key: cord-282927-jhma20de
authors: Mondal, Chittaranjan; Adak, Debadatta; Majumder, Abhijit; Bairagi, Nandadulal
title: Mitigating the transmission of infection and death due to SARS-CoV-2 through non-pharmaceutical interventions and repurposing drugs
date: 2020-09-28
journal: ISA Trans
DOI: 10.1016/j.isatra.2020.09.015
sha: 
doc_id: 282927
cord_uid: jhma20de

The Covid-19 pandemic has put the world under immeasurable stress. There is no specific drug or vaccine that can cure the infection or protect people from the infection of coronavirus. It is therefore prudent to use the existing resources and control strategies in an optimal way to contain the virus spread and provide the best possible treatments to the infected individuals. Use of the repurposing drugs along with the non-pharmaceutical intervention strategies may be the right way for fighting against the ongoing pandemic. It is the objective of this work to demonstrate through mathematical modelling and analysis how and to what extent such control strategies can improve the overall Covid-19 epidemic burden. The criteria for disease elimination & persistence were established through the basic reproduction number. A case study with the Indian Covid-19 epidemic data is presented to visualize and illustrate the effects of lockdown, maintaining personal hygiene & safe distancing, and repurposing drugs. It is shown that India can significantly improve the overall Covid-19 epidemic burden through the combined use of NPIs and repurposing drugs though containment of spreading is difficult without serious community participation.

The world has been passing through an extraordinary crisis period since late December 2019 due to an extraor- 25 dinary pathogen Covid-19 [1] . This virus is extraordinary because our immune system is unable to fight against this 26 virus, and it has stunning transmission capability from human-to-human. The crisis is extraordinary because it has 27 not only caused a global medical crisis but also put the world into unprecedented economic and social crises [2] . The was used to estimate the final size and its peak time for China, South Korea, and the rest of the World [35] . In [36] , 97 a simple SIR model was used to estimate various epidemic parameters by fitting the epidemic data of the Republic 98 of Korea. Nonlinear incidence was used in an epidemic model to show that lockdown can make significant delay to 99 attain the epidemic peak but unable to eradicate the disease [37] . A simple iteration model, which uses only daily 100 values of confirmed cases, was considered to forecast the covid positive cases for the United States, Slovenia, Iran, In this paper, we have proposed a minimal epidemic model to capture the dynamics of observed data of detecting, 108 recovered and death cases of any Covid-19 affected country. The model is then extended to include three control 109 strategies to mitigate the transmission of Covid-19 and to reduce its related deaths. Human-to-human transmission 110 of infection depends mainly on two things: the number of per capita daily contacts between susceptible and infec-111 tives, and the probability of disease transmission per effective contact [42] . As mentioned before, per capita daily 112 contacts may be significantly reduced through lockdown, while the probability of virus transmission can be reduced 113 by using face masks and other good hygiene practices. We, therefore, used these nonpharmaceutical interventions as 114 two control measures. The third control is used to reduce the death rate of severely infected Covid-19 patients using 115 repurposing drugs. The proposed epidemic model is then analyzed to determine the criteria for persistence and extinc-116 tion of the disease. It is shown that repurposing drugs is very useful in saving lives and increasing the recovery rate.

Through a case study, it is shown that India can significantly improve the overall Covid-19 epidemic burden through 118 the combined use of NPIs and repurposing drugs and it is true for any other country. India can save lives of 4794 119 corona infected patients in the next one month (August 28 to September 26) if the repurposing drug has efficacy 0.3.

The rest of the paper is organized as follows. In Section 2, we present the mathematical model to be analyzed.

