key: cord-0784341-gei2r6aj
authors: Alemneh, H. T.; Telahun, G. T.
title: Mathematical Modeling and Optimal Control Analysis of COVID-19 in Ethiopia
date: 2020-07-24
journal: nan
DOI: 10.1101/2020.07.23.20160473
sha: 424f415b2ae42773e2a1131a8df25cd7d94a2bd4
doc_id: 784341
cord_uid: gei2r6aj

In this paper we developed a deterministic mathematical model of the pandemic COVID-19 transmission in Ethiopia, which allows transmission by exposed humans. We proposed an SEIR model using system of ordinary differential equations. First the major qualitative analysis, like the disease free equilibruim point, endemic equilibruim point, basic reproduction number, stability analysis of equilibrium points and sensitivity analysis was rigorously analysed. Second, we introduced time dependent controls to the basic model and extended to an optimal control model of the disease. We then analysed using Pontryagins Maximum Principle to derive necessary conditions for the optimal control of the pandemic. The numerical simulation indicated that, an integrated strategy effective in controling the epidemic and the gvernment must apply all control strategies in combating COVID-19 at short period of time.

The outbreak was first identified in Wuhan, China, in December, 2019, which has been spreading worldwide [5, 6] . Following the day of the outbreak of the pandemic to present more than seven million globally confirmed cases and half million deaths as well as four million recovered numbers [7] . In Africa also the virus spread to 57 countries [1, 7] . Through out Africa up to Jun 14, 2020 confirmed cases are 235,707, number of deaths are 6,283 and recoveries are 36,850 [8] .

In order to combat this pandemic, different preventive measures are recommended, such as avoiding close contact with sick people, avoiding touching the eyes, nose and mouth with unwashed hands, washing hands often with soap and water for at least 20 seconds, using an alcohol-based hand sanitizer containing at least 60% alcohol when soap and water are not available [1, 3, 4] .

Limit the number of people you have close contact with, or are visited by, to a few at a time. For infected individuals, staying home, covering cough or sneeze with a tissue, then throw the tissue in the trash, and clean and disinfect frequently touched objects are recommended for controlling COVID-19, perform home quarantine for 14 days after the last contact with a patient with confirmed COVID-19 [9, 10] .

Mathematical models with optimal control analysis has become an important tool in order to understand the dynamics of disease transmission and decision making processes regarding intervention programs for the disease control. COVID-19 have been modeled by very few researchs after the outbreak of the disease. The study by [11] , used an SIR model to predict the magnitude of the COVID-19 epidemic in Pakistan and compared the numbers with the reported cases on the national database. They predicted that 90% of the population will have become infected with the virus if policy interventions seeking to curb this infection are not adopted aggressively. The study presented in [12] used SEIR compartments by considering limited parameters, from January 22, 2020 to March 3, 2020 and Prediction SEIR forecasted by using SEIR model They also looked at the feelings, current disease trends and economic and political impacts. The article [13] , proposed conceptual models for the COVID-19 outbreak in Wuhan with the consideration of individual behavioural reaction and governmental actions, e.g., holiday extension, travel restriction, hospitalisation and quarantine. The article published by [12] also proposed SEIR model by incorporating the intrinsic impact of hidden exposed and infectious cases on the entire procedure of epidemic, which is difficult for traditional statistics analysis. The other study is done in China using SIR by considering logistic growth [11] . Using their model, the study approximated future number of cases in china and proposed possible controlling mechanisms. The study [14] simulated the ongoing trajectory of Covid 19 outbreak in Wuhan using an age-structured SEIR model for several physical distancing measures. The article presented by [15] , proposed a compartmental model by dividing the quarantined individuals in to two sub-groups. They developed an SEIRU model where R and U are quarantine-infected individuals expected to recover and meet undetectable criteria. A more detail model is proposed by [16] , and studied a mathematical modeling of the spread of the COVID-19 taking into account the undetected infections, in China. The study in [17] , proposed a new epidemic model in the case of Italy that discriminates between infected individuals depending on whether they have been diagnosed and on the severity of their symptoms. However, all of the above models studied the dynamics of disease transmission, they did not study finding an optimal control strategy using the maximum princeiple of Pontryagn.

