ω. Therefore the COVID-19 model is well posed, Mathematically, Biologically and Epidemiologically valid in the region Ω. □

Existence of equilibrium points for COVID-19 model
In this section, we examine the COVID-19 model (2) objectively with the condition of existence of the equilibrium points. To verify the existence of the equilibrium points, the COVID-19 free equilibrium (CFE) point can be obtained by setting the derivatives on the right-hand side equations of COVID-19 model (2) to be zero. It is also achieved by substituting humans infected with COVID-19 (I) to be zero. Thus, Similarly, setting the derivatives on the right-hand side of COVID-19 model (2) to be zero and substituting S = S * , E = E * , I = I * , Q = Q * and R = R * . Endemic equilibrium point (E 1 ) exists when I is non-zero and we obtain , By simplifying each of the equation in eq.(9) further to obtain where a 1 = (α +υ 2 +δ +μ), a 2 = (σ +μ), a 3 = (θ +μ) and

Basic Reproduction Number (R 0 ) of COVID-19 Model
Whenever R 0 < 1, each human discharges or liberates on average that is below one infected human then COVID-19 dies out. Otherwise, COVID-19 turns out to be endemic or affects the susceptible population. We derive R 0 of the COVID-19 free equilibrium (CFE) by using next generation matrix approach [33,34]. Thus, and Multiplying F with the inverse matrix of V, the basic reproduction number (R 0 ) is the dominant or largest eigenvalue corresponding to the Spectral radius of the matrix (FV − 1 ). This is denoted by where ρ is the Spectral radius. Therefore, the basic reproduction number is given by

Global Stability of the COVID-19 Free Equilibrium (CFE)
Proof. We define the Lyapunov function It is easy to establish that x 1 and x 2 are positive. Differentiating T with respect to time, This can be written as From the result obtained in eq.(16), we can see that Ṫ ⩽0 provided R 0 ⩽0 as well as Ṫ = 0 provided that R 0 = 1 or I = 0. This implies that, the highest invariance set in and DFE (E 0 ) is globally asymptotically stable in R 5 + [35,36]. Epidemiologically, the proof of Theorem 4 show that, COVID-19 will die out in the neighbourhood whenever R 0 ⩽0 regardless of the number in eq.(2) at the initial stage of the population. □

Global stability analysis of endemic equilibrium
Proof. The existence of endemic equilibrium points E 1 = (S * , E * , I * , Q * , R * ) defined in eqs. (9) and (10), where I is written as By substitution method and making I subject of formula, eq.(17) becomes , and endemic equilibrium does not exist. Moreover, Therefore, there exists the endemic equilibrium only at R 0 > 1. This shows that model (2) has a unique endemic that is positive equilibrium whenever R 0 > 1. □ Theorem 6. If R 0 > 1 then the endemic equilibrium of model (2) given by Proof. Following [35][36][37], the equation below is made of the following Goh-Volterra type Lyapunov function: where k1=ϕcS * I * σE * and k 2 = ϕ c S * I * αI * . Differentiating eq.(19) with respect to time, we have; If R h →R * h as time, t→∞ in eq.(9) and model (2) then by using substitution method and simplifying to obtain, Lastly, since arithmetic mean is more than geometric mean, then the following inequalities hold; Therefore, L ⩽0 for R 0 > 1. Since all the parameters are positive Also R→R * h as time, t→∞ and by LaSalle's invariance principle [36], the endemic equilibrium E 1 is globally asymptotically stable whenever R 0 > 1. Epidemiologically, the proof of Theorem 6 implies that, COVID-19 would establish itself in the neighbourhood whenever R 0 > 1 regardless of the number of infectious humans at the initial stage of the population (see Table 1).

Model fitting and estimation of parameters
In this section, we carry out the numerical simulation for the model fitting using least square method. This enables us to compare our model with the daily data of COVID-19 released by NCDC from December 1, 2020, to January 31, 2020 as shown in Fig. 2. Also, we consider the NCDC report released on December 1, 2020, for the initial conditions E(0) = 2003 which is the sample tested, I(0) = 416 which is the new confirmed cases, Q(0) = 404 which is the quarantine of humans (that is, the differences between new confirmed cases and new confirmed fatalities), R(0) = 115 which is the new discharged cases and an estimated S(0) = 5000 which is the susceptible humans. using the initial conditions and some estimation of parameters (that is, ϕ c , σ and δ), we are able to fit other parameters (that is, μ, υ 1 , υ 2 , υ 3 , α and θ) as well as calculating the basic reproduction number R 0 = 0.523984 in eq. (13) as shown in Table 2.

