key: cord-320262-9zxgaprl
authors: Asamoah, Joshua Kiddy K.; Owusu, M.A.; Jin, Zhen; Oduro, F.T.; Abidemi, Afeez; Gyasi, Esther Opoku
title: Global stability and cost-effectiveness analysis of COVID-19 considering the impact of the environment:using data from Ghana
date: 2020-07-10
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110103
sha: 
doc_id: 320262
cord_uid: 9zxgaprl

COVID-19 potentially threatens the lives and livelihood of people all over the world. The disease is presently a major health concern in Ghana and the rest of the world. Although, human to human transmission dynamics has been established, not much research is done on the dynamics of the virus in the environment and the role human play by releasing the virus into the environment. Therefore, investigating the human-environment-human by use of mathematical analysis and optimal control theory is relatively necessary. The dynamics of COVID-19 for this study is segregated into compartments as: Susceptible (S), Exposed (E), Asymptomatic (A), symptomatic (I), Recovered (R) and the Virus in the environment/surfaces (V). The basic reproduction number [Formula: see text] without controls is computed. The application of Lyapunov’s function is used to analyse the global stability of the proposed model. We fit the model to real data from Ghana in the time window 12th March 2020 to 7th May 2020, with the aid of python programming language using the least-squares method. The average basic reproduction number without controls, [Formula: see text] is approximately 2.68. An optimal control is formulated based on the sensitivity analysis. Numerical simulation of the model is also done to verify the analytic results. The admissible control set such as: effective testing and quarantine when boarders are opened, the usage of masks and face shields through media education, cleaning of surfaces with home-based detergents, practising proper cough etiquette and fumigating commercial areas; health centers is simulated in MATLAB. We used forward-backward sweep Runge-Kutta scheme which gave interesting results in the main text, for example, the cost-effectiveness analysis shows that, Strategy 4 (cleaning of surfaces with home-based detergents) is the most cost-effective strategy among all the six control intervention strategies under consideration.

that other optimal control model on COVID-19 have been studied (see for example [27, 28, 29 , 30, 31, The model further assumes that, no exposed individual transmits the disease.

The proportion of those in E class into both A and I classes is given as k 2 (1 − γ)E and k 1 γE respectively. 68 Similarly, individuals recovering from the A class is v 2 (1 − φ)A and those joining the I class from the A 69 compartment is v 1 φA. Though, not much had been said with regards to the possibility of the recovered 70 individuals joining the susceptible population. All the same we included it so to ascertain the impact short 71 and long-term immunity on the dynamics of COVID-19, denoted as ρ, where ρ ≥ 0. Table 1 Progression rate from exposed to the symptomatic (severely infected) class k 2 Progression rate from exposed to the asymptomatic class 

given that S ≥ 0, E ≥ 0, A ≥ 0, I ≥ 0, R ≥ 0 and V ≥ 0. 

We simplify equation (1) to get the total differential equation as;

where N = S + E + A + I + R. non-negative such that the initial conditions are also given as

The proof of Theorem 1 can be obtained using the procedures presented in [33] , as shown below.

Proof. We let Y = (S, E, A, I, R, V ) T and K 0 = βI, K 1 = βA, K 2 = β 1 V where T is transposition, then our differential equation (1) can be rewritten in a matrix form as dY

Now, using the third equation in model (1), thus

and deploying the method of integration factor and change of variables [33] , yields

t 0 E(s)e −(ω+v1φ+v2(1−φ))s ds .

Next, we consider the fourth, fifth, sixth equation of model (1) and using the same above process, we have written as

where m = (m 1 + m 2 ).

Proof. From the first equation of (2) we have

Now, take M 2 to be a solution which is unique to the initial value problem

Which when solved gives

Hence, by the comparison principle (see for instant Theorem 5 of [38]), it accompanies that

Also from the second equation of (2), we let m = (m 1 + m 2 ) with the assumption that 0 < A + I ≤ Λ ω . Then,

Now, let M 3 to be a solution which is unique to the initial value problem

Which when solved gives

and, by the comparison principle (see for instant Theorem 5 of [38]), it accompanies that

From (5) and (8) Here, we first focus on equilibrium points when there is no disease in the system. Considering equation

(1), we put E = A = I = R = V = 0. This indicates that, there is no disease in the system at this stage.

Therefore, solving for the stationary points, we have E = (S * , 0, 0, 0, 0, 0) where S * = Λ ω . For the R 0 , we use the concept of the next generation approach. Here, we seek to find the average number of new infections given that an infected individual is introduced into the population under study [39] . We let G be the next generation matrix which consists of

is the rate at which a new infection occurs in compartment i. Also, v + i and v − i are the rate of immigration into compartment i and the rate at which new individuals are transferred from compartment i respectively.

We note that, all the functions are continuously differentiable at least twice [39] .

