key: cord-291227-dgjieg7t
authors: Mandal, Manotosh; Jana, Soovoojeet; Nandi, Swapan Kumar; Khatua, Anupam; Adak, Sayani; Kar, T.K.
title: A model based study on the dynamics of COVID-19: Prediction and control
date: 2020-05-13
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.109889
sha: 
doc_id: 291227
cord_uid: dgjieg7t

As there is no vaccination and proper medicine for treatment, the recent pandemic caused by COVID-19 has drawn attention to the strategies of quarantine and other governmental measures, like lockdown, media coverage on social isolation, and improvement of public hygiene, etc to control the disease. The mathematical model can help when these intervention measures are the best strategies for disease control as well as how they might affect the disease dynamics. Motivated by this, in this article, we have formulated a mathematical model introducing a quarantine class and governmental intervention measures to mitigate disease transmission. We study a thorough dynamical behavior of the model in terms of the basic reproduction number. Further, we perform the sensitivity analysis of the essential reproduction number and found that reducing the contact of exposed and susceptible humans is the most critical factor in achieving disease control. To lessen the infected individuals as well as to minimize the cost of implementing government control measures, we formulate an optimal control problem, and optimal control is determined. Finally, we forecast a short-term trend of COVID-19 for the three highly affected states, Maharashtra, Delhi, and Tamil Nadu, in India, and it suggests that the first two states need further monitoring of control measures to reduce the contact of exposed and susceptible humans.

more significant compare to quantitative analysis. Hence a suitable math-27 ematical model would not only able to represent the whole disease system 28 but also the study of the model would undoubtedly derive the precise nature 29 of the disease. It may forecast the behavioral aspect of the disease shortly. 30 Although the primitive mathematical models on theoretical epidemiology 31 (see Bernoulli [1] , Hamer[13] , Ross[40] . Kermack According to the information received, it may take around one week to two 41 weeks for the exposure of symptoms of COVID-19 of an infected person, 42 although during this period, that person able to infect other susceptible per-43 sons. However, there may be some infected persons whose infection is so mild 44 that the person would recover due to innate immunity even before the hos-45 pitalization. Thus in this article, by the term 'infected' person, we will mean 46 those persons who are hospitalized. Further, we assume that the medical 47 personals assisting COVID-19 positive hospitalized individuals have taken 48 necessary protective items. Thus to keep simplicity, we believe that only 49 exposed persons and asymptomatic infected persons can spread the disease. Since here, we have assumed that the virus of COVID-19 is spreading when 122 a vulnerable person comes into contact with an exposed person; therefore 123 we think that ρ 1 (0 < ρ 1 < 1) portion of susceptible human would maintain 124 proper precaution measure and ρ 2 (0 < ρ 2 < 1) portion of the exposed class 125 would take proper precaution measure for disease transmission (i.e., use of 126 face mask, social distancing and implementing hygiene). Therefore the dis-127 ease can only be transmitted to the (1−ρ 1 )S portion of susceptible individuals 128 due to the contact of (1−ρ 2 )E portion of exposed individuals with a bi-linear 129 disease transmission rate β. We know that a person is whether infected by 130 the SARS-CoV-2 virus or not can be clinically detected using RT-PCR ex-131 amination and a person with negative results in the RT-PCR test may still 132 be COVID-19 positive as it may take some days (from 7 to 21 days) to ex-133 press infection. Therefore, the portion with positive COVID-19 of the class 134 of population E is considered as infected, and they are hospitalized. Let α 135 and b 2 be the portions of the exposed class goes to the infected class and 136 quarantine class, respectively. It should be noted that 0 < α + b 2 < 1 since 137 it would take quite a long time to get the output of the RT-PCR test, and 138 sometimes it requires more than one RT-PCR analysis for a single person for confirmation of COVID-19. Let among the quarantine classes of populations, 140 cQ portion of communities move to infected level, and the b 1 Q part would 141 become susceptible to the disease after the quarantine period. Let η and σ 142 be respectively recovery rate of the hospitalized infected populations I and 143 exposed class E. Let d be the natural death rate, which is common to all 144 classes of communities and δ be the COVID-19 induced death rate. Also, it 145 is statistically observed a person once recovered from the disease COVID-19 146 has very little chance to become infected again for the same disease. Hence, 147 we assume that no portion of the recovered population moves to the sus- consisting of five first order differential equations shown as below: Proof We assume that P = S + E + Q + I + R.