Optimal control analysis of COVID-19
Optimal control is one of the important tools used in mathematical biology to eradicate or minimize the rate of infection in exposed and infected humans in the population. In order to minimize COVID-19 transmission, we propose a model which is an extension of the model (1) by incorporating three control measures into the model. These are; Preventive effort being made by NCDC to ensure that the public follows the guidelines such as, use of face mask, hand sanitizer and social distancing (u 1 (t)), treatment of COVID-19 patients, active screening along with testing (u 2 (t)), and control against recurrence and reinfection of humans who have recovered from COVID-19 (u 3 (t)). Therefore, the reduction rate of COVID-19 through NCDC guidelines is given by factor Also, reduction in relapse rate of humans who have recovered from COVID-19 is given by factor (1 − u 3 (t)) and from the autonomous equations in eq.(1), the proposed optimal control of COVID-19 model can be written as Following eq.(2), we normalize the above eq.(21) as dS dt

Description of Optimal Control
In this section, we minimize the total number of exposed and infected humans to COVID-19 in the population using the control variables u 1 (t) , u 2 (t) and u 3 (t) in the model. Also, we show that all variables u 1 (t), u 2 (t) and u 3 (t) in the model dynamics of the population are positive. According to [42], we define the objective function as subject to model (22), where A 1 and A 2 are positive weight constants of exposed and infected humans respectively. B 1 , B 2 and B 3 are positive weight constants of control variables u 1 (t), u 2 (t) and u 3 (t) respectively. Meanwhile, the quadratic costs B 1 u 2 1 (t), B 2 u 2 2 (t) and P 3 u 2 3 (t) associated  with the use of preventive effort by NCDC to ensure that the public follow the guidelines such as, use of face-mask, hand sanitizer along with social distancing, treatment of COVID-19 patients, active screening with testing and control against recurrence and reinfection of humans who have recovered from COVID-19 respectively. We choose the quadratic cost in line with the literature on epidemic controls [38][39][40]. The purpose is to find an optimal control of u * 1 (t), u * 2 (t) and u * 3 (t) such that; where Ω = {u i : 0⩽u i (t)⩽1, Lebesgue measurable t = [0, t f ] for i = 1, 2, 3} is the control set subject to the model (22) with initial conditions.

Existence of the optimal control
In this part, we show the existence of the optimal control with initial conditions t = 0 viz state and prove the following theorems, and analyze the properties of the model (22) with all non-negative initial conditions ∀t > 0. We will also use model (22) to see the existence of optimal control with the necessary conditions that satisfy the Pontryagin's Maximum Principle [41]. This can be done by applying Pontryagin's Maximum Principle to convert Eqs. (22)-(24) into a problem of minimizing point-wise Lagrange, L, with respect to u 1 , u 2 and u 3 . The Lagrangian of the control problem is given by; This would be used to find minimal value of the Lagrangian. It could be achieved by defining Hamiltonian, H, for the control problem as [41] Substituting eq.(25) and model (22) into eq. (26), we obtain Where λ 1 , λ 2 , λ 3 , λ 4 and λ 5 are the adjoint variables. The existence of optimal control of the model (22) would be considered by applying the theorem.

Theorem 7. There exists an optimal control u
∈ Ω such that; the control model (22) with initial conditions at t = 0 and Proof. The state and control variables of the model (22) are positive values and the control set Ω is close and convex. Therefore the integrand of the objective function J in which it is expressed in model (22) is a convex function of (u 1 , u 2 , u 3 ) on the control set Ω. Since the state solutions are bounded, then Lipschitz property of the state system with respect to the state variables is satisfied. It can also be seen that ∃ positive numbers η 1 , η 2 and a constant ∊ > 1 such that, Therefore, the state variables are bounded and the existence of optimal control of the model (22) is concluded. □

