key: cord-0999544-gfqp659j
authors: Nabi, Khondoker Nazmoon; Kumar, Pushpendra; Erturk, Vedat Suat
title: Projections and fractional dynamics of COVID-19 with optimal control strategies
date: 2021-01-28
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2021.110689
sha: 201e28f8d42603c7b222d0e7f176e4b9c1a917fa
doc_id: 999544
cord_uid: gfqp659j

When the entire world is eagerly waiting for a safe, effective and widely available COVID-19 vaccine, unprecedented spikes of new cases are evident in numerous countries. To gain a deeper understanding about the future dynamics of COVID-19, a compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention strategies. Model parameters have been calibrated using sophisticated trust-region-reflective algorithm and short-term projection results have been illustrated for Bangladesh and India. Control reproduction numbers [Formula: see text] have been calculated in order to get insights about the current epidemic scenario in the above-mentioned countries. Forecasting results depict that the aforesaid countries are having downward trends in daily COVID-19 cases. Nevertheless, as the pandemic is not over in any country, it is highly recommended to use efficacious face coverings and maintain strict physical distancing in public gatherings. All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives. In addition, optimal control strategies for fractional system have been designed and the existence of unique solution has also been showed using Picard-Lindelof technique. Finally, unconditional stability of the fractional numerical technique has been proved.

• A compartmental mathematical model has been proposed in this paper incorporating all possible non-pharmaceutical intervention policies

• Short-term projection results have been illustrated for Bangladesh and India

• All necessary graphical simulations have been performed with the help of Caputo-Fabrizio fractional derivatives

• Optimal control problem for fractional system has been designed and the existence of unique solution has also been showed by using Picard-Lindelof technique 1 . Introduction 1 Mass-vaccination campaigns have already been launched in several countries amid coronavirus surges. 2 However, many scientists have expressed their concerns regarding the efficacy of approved vaccines potential 3 emergence of virus variants. As worldwide distribution of COVID-19 vaccines is indeed a tedious process, 4 non-pharmaceutical intervention strategies are the realistic and effective solutions to control the new spikes 5 technique. We used the numerical data of the two given countries and perform the all necessary graphical 40 simulations.

The entire chapter is organized as follows. Materials and methods are presented in Section 2. Section 42 3 is solely devoted to properties of solutions and asymptotic stability of the proposed model. In section 43 4, estimation of model parameters and projection results have been discussed using daily COVID-19 data 44 of Bangladesh and India. In section 5, numerical and graphical simulations have been illustrated using 45 Caputo-Fabrizio fractional derivatives. Later, optimal control problem has been designed in fractional sense 46 in section 6. The chapter ends with some insightful findings and strategies, which could significantly control 47 the transmission dynamics of COVID-19. tion status, the entire human population (denoted by N (t) at time t) has been stratified into nine mutuallyexclusive compartments of susceptible individuals (S(t)), early-exposed individuals (E 1 (t)), pre-symptomatic individuals (E 2 (t)), symptomatically-infectious (I(t)), asymptomatically-infectious or infectious individuals with mild-symptoms (A(t)), quarantined infectious (Q(t)), hospitalised or isolated individuals (L(t)), recovered individuals (R(t)), disease-induced death cases (D(t)). Hence, N (t) = S(t) + E 1 (t) + E 2 (t) + I(t) + A(t) + Q(t) + L(t) + R(t) + D(t)

The schematic diagram of the proposed model is illustrated in Figure 1 , where susceptible individuals can become infected by an effective contact with individuals in the pre-symptomatic (E 2 (t)), symptomaticallyinfectious (I(t)), asymptomatically-infectious (A(t)), quarantined-infectious (Q(t)) and isolated-infectious (L(t)). Effective contact rates are λ E2 , λ I , λ A , λ Q , and λ L respectively and the expressions are defined in (2) . Importantly, the compartment E 1 (t) consists of early-infected individuals who are still not infectious, whereas the individuals in pre-symptomatic cohort E 2 (t) have the capability of transmitting coronavirus before the end of the disease incubation period. A proportion of individuals in newly-exposed compartment (E 1 (t)) progress to pre-symptomatic class (E 2 (t)) at a rate κ 1 . After the completion of disease mean incubation period, at a rate ρκ 2 , a fraction of individuals who have clear clinical symptoms of COVID-19 progress to I(t) compartment. Individuals in E 2 (t) class who do not have any clear symptoms progress to A(t) class at a rate (1 − ρ)κ 2 . Pre-symptomatic individuals are assumed to be self-quarantined at a rate q.