Now the next generation matrix is defined as;

Hence, the R 0 is given as the maximum absolute eigenvalue of the next generation matrix (G) given that, σ contains all the eigenvalues of G. This eigenvalue is known as the spectral radius (ρ). This is represented as

Among the infected classes (E, A, I, V), we have f i as

Finding the Jacobian of the matrix f i gives

Considering the same compartments (E, A, I, V), we get the matrix V as

Finding the Jacobian matrix of V i gives

After computing for the eigenvalues of the matrix G, we have that the maximum absolute eigenvalue, R 0 is given as;

secondary infection seeded by I state through direct contact

secondary infection seeded by I state through the environment (9) which can be written as

where R A is the secondary infections generated by asymptomatic persons through direct contact; R I is the secondary infections generated by symptomatic persons through direct contact; R Ae is the secondary infection seeded by asymptomatic persons through the environment; and R Ie is the secondary infection seeded by symptomatic persons through the environment. We can also express R 0 in terms of (T 1 , C 1 , Q 1 ,

where 

Now, expressing the other state variables in terms of E, it implies that

We now substitute A, I and V into second equation (12) and factorizing E out gives

From the initial hypothesis, E = 0 and this implies that,

Making S * in equation (14) the subject gives

Now, adding first equation and second equation of (12), we substitute S * from equation (15) and simplify E * we get

where

But, we know that the basic reproduction number is expressed as

Therefore, we express the endemic equilibrium points, (S * , E * , A * , I * , R * , V * ) in terms of R 0 as; 

Based on the preliminary notes above, we now state the Lyapunov stability theorem.

Theorem 3 (Lyapunov Stability Theorem). The equilibrium, y * is globally stable if the function, L(y) is radially unbounded and positive definite globally such that it has globally negative time derivative, L(y) < 0 ∀y = y * .

We say that; the function L(y) is a Lyapunov function if it satisfies the above theorem, the proof can be 109 found in [41] .

Another important theorem which also plays a key role here is the Kransovkii-LaSalle theorem. This is an extension of Lyapunov function. In summary, this theorem puts forward that; considering an autonomous system, y = f (y) which has equilibrium, y * and that f (y * ) = 0, we assume there is a continuously differentiable positive definite and radially unbounded function L : R n → R which meets the conditionL(y) ≤ 0 ∀ t, y ∈ R n . We then define the invariant set as Proof. We employ the approach in [43] to analyze both the stability at disease free and endemic equilibrium.

We define a Lyapunov, L for the disease-free equilibrium point as follows;

Differentiating L with respect to t gives;

We substituteĖ,Ȧ,İ,V from equation (1) intoL gives;

After further simplification we havė 

Differentiating the function above gives;

Substituting equation (1) into equation (18) with further simplification gives;

Considering the expression

we have that, h 1 = 1. This implies that the coefficients of x 1 x 4 , x 3 x 1 and x 6 x 1 are all 0. Equating the coefficients of x 2 , x 3 , x 4 , x 5 and x 6 to 0 and solving for h 2 , h 3 , h 4 and h 5 gives;

Therefore, T can be rewritten as

It then follows that, T ≤ 0 if x 1 = 1, x 2 = 1, x 3 = 1, x 4 = 1, x 5 = 1 and x 6 = 1. Hence we may conclude that;L

By LaSalle theorem, the invariant set is defined as

Since the invariant set, ζ 1 only contains the endemic equilibrium (S * * , E * * , A * * , I * * , R * * , V * * ), then the 118 endemic equilibrium is said to be globally asymptotically stable under the given region D. In this section, our focus is to verify the validity of the model. This is achieved by fitting and comparing the proposed model with a real data to know its degree of accuracy. are shown in Table 2 and Figure 3 respectively. The blue points in Figure 3 represent the cumulative number In computing for the normalized sensitivity index ( p R0 ) on the R 0 for each of the parameters p, we use the formulae below [41];

Applying the formula above gives the parameters with their sensitivity index in the Table 3 . From Table 3 , asymptomatic class. The model shows that, asymptomatic individuals can join the severely infected class.

However, the estimated parameter value shows that, only few people experience such situation, that is, 0.5%. Considering the system x (t) = f (x(t)) with x(0) = x 0 , such that x 0 ∈ R n where f : R n → R n and

x : [0, ∞) → R n . We introduce variables responsible for the control u i , i = 1, 2, 3...n ∈ N. We then have;

The admissible control set is given as; U * = {u(t) ∈ L (t 0 , t f )|u(t) ∈ A}. The aim 198 here is to target the best control variables, u i , which can efficiently reduce the rate of secondary transmission 199 at a minimum cost of their implementation at any time t (0) ≤ t ≤ t (f ) [41] . That is, we seek to achieve a 200 reduction in the number of individuals in the susceptible, exposed, asymptomatic, severely infected classes 201 and also reduce the content of the virus in the system at a minimum cost simultaneously. To achieve the 202 above objective demands a lot of constructive considerations. For example, implementing total lock down for 203 about two months as a control strategy in this context might highly prove not to be feasible. The loop hole 204 here is; is the country adequately prepared both financially and technically to provide to the satisfaction of 205 its inhabitants the basic needs such as food, water and others throughout the whole period assigned for this 206 measure? It is highly probable the answer might be a big no. It is an undeniable fact that, this measure 207 might prove impractical in controlling the spread in this country. This is why there is a need to objectively 208 sort for more dense restrictive measures with flexible and feasible approaches to be employed in this setting 209 so as to control the disease. We rely on the Pontryagin's maximum principle as applied in [48] for this 210 analysis. We base on the premises above to set below likely control strategies: The objective functional under discussion, Q, which is to be minimized is given as;