Uniqueness of the optimal control
Pontryagin's Maximum Principle is used to reveal the necessary conditions for this optimal control. This is as a result of the fact that, minimizing the cost functional in eq. (23) subject to the model (22) is the existence of an optimal control. According to [19][20][21], If (x, u) is an optimal solution of an optimal control problem then ∃ a non-trivial vector function λ = (λ 1 , λ 2 , λ 3 , λ 4 , …, λ n ) thus, satisfying the following equations Therefore, we can now apply the necessary conditions to the Hamiltonian, H, in eq. (27). Theorem 8. Let S * , E * , I * , Q * and R * be optimal state solutions associated with optimal control (u * 1 , u * 2 , u * 3 ) for the optimal control problem in model (22) and eq. (23). There exist the co-states λ i which verify eq.(31) with the transversality conditions λ i (t f ) = 0 in eq.(32) for i = 1, 2, 3, 4, 5 and in eq.
Proof. Consider eq.(27) by differentiating Hamiltonian, H, with respect to S, E, I, Q and R. Also, considering the state variables by applying the first and third equations in eq.(30) into eq. (27), we have the following: with the transversality conditions To evaluate the optimal control of the control variable set, where u i = (0, 1). let S = S * , E = E * , I = I * , Q = Q * and R = R * and apply the second equation in eq.(30) by differentiating Hamiltonian, H, in eq. (27) with respect to control variable u 1 , u 2 and u 3 to obtain make u * i for i = 1, 2, 3 subject of formula to obtain , .
Therefore, we have This show that, the uniqueness of the optimal control of the model has been achieved for small t f based on prior boundedness of the state variables as well as adjoint variables. This is made possible through the use of Lipschitz property of the ordinary differential equations. Graph of COVID-19 data and model is provided in Fig. 3. □

Category A: Use of face-mask, sanitizer and keeping social distancing control
In this category, Fig. 4 illustrates the impact of face-mask, sanitizer and social distancing control (u 1 ) in eradicating COVID-19 in Nigeria. As shown in Fig. 4 that is, in Fig. 4(a), it is shown that, the number of exposed humans increases from 2003 very fast above 87509 at time t = 200 days when the use of face-mask, sanitizer and social distancing control are not implemented. But when the use of face-mask, sanitizer and social distancing control are implemented, it decreases gradually from 2003 to 1132 at time t = 200 days and it will continue to decrease until there is no exposed humans as time increases. Fig. 4(b) shows that the number of infected humans increases from 416 very fast until it reaches 2899 within time t = 52 days before gradual reduction of number of infected humans when the use face-mask, sanitizer and social distancing control is not implemented. But when the use of face-mask, sanitizer and social distancing control is implemented, the number of infected humans increases gradually from 416 until it reaches 692 within time t = 30 days before decreasing gradually until there is no infected humans as time increases. The control profile in Fig. 4(c) shows that, the use of face-mask, sanitizer and social distancing control (u 1 ) is at its maximum value from the starting point until time t = 200 days before decreasing to the minimum value.

Category B: Treatment of COVID-19 patients, active screening and testing
In this category, Fig. 5 shows the influence of treatment of COVID-19 patients, active screening and testing control (u 2 ) on COVID-19 spread in Nigeria. As shown in Fig. 5 that is, in Fig. 5(a), it is verified that the number of exposed humans keeps on increasing from 2003 very fast above 87510 at time t = 200 days in the absence of treatment of COVID-19 patients control. Also, when it involves treatment of COVID-19 patients, active screening and testing control, the number of exposed human increases from 2003 not as fast as without control above 35545 at time t = 200 days. In Fig. 5(b), it is verified that the number of infected humans increases from 416 very fast until it reaches 2899 within time t = 52 days before gradual reduction of number of infected humans when treatment of COVID-19 patients, active screening and testing control is not implemented. But when the treatment of COVID-19 patients, active screening and testing control are implemented, it increases gradually from 416 until it reaches 580 within time t = 30 days before decreasing gradually until there is no infected human as time increases. The control profile in Fig. 5(c) show that, treatment of COVID-19 patients, active screening and testing control (u 2 ) is at its minimum  value from the starting point until time t = 200 days before increasing to the maximum value.