With the help of diagnostic or surveillance testing approaches, symptomatically-infectious individuals and asymptomatically-infectious individuals are brought under institutional or home isolation at rates τ A and τ I respectively. Moreover, the parameter γ I (γ A )(γ Q )(γ L ) represents the recovery rate for individuals in the I(A)(Q)(L) class. Finally, the disease-induced mortality rate for individuals in the I(Q)(L) compartment is defined by the parameter δ I (δ Q )(δ L ). Considering all the above-mentioned interactions, the transmission dynamics of COVID-19 can be described by the following system of nonlinear ordinary differential equations.

where the forces of infection are defined below

The parameters are described in Table 1 .

Parameter Description

Effective contact rate (a measure of physical or social distancing) m Proportion of individuals who use face coverings or surgical masks ζ Efficacy of face coverings at reducing outward transmission by infected individuals as well as preventing acquisition κ 1 Rate of progression from early-exposed class (E 1 (t)) to pre-symptomatic class (E 2 (t)) ρκ 2 Rate of progression from pre-symptomatic class (E 2 (t)) to symptomatically-infectious class (I(t))

(1 − ρ)κ 2 Rate of progression from pre-symptomatic class (E 2 (t)) to asymptomatically-infectious class (A(t)) 

) is a solution of (1) with positive initial conditions. Let us consider E 2 (t) for t ≥ 0. It follows from the third equation of system (1) that

Since E 2 (0) ≥ 0, it follows that E 2 (t) ≥ 0 for t ≥ 0. We proceed with the same for S(t), E 1 (t), I(t), A(t), 68 Q(t), L(t), R(t) and D(t). 

where N 0 is equal to the total population. It follows that for all t ≥ 0, we have

In what follows, we study the model (1) in the following set D = (S, E 1 , E 2 , I, A, Q, L, R, D) ∈ R 9 + : S + E 1 + E 2 + I + A + Q + L + R + D ≤ N 0 which is positively-invariant and attracting region for the model (1). The disease-free equilibrium point denoted by x 0 can be defined as follows:

x 0 = (S 0 , 0, 0, 0, 0, 0, 0, 0, 0) = (N 0 , 0, 0, 0, 0, 0, 0, 0, 0)

Using notations in [24], matrices F and V for the new infection terms and the remaining transfer terms are, respectively, given by

Then, the control reproduction ratio is defined, following [25, 24], as the spectral radius of the next generation matrix, F V −1 :

where,

where ρ(·) represents the spectral radius operator.

The formula for control reproduction number has been formulated. Indeed, the insightful epidemic 19, and the disease-equilibrium E 0 is unstable.

Lemma 3. If R c < 1, the disease-free equilibrium x 0 is locally asymptotically stable and unstable if R c > 1.

[5] 96 Remark 2. Lemma 3 implies that if R c < 1, then a sufficiently small flow of infected individuals will 97 not generate an outbreak of COVID-19, whereas for R c > 1, epidemic curve reaches a peak by growing 98 exponentially and then decreases to zero as t → ∞.

The better control of the COVID-19 can be established by the fact that the DFE x 0 is globally asymp-100 totically stable (GAS). In this context, we claim the following result.

Theorem 3. if R c < 1, then the manifold, W, of disease-free equilibrium points of the model (1) is GAS in 102 D.

In the absence of use of face coverings, i.e. m = 0, R c converges to the basic reproduction number, R 0 . Now, we will study the global stability of the disease-free equilibrium whenever the basic reproduction number is less than one (R c < 1). For this, we use the following Lyapunov function

By deriving this function along the trajectories of the system (1), we obtaiṅ

We choose a i , i = 1, 2, ..., 6, such that coefficients of E 1 , I, A, Q, and L become zero. That is

which the non-zero solution is given by

τI +δI +γI

Plugging (9) into (7) giveṡ

Setting a 1 = 1 k2+q , we finally obtaiṅ

0. Therefore, L is a Lyapunov function for system (1) . Moreover, the maximal invariant set contained Theorem 4. If R c ≤ 1, then the disease-free equilibrium E 0 is globally asymptotically stable on Ω. data from January 30, 2020 to January 05, 2021 have been considered to calibrate the model parameters. 