integrating the above inequality and by applying the theorem of differential equation due to Birkhoff and Rota [2], we get

Now for t → ∞,

Hence all the solutions of (1) that are initiating in {R 5 + } are confined in the region

for any > 0 and for t → ∞. Hence the theorem. interval" see (van den Driessche and Watmough [6] ). Therefore the dimen-195 sionless quantity R 0 refers as the expectation of the spreading disease.

There are several techniques are available for the evaluation of R 0 for an 197 epidemic spread. In our present research article we use the next generation 198 matrix approach [5, 9, 22] . Now the classes which are directly involved for 199 spread of disease is only E, Q, I. Therefore from system (1) we have

(2)

The above system can be written as dy dt = Φ(y) − Ψ(y),

equilibrium. Now the Jacobian matrix of Φ and Ψ at the disease free equilib-204 rium are respectively given by,

The basic reproduction number (R 0 ) is the spectral radius of the of the matrix 207 (F V −1 ) and for the present model it is given by

3.3. Equilibria

The system has two possible equilibria. One is disease free equilibria 210 where infection vanishes from the system. It is given by

where infection is always present in the system is called endemic equilibria,

Note It is observed from the expression of the above two equilibrium point 217 is that the disease free equilibrium E 0 is always feasible but the endemic Theorem 3.2. The disease free equilibrium E 0 is locally asymptotic stable if

Proof. The Jacobian matrix at the disease free equilibrium of the system (1) is given by

Now the characteristic equation of the system (1) at its disease free equilib-227 rium is given by

(4) Clearly all the eigen value of the Jacobian matrix are negative if and only if 229 R 0 < 1. Hence the system is locally asymptotically stable if R 0 < 1 and it is 230 unstable if R 0 > 1. Hence the theorem.

Note Here we see that the disease free equilibrium E 0 losses its stability 232 when the R 0 increases to its value greater than 1. So, we may conclude that 233 at R 0 the system (1) passes through a bifurcation around its disease free 234 equilibrium which are discussed in the next theorem..

Theorem 3.3. The system (1) passes through a transcritical bifurcation 236 around its disease free equilibrium when R 0 = 1.

Proof. From the above analysis, it has been observed that when R 0 < 1 238 between the two equilibria, only the disease free equilibrium exists and lo-239 cally asymptotically stable where as R 0 > 1 is the threshold condition for Proof. The jacobian matrix for the system (1) is given by

It is clear from (5) that first two root are negative real and remaining roots 260 are the roots of the cubic polynomial. It is also observe that here C 1 , C 2 , 261 C 3 and C 1 C 2 − C 3 all are positive for any parametric value. Hence following 262 the Routh-Hurwitz criterion we may conclude that the system (1) is locally 263 asymptotically stable around its endemic equilibrium E 1 . 

subject to the proposed model (1) . The parameters c 1 and c 2 corresponds as the weight constraints for the infected population and the control respectively. Here the objective functional is linear in the control with bounded states. Therefore it can be be showed by using standard results that an optimal control and corresponding optimal states exist [8] . Now we need to find out the value of the optimal control M * (t) such that

Here we use the Pontryagin's Maximum Principle [8, 28, 37 ] to derive the 275 necessary conditions for our optimal control and corresponding states. The

Lagrangian is given by

The Hamiltonian is defined as follows 

We minimize the Hamiltonian with respect to the control variable M * (t). 

Using the equations of the system (2) and (5), we obtain

(13) We observe that the control parameter M does not explicitly occur in the 300 above expression, so next we calculate the second derivative with respect to 301 time.

where

Using the state and co-state equations of systems (1) and (9), we simplify 304 the equation (14) and finally obtain

The above equation can be written in the form

and then we can solve the singular control as

Moreover in order to satisfy the Generalized Legendre-Clebsch Condition for the singular control to be optimal, we require d dM d 2 dt 2 ∂H ∂M = Φ 1 (t) to be negative [25] . Therefore we summarize the control profile on a nontrivial interval in the following way:

Hence the control is optimal provided Φ 1 (t) < 0 and a ≤ − Φ 2 (t) Φ 1 (t) ≤ b.

