key: cord-352990-0uglwvid
authors: Nadim, Sk Shahid; Chattopadhyay, Joydev
title: Occurrence of backward bifurcation and prediction of disease transmission with imperfect lockdown: A case study on COVID-19
date: 2020-08-17
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.110163
sha: 
doc_id: 352990
cord_uid: 0uglwvid

The outbreak of COVID-19 caused by SARS-CoV-2 is spreading rapidly around the world, which is causing a major public health concerns. The outbreaks started in India on March 2, 2020. As of April 30, 2020, 34,864 confirmed cases and 1,154 deaths are reported in India and more than 30,90,445 confirmed cases and 2,17,769 deaths are reported worldwide. Mathematical models may help to explore the transmission dynamics, prediction and control of COVID-19 in the absence of an appropriate medication or vaccine. In this study, we consider a mathematical model on COVID-19 transmission with the imperfect lockdown effect. The basic reproduction number, R(0), is calculated using the next generation matrix method. The system has a disease-free equilibrium (DFE) which is locally asymptotically stable whenever R(0) < 1. Moreover, the model exhibits the backward bifurcation phenomenon, where the stable DFE coexists with a stable endemic equilibrium when R(0) < 1. The epidemiological implications of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is only a necessary, but not sufficient for effectively controlling the spread of COVID-19 outbreak. It is observed that the system undergoes backward bifurcation which is a new observation for COVID-19 disease transmission model. We also noticed that under the perfect lockdown scenario, there is no possibility of having backward bifurcation. Using Lyapunov function theory and LaSalle Invariance Principle, the DFE is shown globally asymptotically stable for perfect lockdown model. We have calibrated our proposed model parameters to fit daily data from India, Mexico, South Africa and Argentina. We have provided a short-term prediction for India, Mexico, South Africa and Argentina of future cases of COVID-19. We calculate the basic reproduction number from the estimated parameters. We further assess the impact of lockdown during the outbreak. Furthermore, we find that effective lockdown is very necessary to reduce the burden of diseases.

An outbreak of 2019 coronavirus disease (COVID-19) has resulted in 30,90,445 con- 10 firmed cases and 2,17,769 deaths as of April 30, 2020 according to WHO [6] . The out-11 break was first taken place in Wuhan, China, in December 2019, with the majority of 12 early cases reported in the city. Coronaviruses are single-stranded, positive-sense RNA 13 viruses belonging to the Coronaviridae family [9] . It has been confirmed that 27 people 14 have been infected due to viral pneumonia, including seven critically ill cases [30] , and 15 this epidemic has drawn tremendous attention worldwide. It causes variety of symptoms, 16 including dry cough, fever, weakness, trouble breathing, and bilateral lung infiltration, be the sixth public health emergency of international concern. Since the first discovery 26 and identification of coronavirus in 1965, there are three major outbreaks occurred due to 27 coronaviruses and the outbreak was called 'Severe Acute Respiratory Syndrome (SARS) 28 outbreak (2003) in China [23] . Saudi Arabia suffered from 'Middle East Respiratory 29 Syndrome' (MERS) outbreak (2012) [11] and South Korea (2015) [10] . 30 The Indian government reported that on 30 January 2020 in the state of Kerala, across the world and an unprecedented threat to the community's health care, economy 44 and lifestyle. For all, there is a huge worry as to how long this condition can continue 45 and whether the epidemic can be handled. 46 We also study the cases of Mexico, South Africa and Argentina as the 47 lockdown was carried out partially or in a less severe form in these countries. 48 In February 2020 the virus was confirmed to reach Mexico. However, two 

Recently lockdown measure has been used successfully to control COVID-19 spread.

The aim of this study is to investigate the qualitative effect of the imperfect lockdown 74 on the spread of disease dynamics. To achieve this goal, a mathematical model for 75 COVID-19 with the lockdown is proposed and analyzed. In this model, we implemented 76 the imperfect lockdown, which means that the lockdown susceptible population also gets 77 infected during the lockdown period by unnotified infected individuals. We looked at In-78 dia's situation during the outbreak period and fitted our model with the newly daily cases 79 reported from March 14 th , 2020 to April 19 th , 2020. We also looked at the situation 80 of Mexico, South Africa and Argentina during the outbreak period and fitted 81 our model with the new daily cases reported for a certain outbreak period. 82 We are providing a short-term prediction for India, Mexico, South Africa and 83 Argentina of future cases of COVID-19 using the estimated parameters for 84 the period March 14, 2020, to May 21, 2020, March 23 to July 9, March 17 85 to July 9 and March 14 to July 9 respectively. For the above-mentioned period we 86 aslo estimate the basic reproduction number. It is common for classical epidemic models 87 that a basic reproduction number is a threshold in the context that if the basic repro-88 duction number is greater than one, a disease will persist, and dies out if it is less than 89 one. In this case, for imperfect lockdown, the basic reproduction number does not rep-90 resent the required elimination effort; rather, the effort at the turning point is described The paper is organized as follows: Our proposed mathematical model which incorporates 108 the lockdown of susceptible individuals and imperfect lockdown efficacy is described in 109 Section 2. The model is analyzed specifically for the existence of backward bifurcation 110 in Section 3. In Section 4 we fitted our model to daily new cases. We provided a 

All the parameters and their biological interpretation are given in Table 1 respectively. 146 Proof. The system (2.1) can be written as follows

.., f 6 (x)) denotes the right hand side functions. It is very obvious that for every j = 1, ..., The basic reproduction number R 0 is a threshold value that is epidemiologically significant and determines the potential of an infectious disease to enter a population. To obtain the basic reproduction number R 0 of the system (2.1), we apply the next generation matrix approach. The system has a disease-free equilibrium given by

, 0, 0, 0, 0 .