From all the above studies we can understand that, COVID-19 have been modeled by considering different situations. In the case of Ethiopia, the proposed model can't reflect the real situation of the country. Currently Ethiopia is implementing different measure to control COVID-19 including implementing state of emergency, counry border closing and mandatory quarantine of new arrivals for 14 days. herefore, in the present paper we extend the SEIR model with the aforementioned control measures and propose an optimal control strategy.

The paper is organized as follows. In Section 2 was devoted to the model description and the underlying assumptions. In Section 3 we carry out mathematical analysis of this COVID-19 model. In Section 4 we propose an optimal control problem and and the optimal control analysis are presented. In Section 5 numerical simulation and calibration of the model was implemented for the various strategies considered in this work. The conclusion was presented in Section 6.

In this model the entire population is divided into four sub-populations: Susceptible individuals (denoted by S) are those who are not infected by the disease pathogen but there is a possibility to be infected. Exposed individuals (denoted by E) are individuals who are in the incubation period after being infected and have no visible clinical signsand these individuals could infect other people with a higher probability than people in the infected compartments. After the incubation period, the person passes to the infected compartment; infected individuals (denoted by I) are individuals who developed the symptom of the disease. Recovered individuals (denoted by R )are those individuals who recovered from the disease.

Individuals are recruitment at a rate π is either through immigration or birth. The susceptible individuals got infection of COVD-19 disease at a contact rate of β either from infected with probability of σ 1 or from exposed individuals with probability of σ 2 and move to the exposed compartment. The exposed individuals become infectious and join the infected compartment at τ δ and the remaining proportion of this exposed individuals develop natural immunity and recovered from the disease. By the treatment given, the infected individuals will recover and move to the recovered compartment at a rate of , or may die due to the disease at a rate of ρ . The recovered individuals become again susceptible to the disease at a rate of η. The whole population have an average death rate of µ . Table 1 shows the description of model parameters. The flow diagram of the model is shown in Figure 1 below.

With regards to the above asumptions, the model is governed by the following system of differential equation:

With the initial condition is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) 

If there is no death due to the disease, we get

After solving equation (3) and evaluating it as t −→ ∞, we got

Which is the feasible solution set for the model (1) and all the solution set is bounded in it. Proof. : From the first equation of the system

. CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160473 doi: medRxiv preprint which can be taken as

After evaluating equation (4), we obtain

Similarly, we obtain

Therefore, all the solution sets are positive for t ≥ 0.

When there is no infectious person of the disease in the population, I.e E = I = 0, the disease free equilibrium occur and is obtained by taking the right side of Eq. (1) equal to zero. Therefore the disease free equilibrium point is given by:

We calculate the basic reproduction number 0 of the system by applying the next generation matrix method as laid out by [18] . The first step is rewrite the model equations, starting with newly infective classes:

Then by the principle of next-generation matrix, the Jacobian matrices at DFE is given by

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Therefore, the basic reproduction number is given us

0 is a threshold parameter that represents the average number of infection caused by one infectious individual when introduced in the susceptible population [18] .

Proof. The Jacobian matrix, evaluated at the disease-free equilibrium E 0 , we get:

The characteristic polynomial from the Jacobian matrix is

Where

From the last expression, that is

We applied Routh-Hurwitz criteria and by the principle Eq.(8) has strictly negative real root iff

Clearly we see that ψ 1 > 0 because it is the sum of positive 7 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. . parameters.

Hence the DFEP is locally asymptotically stable if 0 < 1. (1) is globally asymptotically stable if 0 < 1.

Proof. Consider the following Lyapunov function

Differentiating equation (10) with respect to t gives

Substituting dE dt and dI dt from the model (1), we get:

Here take c 2 = − βσ 2 S−(δ+µ) τ δ c 1 , then we have

Taking c 2 = 1, and substituting 0 we get

for S ≤ S 0 = π µ and dV dt ≤ 0 for 0 < 1 and dV dt = 0 if and only if I = 0. This implies that the only trajectory of the system (1) on which dV dt ≤ 0 is E 0 . Therefore by Lasalle's invariance principle, E 0 is globally asymptotically stable in Ω.

In the presence of disease in the population,(S(t) ≥ 0; E(t) ≥ 0; is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. .

Proof. It can be obtained by equating each equation of the model equal to zero. i.e

Then we obtain

Where

Theorem 3.5. The EEP is locally asymptotically stable if 0 > 1 and unstable if otherwise.

