key: cord-1032664-eydfe6n5
authors: Kumar Das, Dhiraj; Khatua, Anupam; Kar, T.K.; Jana, Soovoojeet
title: The effectiveness of contact tracing in mitigating COVID-19 outbreak: A model-based analysis in the context of India
date: 2021-03-19
journal: Appl Math Comput
DOI: 10.1016/j.amc.2021.126207
sha: e6484f178e4e2f27ef51a6a9d6a06cad64fb2f7b
doc_id: 1032664
cord_uid: eydfe6n5

The ongoing pandemic situation due to COVID-19 originated from the Wuhan city, China affects the world in an unprecedented scale. Unavailability of totally effective vaccination and proper treatment regimen forces to employ a non-pharmaceutical way of disease mitigation. The world is in desperate demand of useful control intervention to combat the deadly virus. This manuscript introduces a new mathematical model that addresses two different diagnosis efforts and isolation of confirmed cases. The basic reproductive number, [Formula: see text] is inspected, and the model’s dynamical characteristics are also studied. We found that with the condition [Formula: see text] the disease can be eliminated from the system. Further, we fit our proposed model system with cumulative confirmed cases of six Indian states, namely, Maharashtra, Tamil Nadu, Andhra Pradesh, Karnataka, Delhi and West Bengal. Sensitivity analysis carried out to scale the impact of different parameters in determining the size of the epidemic threshold of [Formula: see text]. It reveals that unidentified symptomatic cases result in an underestimation of [Formula: see text] whereas, diagnosis based on new contact made by confirmed cases can gradually reduce the size of [Formula: see text] and hence helps to mitigate the ongoing disease. An optimal control problem is framed using a control variable [Formula: see text] projecting the effectiveness of diagnosis based on traced contacts made by a confirmed COVID patient. It is noticed that optimal contact tracing effort reduces [Formula: see text] effectively over time.

• An optimal control problem is framed to measure the efficacy of contact tracing.

• The model is calibrated with the data from the six most affected states of India.

• A short-term prediction is given with possible variation in contact tracing level. introduced to represent the effect of these policies with an associated rate 148 p at which it is implicated [11] . Hence, the total rate at which susceptible 149 6 becomes immune to the disease due to the government policy is pLS, and they directly move to the recovered class. Therefore, the rate of change in 151 susceptible class is monitoring this sub-population is given by

The above framework of each sub-classes gives the following dynamical 184 system describing the transmission of COVID-19 in human.

(1)

The initial history are given by Proof. From the first equation of system (1), we obtain

Now, if we assume that

then from the second equation of the system (1), we have

Again, by similar argument one can show that from the third equation of the 206 system (1), whenever

then we have

Finally, from the fourth and fifth equation yields

respectively. Therefore, combining conditions (2) and (3) we conclude that 208 all the solution trajectories remain positive as t > 0 whenever

Hence the theorem follows. 

A biological population must be confined in a region due to its natural 220 constraints. The next theorem establishes this requirement of the model Proof. Let us define N (t) = S(t) + I p (t) + I s (t) + H(t) + R(t). Then adding all the equations of the model system (1), we obtain

Now, by applying the differential inequality due to Birkoff and Rota [30], we obtain

Thus as t → ∞, we have 0 ≤ limsup t→∞ N (t) ≤ Π µ . This implies that all solutions of the system (1) initiating from R 5 + are uniformly bounded in the region

for some > 0. Hence the theorem follows. The disease free equilibrium point of the system (1) can be obtained by putting I p (t) = 0, I s (t) = 0 and H(t) = 0 and finally takes the form

.

. The basic reproduction number is a central concept in the study of the For this, we identify the compartments which are infected from the system (1) as (I p , I s , H) and decomposing the right hand side as F −V, where F is the transmission part denoting the addition of new infection, and the transition part is V, which indicates the change in state. Let us consider the system. Following this technique we re-write the system (1) as

Now, we calculate the Jacobian of F and V at the disease free steady state E 0 as

Now according to the [34] , the spectral radius (ρ), i.e., the maximum eigenvalue of the matrix F V −1 gives the basic reproduction number R 0 of the system (1). After some algebraic simplifications, we obtain

The model system (1) always exhibits a disease free equilibrium E 0 = ( Π µ+pL , 0, 0, 0, ΠpL µ(µ+pL) ) and an endemic equilibrium point

It is to be noted that the infected steady state of the model system (1) is 237 feasible provided Π > (µ + pL)S * and γ + d + µ > ξ 1 + ξ 2 . Now after some 238 algebraic simplifications, we obtain that the condition Π > (µ + pL)S * is 239 equivalent to R 0 > 1. Hence we conclude that the model system (1) has a 240 unique endemic steady state E * = (S * , I * p , I * s , H * , R * ) when γ +d+µ > ξ 1 +ξ 2 241 and R 0 > 1.