We study numerical results in two different cases, first for fixed control 308 and second when the control has been applied optimally. First, we consider 309 the values of parameters in Table 1 , for numerical simulations. Since δ is the 310 disease induced mortality rate and d is the natural death rate, hence δ > d.

Using these parameters and the initial conditions as S(0) = 500, E(0) = R 0 with respect to the relative change in its parameter ( Table 1 ) . F (x 1 , x 2 , · · · , x n ) , for the parameter,

To find the sensitivity of R 0 , we consider the parameters A, β, ρ 1 , ρ 2 , α, d, p, M, b 2 , σ 334 as R 0 is the functions of these parameters. The sensitivity index of R 0 with re-335 spect to the parameter β is given by

Similarly, we can find the sensitivity indices of R 0 with respect to the other 337 parameters.

Positive index indicates that R 0 is an increasing function of the corre-339 sponding parameter and negative index implies that R 0 is a decreasing func-340 tion of that parameter. For example, as Γ R 0 β = 1 , it shows that if β is in-341 creased by 10% then the R 0 is also increased by 10%. Again, as Γ R 0 d = −0.409 342 implies that 10% increment in d will decrease R 0 by 4.09%. From Table 2 , increases, which has been demonstrated in Fig. 7 . 353 We know that the numerical value of basic reproduction number R 0 deter-354 mines the exact nature of the disease. From the table 1 and Fig. 5, Fig. 6 , We fit the proposed model (1) to the daily active infected, confirmed 413 (cumulative) infected, and recovered COVID-19 cases in those three states 414 of India using the set of parameters as given in Table 4 and the initial size of 415 the population from the Table 5 . To fit these real data, we use the software 416 Mathematica and then predict the behavior of COVID-19 for those three 417 states on a short term basis. In Fig. 11, Fig. 12, and Fig. 13 , we respectively 418 present the active COVID-19 cases in Maharashtra, Delhi, and Tamil Nadu 419 for 91 days starting from 2nd March, 2020, till the 31st May 2020. Also, in 420 Fig. 14, Fig. 15 and in Fig. 16 , we present the cumulative confirmed (i.e., the 421 sum of active cases, recovered and death) COVID-19 cases of Maharashtra, 422 Delhi, and Tamil Nadu, respectively, for the same period. 2  19  48  0  0  48  14  0  0  14  3  0  0  3  20  52  0  0  52  14  0  0  14  3  0  0  3  21  64  0  0  64  18  0  0  18  6  0  0  6  22  74  0  0  74  26  0  0  26  7  0  0  7  23  95  0  0  95  29  0  0  29  8  0  0  8  24  104  0  0  104  30  0  0  30  14  0  0  14  25  124  1  3  128  35  0  0  35  21  0  0  21  26  120  6  3  129  38  0  0  38  27  0  0  27  27  111  15  4  130  38  0  0  38  36  0  0  36  28  150  25  5  180  38  0  0  38  39  2  0  41  29  155  25  6  186  47  0  0  47  47  3  0  50  30  165  25  8  198  95  0  0  95  62  4  0  66  31  168  39  9  216  95  0  0  95  117  6  0  123 Here, in both table 3A and 3B, the phrases 'Re' and 'Conf' represents recovered and confirmed infected class respectively.

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CRediT Author Statement Manotosh Mandal: Conceptualization, Methodology, Software, Formal analysis, Investigation, Validation, Writing -Original Draft Soovoojeet Jana: Conceptualization, Validation, Methodology, Writing -Original Draft, Writing -Review & Editing, Visualization Swapan Kumar Nandi: Validation, Formal analysis, Investigation Anupam Khatua: Investigation, Resources, Data Curation Sayani Adak: Resources, Data Curation T. K. Kar: Conceptualization, Writing -Review & Editing, Visualization

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