Proof. The Jacobian matrix of system (1) at the DFEP is given by

To determine the local stablity of endemic equilibrium, we used the center manifold theory [19] , by taking β as a bufiracation parameter. let S = z 1 , E = z 2 , I = z 3 and R = z 4 . In addition, using

. Then the value of β as a bifurcation parameter and solve 0 = 1, which leads to

The right eigenvector, w = (w 1 , w 2 , w 3 , w 4 ) T , associated with this simple zero eigenvalue can be 9 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160473 doi: medRxiv preprint obtained as:

, associated with this simple zero eigenvalue is given

Since the first and fourth component of v are zero, we don't need the derivatives of f 1 and f 4 . From the derivatives of f 2 and f 3 , the only ones that are nonzero are the following:

and all the other partial derivatives of are zero. The direction of the bifurcation at 0d = 1 is determined by the signs of the bifurcation coefficients a and b, obtained from the above partial derivatives, given respectively by

Do to the sign of the coefficient a < 0 and b > 0 at β * , by the theorem 4.1 stated in [19] system (11) undergo forward bifurcation at 0 = 1 and the unique endemic equilibruim is locally assymptotically stable for 0 > 1.

We carried out sensitivity analysis, on the basic parameters, to identify their effect to the transmition of the disease. To go through sensitivity analysis, we used the normalized sensitivity index definition as defined in [20] . The Normalized forward sensitivity index of a variable, 0 , that depends differentiably on a parameter, p, is defined as: Λ 0 p = ∂ 0 ∂p × p 0 for p represents all the 10 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160473 doi: medRxiv preprint basic parameters. Here we have 0 = β Π(σ 1 τ δ+σ 2 ( +ρ+µ)) µ ( +ρ+µ) (δ+µ) . For the sensitivity index of 0 to the parameters:

The sensitivity indices of the basic reproductive number with respect to main parameters are found in Table 2 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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Here we extend the basic model in (1) in to optimal control. As we indicated in section 1, Ethiopian goverment have started a lot of activities to combat COVID-19. Some of preventive activities are; forced isolation for 14 days for new arrivals, closing country border, quarantining and supporting infected individuals with medication, restricting number of sits by half in all transport system, convincing all media outlets to campaigning on COVID-19, announcing state of emergency and others. As we observe the started activities are not done in optimal level. Therefore, we want to show the concerned body the effectiveness of those activities if they are implemented in an optimal level. To perform optimal control, we categorize those activities in to three broad control strategies listed below. After incorporating the three controls in model (1) gives the following optimal control model in equation (15) below.

The purpose of introducing controls in the model is to find the optimal level of the intervention strategy required to reduce the spreads the pandemic in the population. Here we want to find the optimal values u 1 , u 2 and u 3 that minimizes the objective functional subject to the differential equations (15) . The objective functional is given as

where t f is the final time, a 1 and a 2 are weight costs of the Exposed humans and infected humans respectively while w 1 , w 2 and w 3 are weight costs for each individual control measure. In this paper, a quadratic function which satisfies the optimality conditions is considered for measuring 12 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. . the cost of the controls as applied by [21, 22, 23] . The goal is to find the optimal control (u * 1 ,u * 2 , u * 3 ) such that.

We used Pontryangin's Maximum Principle [24] to drive the necessary conditions that an optimal control must satisfy. This principle converts equation (15) and (16) into a problem of minimizing point-wise Hamiltonian (H), with respect to u 1 (t), u 2 (t) and u 3 (t) as:

Where λ i , i = 1, 2, 3, 4 are the adjoint variable associated with S, E, I, and R to be determined suitably by applying Pontryagin's maximal principle [24] and also using [25], the existence of an optimal control is guaranteed.