The infected compartments of the model (2.1) consist of (E(t), I(t), J(t)) classes. Fol-153 lowing the next generation matrix method, the matrix F of the trransmission terms and 154 the matrix, V of the transition terms calculated at ε 0 are,

So, the next generation matrix F V −1 is,

Calculating the dominant eigenvalue of the next generation matrix F V −1 , we obtain the basic reproductive number as follows [13; 33]

The basic reproduction number R 0 is defined as the expected number of secondary cases 157 generated by one infected individula during its lifespan as infectious in a fully susceptible 158 population. The basic reproduction number R 0 of (2.1) given in 3.2.

Using Theorem 2 in [33], the following result is established.

Lemma 3.1. The disease-free equilibrium ε 0 of system (2.1) is locally asymptotically 161 stable whenever R 0 < 1, and unstable whenever R 0 > 1. 3.3. Existence of endemic equilibrium 163 We are now exploring the existence of endemic equilibrium. Let ε * = (S * , L * , E * , I * , J * , R * ) be any endemic equilibrium of system (2.1). Let us denote

Further, the force of infection be

By setting the right equations of system (2.1) equal to zero, we have

Substituting the expression in 3.4 into 3.3 shows that the non-zero equilibrium of the model (2.1) satisfy the following quadratic equation, in terms of λ * h :

The endemic equilibrium of the model (2.1) can be obtained by solving for λ * h from In order to check the possibility of backward bifurcation in (2.1), the discriminant B 2 − 4AC of the quadratic 3.5, is set to zero and the result solved for the critical value (denoted by R c 0 ) of R 0 . This gives:

from which we have seen that backward bifurcation occurs for values of R 0 such that 184 R c 0 < R 0 < 1. We explore the details analysis of backward bifurcation in the next a stable endemic equilibrium co-exists with a stable disease-free equilibrium for R 0 < 1.

Clearly, this phenomenon has significant public health consequences, as it makes the 193 classical requirement of the associated basic reproduction number being less than unity 194 to be necessary, but not sufficient to eradicate the disease.

In this section, we explore the phenomenon of backward bifurcation in system (2.1).

First, we execute bifurcation analysis by using the center manifold theorem as follows:

The Jacobian of the system (3.7) at the DFE ε 0 is given as,

Choose β as the bifurcation parameter, then setting R 0 = 1 gives

The system (3.7) at the DFE ε 0 evaluated for β = β * has a simple eigenvalue with zero 201 real part, and all other eigenvalues have negative real part. We therefore apply the Center

Manifold Theorem in order to analyze the dynamics of (3.7) near β = β * .

The Jacobian of (3.7) at β = β * , denoted by J ε 0 |β = β * has a right eigenvector (corresponding to the zero eigenvalue) given by w = (w 1 , w 2 , w 3 , w 4 , w 5 , w 6 ) T , where

Similarly, from J ε 0 |β = β * , we obtain a left eigenvector v = (v 1 , v 2 , v 3 , v 4 , v 5 , v 6 ) T (corresponding to the zero eigenvalue), where

(3.10)

We calculate the following second order partial derivatives of f i at the disease-free equilibrium ε 0 to show the existence of a backward bifurcation and obtain Since the coefficient b is always positive, system (2.1) undergoes backward bifurcation

We have established the following conclusion.

Theorem 3.3. System (2.1) undergoes a backward bifurcation at R 0 = 1 whenever the 206 inequality 3.14 holds.

Furthermore, it should be noted that for the case when lockdown susceptible individuals do not acquire infection during lockdown period (i.e., r = 0), the bifurcation coefficient a becomes

Thus, since a < 0 in this case, it follows from Theorem 4. globally-asymptotically stable (GAS) under some certain conditions, as shown below.

Setting r = 0 in the model (2.1) gives the following reduced model:

It can be shown that the reproduction number associated with the reduced model (3.15), is given by

The model (3.15) has a DFE ε 01 = (S 1 , L 1 , 0, 0, 0, 0).

Theorem 3.4. The DFE (ε 01 ) of the reduced model (3.15), is globally asymptotically

Proof. Consider the following Lyapunov function D = γk 2 θk 3 (k 2 + l) E + k 2 θ(k 2 + l) I

We take the Lyapunov derivative with respect to t,

Since all the variables and parameters of the model (2.1) are non-negative, it follows that 

Therefore by combining all above equations, it follows that each solution of the model 238 equations (2.1), with initial conditions ∈ Ω , approaches ε 0 as t → ∞ for R * 0 ≤ θ < 1.