Different mathematical results that determine the dynamics of the system are given in Section 3. A case study with Here we consider five compartments, viz. susceptible (S ), latent (E), home isolated (I h ), hospitalized (I c ) and 127 recovered (R) classes, to classify the human population of a coronavirus affected country based on their health status. Thus, N(t) = S (t)+E(t)+I h (t)+I c (t)+R(t) represents the total population of the country at any time t. As SARS-CoV-2 129 is a novel virus, every individual is assumed to be susceptible to this virus. After getting an infection, an individual Covid-19 patients join the class R after recovery or succumb to infection. It is further assumed that individuals join S 134 class through birth at a rate G. The average per capita daily contacts of an infected individual is assumed to be n and 135 the probability of disease transmission through contact between an infected individual and a susceptible individual 136 is p. A common practice while writing the incidence term is to express the product of n and p as a single term, are reports that covid patients die in transit due to lack of hospital beds, unavailability of ambulance [43, 44, 45] . We, 142 therefore, assume that some individuals of I h class may also succumb to infection before they are shifted to I c class. It 143 is, however, true that most of the members in mildly infected class (I h ) recover from the infection. Therefore, it would 144 be justified to assume that ν 1 >> ν 2 and m 1 << m 2 . The natural death rate of susceptible individuals is assumed to 145 be σ, giving 1 σ as the average life expectancy of the common people, and considered in all compartments. We also the death class are not included to represent the total population. The dimension of the proposed model was kept low, though this simplified representation reflects the usual coronavirus infection management protocol followed in many 149 countries including India. With these assumptions, we propose the following model for Covid-19 epidemic:

Those who are critically ill are assumed, for simplicity, to be unable to spread the infection further. This assumption 151 may be too strict but simplifies the model to

(2)

A schematic representation of the system (2) is given in Fig. 1 153 Figure 1 : Schematic diagram of the disease progression mechanism considered in the system (2).

J o u r n a l P r e -p r o o f We now introduce three constant controls u i (i = 1, 2, 3), 0 ≤ u i ≤ 1 to the above epidemic model. The control 154 u 1 is introduced to reduce the daily number of contacts due to lockdown (complete or partial). A second control u 2 155 is introduced to diminish the probability of transmission due to using a face mask and maintaining social distance, is applied to reduce the death rate and increase the recovery rate by using various repurposing drugs and convalescent 159 plasma therapy. As the effect of the application of repurposing drugs, a fraction of infected people, who earlier 160 succumbed to infection, now recovers and joins R class; while the remaining fraction is the member of the disease-161 related death class, V. In fact, without the repurposing drugs, infected individuals will die at a rate m 2 and there will 162 be no extra recovery or a reduction in the death class. Introducing these controls, the system (2) reads

(3)

The models (2) and (3) will be analyzed with the initial conditions

It is to be noted that different sub-cases may be deduced from the model (3) depending on the mitigation measures.

For example, an epidemic model of Covid-19 can be deduced for the before lockdown period, when there was no Covid-19 pandemic as well as the recently approved treatment strategy with repurposing drugs.

, then any solution ofẊ = F(X) with given initial 178 condition, say, X(t) = X(t; X 0 ) will remain positive, i.e., X(t) ∈ R n + , ∀t > 0. 

. It can be easily seen from (3) that

Lemma 3.1 then gives that all solutions of the system (3) starting with the initial condition (4) are positive. Again,

Thus, following [47] , R 5 + is invariant. Therefore, all solutions of the system (3) with initial condition (4) are positively invariant. To show the boundedness of the solutions of (3), we define

Then we have

Hence all the solutions of the system (3) are positively invariant and ultimately bounded in the region Γ defined in (5). given by

6 J o u r n a l P r e -p r o o f

The transmission matrix and transition matrix associated with this infection subsystem (9) in a completely susceptible scenario (when S = G σ ) are given by T and Σ, respectively, where

Then the spectral radius of the matrix −TΣ −1 gives the basic reproduction number of the system (3) and is defined by

whenever u i ∈ [0, 1), i = 1, 2. It is straightforward to see that the value of R 0 is higher when there is no control. It is to 197 be noted that whenever u 1 = 1 or u 2 = 1 then the incidence term becomes zero and all subsequent state variables tend 

,

where

Then the Eq. (13) can be reexpressed as

where R 0 is given by (11) with u i ∈ [0, 1), i = 1, 2. Clearly, there exists a unique interior equilibrium E * of the system 208 (3) whenever R 0 > 1. 209 We give below the stability results of these equilibrium points.

Theorem 3.1 The disease-free equilibrium E 1 of the system (3) is globally asymptotically stable if R 0 ≤ 1.