Category C: Control against recurrence and reinfection of humans who have recovered from COVID-19
In this category, Fig. 6 shows the effect of control against relapse of humans who have recovered from COVID-19 (u 3 ). From Fig. 6 that is, in Fig. 6(a), it is confirmed that the number of exposed human continue to increase very rapid above 87509 at time t = 200 days without the control against relapse of humans who have recovered from COVID-19. But with the control against relapse of humans who have recovered from COVID-19, the number of exposed humans increases from 2003 gradually above 19957 at time t = 200 days. In Fig. 6(b), it is confirmed that the number of infected humans increases from 416 very fast until it reaches 2899 within time t = 52 days before gradual reduction of the number of infected humans without the implementation of control against relapse of humans who have recovered from COVID-19. But with the implementation of control against relapse of humans who have recovered from COVID-19, it decreases gradually from 416 to 112 at time t = 200 days and it will continue to decrease until there is no infected human as time increases. The control profile in Fig. 6(c) shows that, the control against relapse of humans who have recovered from COVID-19 (u 3 ) is at its minimum value from the starting point until time t = 200 days before increasing to the maximum value.

Category D:
Use of face-mask, sanitizer and keeping social distancing control, treatment of COVID-19 patients, active screening and testing, and control against relapse of humans who have recovered from  In this category, Fig. 7 shows the impact of face-mask, sanitizer and social distancing control (u 1 ), treatment of COVID-19 patients, active screening and testing (u 2 ), and control against relapse of humans who have recovered from COVID-19 (u 3 ). As shown in Fig. 7 that is, in Fig. 7  (a), it is verified that the number of exposed human increases from 2003 very fast above 87509 at time t = 200 days without control strategies (u 1 , u 2 and u 3 ). But with control strategies (u 1 , u 2 and u 3 ), the number of exposed human decreases gradually from 2003 to 1042 at time t = 200 days and it will continue to decrease until there is no exposed human as time increases. In Fig. 7(b), it is verified that the number of infected human increases from 416 very fast until it reaches 2899 within time t = 52 days before gradual reduction of number of infected humans when all the control strategies (u 1 , u 2 and u 3 ) are not implemented, but when all the control strategies (u 1 , u 2 and u 3 ) are implemented, it decreases gradually from 416 to 15 at time t = 200 days and it will continue to decrease until there is no infected humans as time increases. The control profile in Fig. 7(c) shows that, use of face-mask, sanitizer and social distancing control (u 1 ) is at its maximum value from the starting point until time t = 200 days before decreasing to the minimum value. Also, treatment of COVID-19 patients, active screening along with testing (u 2 ), and control against recurrence and reinfection of humans who have recovered from COVID-19. (u 3 ) are at their minimum values from the starting point until time t = 200 days before increasing to their maximum values.

Conclusion
Mathematical study on the transmission dynamics of the novel COVID-19 pandemic was discussed in order to provide more insights into disease transmission, explore potential prevention and control strategies to minimize the rising tide of disease transmission in the population. A deterministic mathematical model was formulated by subdividing the human population into susceptible human, exposed human, infected human, quarantine human and recovered human. The theoretical study carried out on the basis of the stability analysis showed that the model has a globally asymptotically stable disease-free equilibrium if the basic number of reproductions of new transmissions of coronaviruses is less than one. The biological interpretation of this findings revealed that: • COVID-19 can be effectively managed or even eliminated if the control measures implemented are capable of taking or sustaining the basic reproductive number R 0 to a value below unity. This is an indication that, regardless of the number of infectious individuals initially introduced into the fully susceptible population. • The spread of COVID-19 in the population can be efficiently managed. In addition, three separate simulation cases were conducted to obtain the best control strategies in a comparative manner. • The results from our analysis revealed that the impacts of three interventions with optimal implementation costs out-perform the reduction of the disease epidemic.
Therefore, the fight against the spread of COVID-19 requires a multifaceted approach. We have not considered other control measures such as personal hygiene and public awareness, this gives space for future research. The findings obtained in this work can be a valuable reference for COVID-19 Local National Control Program and the basis for the preparation and design of the best intervention strategies to eradicate COVID-19.  t)), treatment on COVID-19 patients, active screening and testing (u 2 (t)) and control against recurrence and reinfection of humans who have recovered from COVID-19. (u 3 (t)).