Theorem 4.1. For an optimal control set u 1 , u 2 , u 3 that minimizes J over U, there are adjoint variables, λ 1 , ..., λ 4 such that:

With transversality conditions, λ i (t f ) = 0, i = 1, ..., 4. Furthermore, we obtain the control set

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Proof. The adjoint equation and transversality conditions are standard results from Pontryagin's maximum principle [24] . We differentiate Hamiltonian with respect to states S, E, I and R respectively, and then the adjoint system is written as

With transversality conditions, λ i (t f ) = 0, i = 1, ..., 4. Similarly by following the approach of Pontryagin et al. [24] , the characterization of optimal controls u * 1 , u * 2 , u * 3 , that is, the optimality equations are obtained based on the conditions: ∂H ∂u i , f or i = 1, .., 3, which gives,

Setting ∂H ∂u i = 0 at u * i , the results are

When we write by using standard control arguments involving the bounds on the controls, we

In compact notation

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The optimality system is formed from the optimal control system (the state system) and the adjoint variable system by incorporating the characterized control set and initial and transversal 16 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07. 23.20160473 doi: medRxiv preprint In this section, we examine the COVID 19 model and studied the effects of combined strategies on controlling the transmission of the disease. The optimal control set was obtained by solving the optimality system, consisting of the state and adjoint systems. An iterative scheme was used for solving the optimality system. We start to solve the state equations with an initial guess for the controls over the simulated time using the forward fourth order Runge-Kutta scheme. Be- Table 3 . We investigated numerically the effect of the following optimal control strategies on the spread of COVID 19 in a population. Considering strategies that implement one intervention only is not guaranteed to reduce and/or eradicate the disease totally from the community. So that those strategies which incorporate more than one intervention are ordered below and compared pairwise: 17 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. In addition to those parameter values in Table 3 , we used a 1 = 1, a 2 = 5, w 1 = 40, w 2 = 150 and w 3 = 75 for simulation of COVID 19 pandemic disease model.

Here we have investigated the impact of optimal prevention methods and supporting infected individuals in quarantine center with medication treatment. From the simulation results of Figure 2 , we see that the combination of the two method is effective in controlling COVID-19 in the specified period. Moreover, we can see that after implementing this strategy the number of exposed and infected human population goes to zero after 15 months. 

Here also we experimented by combining preventive measure and awareness creation through media campaign. From the numerical result depicted in Figure 3 below (the left hand side) show as it is possible to combat the exposed human population of COVID-19 to zero after 8 months and it will also start to relapse again after 10 months. The right hand side of Figure 3 also indicate that as it is possible to minimize infected-COVID-19 humans but unable to eliminate by this strategy in the stated time. Also this method is not effective once the infectious individuals are entered in the country.

18 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160473 doi: medRxiv preprint 

Here we experimented the model considering optimal support of infected individuals in quarantine center and creating awareness through media. Any activities from preventive technique of the disease is not considered here. From the numerical result depicted in Figure 4 shows that as COVID-19 will relapse for second time after it goes to zero in the specified time. 

Here we have applied allstrategies to optimlize the objective function. The results from Figure 5 shows that bringing down the exposed and infected population in short period of time and it is best 19 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160473 doi: medRxiv preprint compared to all the above three control technique. Therefore, government decision makers and all stakeholders consider in applying all strategies to combat COVID-19 in the specified time. 

In this paper an SEIR deterministic model for the transmission dynamics of the pandemic COVID-19 in the case of Ethipia was formulated. Model analysis demonstrates that its solutions are positive and bounded, and there is a region where the model is well-posed mathematically and epidemiologically meaningful. The basic reproduction number 0 was computed and the stability of equilibria points was investigated. Through Lyapunov's theory, the disease free equilibrium point globally asymptotically stable whenever the 0 < 1 was proven. Using center mainfold theory, bifurcation analysis of the model was proven and the model exhabts forward bifurication at 0 = 1.

Second, by adding three times-dependent controls, we extend the basic model in to an optimal control. By using Pontryagin's Maximum Principle necessary conditions for the optimal control of the transmission of COVID-19 were derived. From the numerical simulation it was found that the integrated control strategy, strategy D: using all technique, is very efficient in short period of time.

Therefore, applying all rounded technique by the EFDRE , decision makers and stakeholders have a significant contribution in combating this pandemic in very short period of time. The result also shows, if the stakeholders could not do all of the intervention strategies together, the pandemic may come agian and will transmit over the country. This paper is still an ongoing research as many more investigations regarding his disease can be 20 . CC-BY-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160473 doi: medRxiv preprint carried out. Yet, it serves as the starting phase to research more in depth on questions that COVID-19 is spreading with incredible speed and have severe consequences. Moreover, to identify those exposed individuals who doesn't develop clinical sign, mass screening in any cost is recommended to combat COVID-19.

All data relevant to this publication will be provided when needed.

Authors have declared that no competing interests exist.

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All authors contributed equally.