The above result shows that, for the case when the lockdown efficacy in preventing tion of this technique for model fitting is given in [26] . The estimated parameters are 264 given in Table 2 the estimated values of unknown initial conditions are given by Table 3 .

The fitting of the daily new hospitalized COVID cases of this four country are displayed 266 in Figure 4 . Using these estimated parameters from Table 2 and the fixed parameters   267 from Table 4 , we calculate the basic reproduction numbers given in Table 5 . In this section, the impact of lockdown is measured qualitatively on the disease transmission dynamics. A threshold study of the parameters correlated with the lockdown of susceptible individuals l is performed by measuring the partial derivatives of the basic reproduction number R 0 with respect to this parameters. We observe that

so that ∂R 0 ∂l < 0 for all 0 < r < 1. We perform the sensitivity of model parameters with respect to the significant re-311 sponse variable and analyze different control parameters to limit COVID cases for the 312 four countries. In order to get an overview of most influential parameters, we compute 313 the normalized forward sensitivity indices of the model parameters with respect to basic 314 reproduction number R 0 . We have chosen parameters transmission rate between hu-315 man population β, rate of transition from exposed to infected class γ, the rate at which 316 symptomatic infected become hospitalized or notified η, recovery rate for symptomatic 317 infected τ 1 , lockdown success rate l and lockdown efficacy r for sensitivity analysis. We 318 use the estimated parameters from Table 2 for the baseline values. The rest of the pa-319 rameter values are the same as in Table 4 . The bar diagram of the normalized forward 320 sensitivity values of R 0 against these parameters is depicted in Figure 7 . However, the 321 mathematical definition of the normalized forward sensitivity index of a variable m with 322 respect to a parameter τ (where m depends explicitly on the parameter τ ) is given as:

The normalized forward sensitivity indices of R 0 with respect to the parameters β, η, l and r for India are found to be

The fact that X β R 0 = 1, means that if we increase 1% in β, keeping other parameters 324 be fixed, will produce 1% increase in R 0 . Similarly, X η R 0 = −0.9415 means increasing Argentina are given in the Table 6 . We have seen that the transmission rate 332 between susceptible humans and lockdown efficacy is positively correlated 333 and the recovery rate of symptomatic infected and lockdown success rate is 334 negatively correlated with respect to basic reproduction number. reproduction number R 0 . We have seen a similar patteren for this four countries.

In cases India, the contour plots in Figure 8 show the dependence of R 0 on dif-339 ferent parameters. In Figure 8 (a) and Figure 8 Table 2 and Table 4 .

From the above finding it follows that lockdown success rate and lockdown efficacy is Table 1 and Table 2. time point which is very slow(See the Figure 12 ). That means extension of 375 lockdown for these two countries is not too much effective. Table 1 and Table 2. backward bifurcation phenomenon, where two stable equilibria, namely the disease-free 387 equilibrium and an endemic equilibrium coexist when the corresponding basic number 388 of reproduction is less than unity. This backward bifurcation phenomena of this study 389 is very important, and this occurs only under imperfect lockdown individuals. This is 390 basically telling us even if the basic reproduction number is less than one, but the disease 391 will persists which is against classical epidemiological theory. In such a situation, the 392 policy makers may stop surveillance, and the results will be disaster. Our model exhibits 393 the non-existence of backward bifurcation when the lockdown is perfect (r = 0). We have 394 seen that the disease-free equilibrium is globally asymptotically stable whenever the as-395 sociated basic reproduction number is less than unity for the perfect lockdown model. Table 1 and Table 2 . Table 1 and Table 2 . 

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 441 The authors declare that they have no conflicts of interest. 442 Appendix A

The center manifold theory [7; 8] is used to determine the existence of the backward 553 bifurcation phenomenon of the model (2.1) theoretically.

Theorem 7.1. Let us consider the following general system of ordinary differential equations with a parameter φWithout loss of generality, it is assumed that x = 0 is an equilibrium for system (A-1) 555 for all values of the parameter φ. (2) Matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let f k be the k-th component of f andThen, the local dynamics of system (A-1) around 0 are totally determined by the sign of (ii) a < 0, b < 0. When φ < 0, with |φ| 1, x = 0 is unstable; when 0 < φ 1, 565 x = 0 is locally asymptotically stable and there exists a negative unstable equilibrium;

(iii) a > 0, b < 0. When φ < 0, with |φ| 1, x = 0 is unstable and there exists a 567 locally asymptotically stable negative equilibrium; when 0 < φ 1, x = 0 is stable and a 568 positive unstable equilibrium appears;

(iv) a < 0, b > 0. When φ changes from negative to positive, x = 0 changes its 570 stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes 571 positive and locally asymptotically stable.

In particular, if a > 0, b > 0 then a backward bifurcation occurs at φ = 0.