subject to the constraints; From equation (4) , we assume that, the weight constant 219 of the exposed, infected (A and I classes) and the virus in the system is 1. Also, to better observe and 220 understand the influence of these control strategies on the model, we assumed that; no recovered individual 221 is vulnerable to be reinfected in equation (4). We accounted for the respective affiliated costs, b 1 u 2 1 , b 2 u 2 2 , 222 b 3 u 2 3 , b 4 u 2 4 , b 5 u 2 5 , which are possible to be incurred during implementation where the square denotes their 223 severity.

It is very necessary to ensure that, the proposed optimal solution exists. For this reason, we employ Filippove-

Cesari theorem as used in [37] . In this case, we show that, the existence of the optimal control solution is 4. The convexity of the integrand of cost functional with respect to u on the set A [37] . 231 We now have the Hessian matrix of the given cost functional as; 

Since the computed Hessian matrix above is everywhere positive definite, it follows that, the objective functional, Q(u 1 , u 2 , u 3 , u 4 , u 5 ) is strictly convex [37] . We also have that,

given that the integrand of the objective functional, 

We now take into accounts the existence of the adjoint function λ i , i = 1, 2, 3...6, such that they satisfy the equations;

with the transversality condition λ i (t f ) = 0 given that ∀u i where i = 1, 2, ..., 5, we have

The optimal control strategies with respect to the befitting variation argument is given as;

We progress with the numerical simulations on the optimal control by using the estimated parameters in 232 Table 2 . dynamics of the subpopulations of exposed, asymptomatic, and symptomatic individuals and the number 254 of virus with control u 2 and without any control implementation is demonstrated by Figure 12 . Figure   255 13- Figure 15 shows the respective dynamics when control u 3 , u 4 and u 5 is used separately. It is revealed 

Here, cost-effectiveness analysis is carried out based on the numerical implementation of the optimality system conducted in Section 4. The cost benefits associated with the implementation of the control strategies can be compared through cost-effectiveness analysis. Thus, following the approach used in several previous studies [53, 54, 55, 56], the incremental cost-effectiveness ratio (ICER) is calculated to determine the most cost-effective strategy of all the different control intervention strategies considered in this work. Most often, ICER is employed to measure up the changes between the costs and the health benefits of any two different control intervention strategies i and j competing for the same limited resources. ICER is defined mathematically as ICER = Difference in costs of control strategies i and j Difference in infections averted by control strategies i and j .

The numerator of ICER in Equation (21) Table 4 . From Table 4 , it is observed that the value of ICER(6) is greater than that of ICER(4). This indicates 285 that Strategy 6 is more costly and less effective than Strategy 4. For this reason, Strategy 6 is excluded from 286 the list of alternative control interventions competing for the same limited resources and ICER is recalculated 287 for Strategies 4 and 3 as illustrated by Table 5 . It can be seen in Table 7 that ICER(1) is greater than ICER(4). This implies that the implementation 296 of Strategy 1 is more costly and less effective than the implementation of Strategy 4. Hence, Strategy 1 is 297 discarded from the list of alternative control intervention strategies competing for the same limited resources.

Now, the ICER is finally recalculated for Strategies 4 and 2 as shown in table 8. Table 8 reveals that ICER(2) is greater than ICER(4). Hence, Strategy 2 is considered to be strongly dom- found to be globally asymptotically stable. We found that, the major transmission parameters β, β 1 , m 1 308 and m 2 contributing to the basic reproduction number of 1.99 − 3.37 were all attributed to humans through 309 personal contact with the susceptible class or activities with the environs. It is further inferred from this 310 study that; applying optimal control strategy on the rate at which the virus is released into the system, m 1 311 and m 2 , and also on the relative transmission rate due to human behaviour will considerably strike down 312 COVID-19 pandemic.

It was also found that, it might be possible the recovered individuals can be reinfected, see Figure 8b . When 314 this happens, then the number of the infected individuals will also increase. Therefore, we highly recommend 315 that, drug manufacturers should aim at drug samples which will induce permanent immunity in the recovered 316 individuals so as to reduce the susceptible population. Cost-effectiveness analysis was carried out based on 317 the numerical implementation of the optimality system conducted in Section 4. This showed that, cleaning of 318 surfaces with home-based detergents is the most cost-effective strategy, followed by: the effective testing and 319 quarantine when boarders are opened, the combination of all the controls then that of intensifying the usage 320 of nose masks and face shields through education. It is highly guaranteed that, this study will help policy The authors declare that they have no known competing financial interests or personal relationships that 328 could have appeared to influence the work reported in this paper. The parameter values (data) used to support the findings of this study have been described in section 3.

Disclosure 338

The authors fully acknowledge that this paper was developed as a result of the first and second author's 339 thesis and project work. 

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