Proof. Consider the Lyapunov function 212 

Noting that

Thus,U 1 ≤ 0 if R 0 ≤ 1. The equality sign occurrs at the disease free equilibrium, E 1 . Therefore, using LaSalle's 

Letting → 0, we obtain lim sup t→∞ I c (t) ≤ 0. Again, using the fact that lim inf t→∞ I h (t) = 0, lim inf t→∞ I c (t) ≥ 0. 

We consider the average per capita daily contacts of an undetected infectious individual, n, as the bifurcating parameter 226 and apply the central manifold theorem to determine the local stability of E * . The critical value n = n * for which R 0 = 1 227 holds is

Now let at n = n * , the Jacobian matrix J E 1 | n=n * has a right eigenvector u = (x 1 , x 2 , x 3 , x 4 , x 5 ) T corresponding to the 229 zero eigenvalue, where 230

Similarly, a left eigenvector corresponding to the zero eigenvalue of the Jacobian matrix J E 1 | n=n * is w = (w 1 , w 2 , w 3 , w 4 , w 5 ), With the transformations S = y 1 , E = y 2 , I h = y 3 , I c = y 4 , R = y 5 , the system (3) can be expressed as

where g i ∈ C 2 (R 5 × R), i = 1, .., 5. Then the second order partial derivatives of g i at E 1 are evaluated as

The signs of the quantities α and β evaluated at n = n * determine the local stability of the system [50], where 234 α = Σ 5 k,m,n=1 w k x m x n ∂ 2 g k (0, 0) ∂y m ∂y n and β = Σ 5 k,i=1 w k x i ∂ 2 g k (0, 0) ∂y i ∂n .

Following the Remark 1 of Theorem 4.1 given in [50], a transcritical bifurcation will occur at R 0 = 1 if α < 0 and 235 β > 0 at n = n * . Substituting all the values of the second order partial derivative evaluate at E 1 with n = n * , we then

implying a transcritical bifurcation at R 0 = 1. Thus, whenever the endemic equilibrium exists, i.e., when R 0 > 1, it 239 becomes locally asymptotically stable. Hence the theorem is proven. The data for India was collected from the online freely available repository Covid19India.Org (https://www.covid19india.org/).

In this website, day-wise data, as well as the cumulative data of corona cases in India and its all states/ union territories Table 2 . that there were many inaccuracies in the initial data. In many cases, there were under-reporting of the actual facts, and The curve R 0 = 1 separates the infection-free state (R 0 < 1) from the endemic state (R 0 > 1) in the plane of control parameters u 1 , u 2 . The present estimated values of u 1 and u 2 are marked with a red dot. Parameters are as in the Table 2 with unlock case.

Basic reproduction of an epidemic model usually delineates the stability and existence of a disease-free state from 329 its endemic state. More precisely, if the basic reproduction number R 0 is greater than unity then an epidemic can 330 spread, otherwise epidemic dies out. We determined the basic reproduction number of our system by next-generation 331 matrix and it showed that the classical properties of the basic reproduction number remained intact for Covid-19 332 epidemic. In the case of India, the basic reproduction number significantly reduced during the lockdown period, 333 however, it remained almost the same during the unlock period ( Fig. 6(a) ). To contain the epidemic, the value of 334 R 0 has to be brought down below 1 through NPIs in absence of vaccine. It is worth mentioning that the value of the 335 nonpharmaceutical control parameters must be significantly high in this case. Fig. 6 (b) shows that any combination of 336 the control parameters u 1 and u 2 that lies above the curve R 0 = 1 for the unlock parameters can eliminate SARS-CoV-2 337 virus from the system. India is trying hard to prevent the transmission of coronavirus by implementing NPIs. People 338 have become more aware of this deadly virus and following the statutory instructions more sincerely than before. Some improve the overall Covid-19 epidemic burden. It is to be mentioned that we have used fixed control in our study, one 349 can, however, consider these controls as time-varying in finding optimal control strategies to reach out to the target.

In this case, the objective should be to minimize the number of individuals to all infection-related classes, the decease 351 related deaths and the related cost. 

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