Now using Theorem 2 in [34] , one can establish the following theorem.

Theorem 3.3. The disease-free equilibrium of the model system (1) is locally 245 asymptotically stable if R 0 < 1, and unstable if R 0 > 1. In order to study the local asymptotic stability of the infected steady state E * = (S * , I * p , I * s , H * , R * ), we calculate the Jacobian matrix of system

(1), as given by

where m = β(I * s + kI * p ), and n 1 , n 2 , n 3 are as defined earlier. Now it is easy 248 to observe that one eigenvalue of the above matrix J E * is −µ. The remaining 249 roots are obtained from the following equation

and P 4 = mσβn 3 S * + mβKS * (n 2 n 3 + ξ 2 η) + σ(ξ 1 η − βS * n 3 )(m + µ + pL) − (m + µ + pL)(n 2 n 3 + ξ 2 η)(βkS * − n 1 ). Now, by applying the well known Routh-Hurwitz criteria, we obtain a set 251 of necessary and sufficient condition for local asymptotic stability of the 252 endemic steady state E * , which are: P i > 0 for i = 1, 2, 3, 4, P 1 P 2 > P 3 and 253 P 1 P 2 P 3 > P 2 3 + P 2 1 P 4 .

Based on the above discussions, we may conclude the following result.

Theorem 3.4. The endemic equilibrium E * is locally asymptotically stable 256 for R 0 > 1, P i > 0 (i = 1, 2, 3, 4), P 1 P 2 > P 3 and P 1 P 2 P 3 > P 2 3 + P 2 1 P 4 . shall follow the approach taken in [38, 39] . After introducing this intervention effort in our model system, the model takes the following form:

The objective of this effort u(t) is to minimize total disease burden (both 

Clearly, the above system (8) [41] as the following theorem.

Theorem 4.1. There is an optimal control u * (t) such that J (I p (t), I s (t), u * (t)) = 289 min u J (I p (t), I s (t), u(t)) subject to the system of differential equations (6) and 

where, Y = [S,

Therefore, using the boundedness of the solutions of (8), we have

where, C 1 is a constant depending upon the coefficients of the system (6).

The above inequality justifies the condition 3.

Next, in order to prove that the integrand of the objective functional (7) i.e. Φ(t, Y, u) = I p + I s + cu 2 is convex in the control set Θ, we have to show

where u and v are two controls and q ∈ (0, 1). Now,

Finally,

Hence the objective function Φ(t, Y, u) is convex in the control set Θ.

To verify the last condition, note that,

where A depends upon the upper bound of I p (t) and I s (t), B = c and β > 1. 322 Hence the existence of the optimal control is proved. Now we need to find out the value of the optimal control u * (t) such that

In the objective functional, the weight constants corresponding to the pre- 

The Hamiltonian is defined as follows:

H 1 (S, I p , I s , H, R, u, λ 1 , λ 2 , λ 3 , λ 4 , λ 5 )

where the adjoint variables can be obtained as the solution of the following 333 system of differential equations

satisfying λ i (T ) = 0, for i = 1, 2, 3, 4, 5, i.e. the transversality conditions. 

Theorem 4.2. The optimal value of the control variable which minimizes the objective functional J in the region Θ is given as u

Proof. The optimal value u * of the control variable u(t) is obtained by using Table 1 with short description and the state-wise es-382 timated parameters sizes can be found in the Table 2 . The best fit initial 383 size are also extracted using the same technique and displayed in the Table 3 .

The best-fit curves are plotted with the cumulative weekly observed con-386 firmed cases under treatment in hospitals and displayed in Figure 4 . Further, 387 using these parameter magnitudes we estimated the epidemic threshold quan-388 tity, that is, the basic reproduction number R 0 for all six different states (see 389 the Table 4 ).

The accuracy of the employed estimation approach is quantified by two 

where, θ(i) denotes the data reported on ith day and Y (i) is the model predic- to a parameter say, η is defined as the following:

Clearly, the above definition is well-defined for any positive parametric indicates that if k is increased by 10% then R 0 is also increased by 13.292% 414 and Γ R 0 p = −0.9999 shows that decreasing p by 10% will reduce R 0 by 9.9%. It is to be observed from Table 6 that contact tracing rates both in (c) Cumulative weekly confirmed cases prediction for the state West Bengal. 

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