[32] The Caputo definition of non-integer order derivative of order > 0 of a function

where n = [ ] + 1 and [ ] is the integer part of . 

The CF non-integer order integral is defined as

Now, we prove the existence of unique solution for the given COVID-19 model in the sense of Caputo-Fabrizio fractional derivative by the application of fixed-point theory. In this concern, the proposed system can be rewritten in the equivalent form as follows:

By applying the CF non-integer order integral operator, the above system (13), reduces to the following integral equation of Volterra type of order 0 < < 1.

Now, we get the subsequent iterative algorithm

Here we assume that we can get the exact solution by taking the limit as n tends to infinity. Let us consider

G 4 (t, I) ,

where

considering the Picard operator as and ∆(t, ζ(t)) = {G 1 (t, S(t)), G 2 (t, E 1 (t)), G 3 (t, E 2 (t)), G 4 (t, I(t)), G 5 (t, A(t)), G 6 (t, Q(t)), G 7 (t, L(t)), G 8 (t, R(t)), G 9 (t, D(t))}. Next we assume that the solution of the non-integer order model are bounded within a time period, ζ(t) ∞ ≤ max{z 1 , z 2 , z 3 , z 4 , z 5 , z 6 , z 7 , z 8 , z 9 , },

where we demand that b < d F . Now by the application of fixed point theorem pertaining to Banach space along with the metric, we obtain

with β < 1. Since ζ is a contraction, we have β < 1, hence the given operator φ is also a contraction.

158 Therefore, the model involving C-F derivative given in Eq. (13) has a unique set of solution. 

We now derive the solution method for I(t) equation of the system (13) and for the rest of the equations it will be similar. The corresponding Volterra integral equation for I(t) is as follows.

We have the following estimations at t k

and at t k+1

Subtracting equation (21) from (20), we obtain

Then by linear interpolation about G 4 (t, I(t)) and applying trapezoid rule for integration on the integral term, we can then write

where ∆t = t k − t k−1 . Hence, we have the numerical approximation for equation of I(t) as

Theorem 5. The numerical approximation (24) is unconditionally stable if G 4 (t k+1 , I(t k+1 )) − G 4 (t k , I(t k )) → 0.

Proof. Let I(t) be the solution of a differential equation as shown in (19) under CF non-integer order derivative operator sense. Then we have to evaluate the norm

For k → ∞, we have

Clearly, the second term of the above inequality approaches zero when k → ∞. Now, if ||G 4 (t k+1 , I(t k+1 ))− 161 G 4 (t k , I(t k ))||→ 0 as k → ∞, we educe that the numerical solution is stable. Table   181 3. In the family of Figure 13 , we analysed the plots of S(t), E 1 (t), E 2 (t), I(t), A(t) and Q(t). We observed 182 that the nature of peaks is mostly same as for other above analysed data, for different fractional order In this concern, our main aim is to decrease the number of infected individuals with COVID-19 at the same time decrease the cost J(v) associated with their strategies. For this purpose, we use a control 

To define the optimal control problem (OCP), we are excluding the death equation D(t), because there 195 is no significance of deaths in optimal controls. Now consider the state system given in (26) in R 8 , 196 with the set of admissible control function.

where the constants k 1 , k 2 and k 3 are a measure of associative cost with the controls v 1 , v 2 and v 3 . Then 199 we find the optimal controls v 1 , v 2 and v 3 to minimize the cost function.

subject to constraint, Now let us take the following modified cost functioñ

where i = 1, 2, 3 and j = 1, 2, 3...8 Hence the Hamiltonian is defined as follows:

where i = 1, 2, 3 and j = 1, 2, 3...8 from Equation 29 and 30, the necessary and sufficient conditions for 206 the functional optimal control problem (FOCP) are given as:

Moreover, θ j (T ) = 0, j = 1, 2, ...8, are the lagseuges multipliers Eqn 31 and 32 express the necessary 208 condition in terms of a Hamiltonian for the OCP defined above. Let us write the Hamiltonian function as follows:

where θ j , j = 1, 2, ...8, representing the lagragars multipliers called co-states.

Theorem:

If v * 1 , v * 2 and v * 3 are optimal controls of the given OCP if S * , E * 1 , ...R * are corresponding optimal paths, then there exists co-state variables θ * 1 , ..., θ * 8 , such that besides the given control system is satisfied, the following conditions are satisfied:

Co-state equations:

D θ * 8 are obtained from the Hamiltonian H as 

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