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. parameter R 0 and found the most sensitive parameter, which has a positive 448 impact on R 0 is the disease transmission rate. The primary finding of this article is that we have derived a mathemati-484 cal model that can be used to study the qualitative dynamics of COVID-19. The basic reproduction number and its sensitivity analysis would determine 486 the controlling procedure of the disease. Also, we have incorporated the gov- using the software Mathematica, we try to fit our model (1) to References 528 Essai dune nouvelle analyse de la mortalite causee par la 529 petite verole. 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Kar: Conceptualization, Writing -Review & Editing, Visualization 01 254 39 9 302 144 6 2 152 227 6 1 234 02 280 42 13 335 207 8 4 219 227 6 1 234 03 277 42 16 335 207 8 4 219 302 6 1 309 04 424 42 24 490 424 15 6 445 403 6 2 411 05 424 42 24 490 478 18 7 503 476 6 3 485 06 647 56 45 748 497 19 7 523 558 8 5 571 07 764 56 48 868 548 21 7 576 608 8 5 621 08 875 79 64 1018 546 21 9 576 664 19 7 690 09 946 117 72 1135 639 21 9 669 709 21 8 738 10 1142 125 97 1364 660 25 13 698 805 21 8 834 11 1276 188 110 1574 684 25 14 723 859 44 8 911 12 1426 208 127 1761 1025 25 19 1069 915 44 10 969 13 1619 217 149 1985 1103 27 24 1154 1014 50 11 1075 14 1948 229 160 2337 1452 30 28 1510 1104 58 11 1173 15 2250 259 178 2687 1501 30 30 1561 1111 81 12 1204 16 2437 295 187 2919 1504 42 32 1578 1110 118 14 1242 17 2711 300 194 3205 1551 51 38 1640 1072 180 15 1267 18 2791 331 201 3323 1593 72 42 1707 1025 283 15 1323 19 3075 365 211 3651 1779 72 42 1893 992 365 15 1372 20 3473 507 223 4203 1668 290 45 2003 1051 411 15 1477 21 3865 572 232 4669 1603 431 47 2081 1046 457 17 1520 22 4248 722 251 5221 1498 611 47 2156 943 635 18 1596 23 4594 789 269 5652 1476 724 48 2248 949 642 18 1629 24 5307 840 283 6430 1518 808 50 2376 911 752 20 1683 25 5559 957 301 6817 1604 857 53 2514 867 866 22 1755 26 6229 1076 323 8068 1702 869 54 2625 838 960 23 1821 27 6538 1188 342 8068 1987 877 54 2918 841 1020 24 1885 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.