key: cord-1026252-dy2tige9
authors: Mishra, A.M.; Purohit, S.D.; Owolabi, K.M.; Sharma, Y.D.
title: A nonlinear epidemiological model considering asymptotic and quarantine classes for SARS CoV-2 virus
date: 2020-06-04
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2020.109953
sha: 74a4c1ae68d030bac3ee41284d9f6b61215b5b09
doc_id: 1026252
cord_uid: dy2tige9

In this article, we develop a mathematical model considering susceptible, exposed, infected, asymptotic, quarantine/isolation and recovered classes as in case of COVID-19 disease. The facility of quarantine/isolation have been provided to both exposed and infected classes. Asymptotic individuals either recovered without undergo treatment or moved to infected class after some duration. We have formulated the reproduction number for the proposed model. Elasticity and sensitivity analysis indicates that model is more sensitive towards the transmission rate from exposed to infected classes rather than transmission rate from susceptible to exposed class. Analysis of global stability for the proposed model is studied through Lyapunov’s function.

In The virus identified as zoonotic, similar to SARS and MERS. The different studies [1] show the basic reproduction number for the disease ranges from 1.4 to 6.49 with a mean of 3.28, a median of 2.79 and interquartile range of 15 1.16. However, Zhang and their collaborators [2] have estimated maximum likelihood value of basic reproduction number: 2.28. Another early dynamics of daily reproduction number estimated as 2.35 which is reduced upto 1.05 after implementing lockdown [3] . The effect of virus is much severe in United States with 1,292,879 infected on 08 May, 2020 [4] . Another study [5] concluded 20 that fatality was highest in persons aged 85 years, ranging from 10% to 27%, followed by 3% to 11% among persons aged 6584 years in United States from February 21-March 16, 2020. In the case of Wuhan city in China, incubation period for the novel coronavirus estimated as 6.4 days and epidemic doubling time 6.4 days [6] . The reproduction number has estimated to 2.56 with 95% 25 confidence interval when unreported cases for the virus has taken into account [7] . In the context of India, Mandal et. al. [8] and Mishra et. al. [9] have found basic reproduction number being 1.5 under the situation in March 2020 by considering SEIR model. They have also suggested control measures in order to prevent from the disease.

A key feature of the disease is significant proportion of asymptotic individuals. A specific study in Japan [10] shows 41.6% of individuals are asymptotic. Mathematical modelling is one of the significant tools to provide valuable insight into the epidemic problems and various other real world problems, for instant see recent work [20] - [25] . This work presents a deterministic compartmental model to explain SARS CoV-2 virus in some extent. We incorporate asymptotic infection and quarantine/isolation into SEIR model.

In the proposed model, S (susceptible) is a healthy population which undergo contagion and move to E class (exposed). After infection from virus, some of them are asymptotic and are categorised into A group (asymptotic).

Others who shows symptoms are moved to I group (infected). Asymptotic are 65 not showing any visible symptom of the disease and it is much difficult to identify them. As per study available, SARS CoV-2 virus has a key feature of asymptotic individuals amounts to 5% − 80% of the exposed people [10] . We denote this probability to p. Asymptotic people are assumed to be infectious with reduced (or enhanced) transmission rate qβ 0 . An individual from asymp-70 totic group either shows symptom after some duration and moves to infected group or remain asymptotic and recovers from disease. Let γ denote the recovery rate of asymptotic individuals. The infected may go under treatment with isolation. In this case, they move to recovered group through Q-group (quarantine/isolation). But some of individuals may not go to isolation and directly 75 move to recovered group. Let ν denote the recovery rate from infected group and δ denote recovery rate from quarantine/isolation. The quarantine rate from infected to isolation group is denoted by α. We are assuming that dead are not infectious. Hence the recovered group contains both dead and recovered people and separated from susceptible after the process. We also assume that once an 80 individual is recovered from disease, he develops the immunity from the virus and will not undergo the cycle.

Mathematically the model is expressed as the following autonomous system:

The total size of population is assumed to be N which are logistically increasing at a rate of Λ and decreasing by natural mortality rate µ. β 0 denotes transmission rate from susceptible to exposed compartment. η is infection rate for the model and θ is isolation rate of individual. Force of infection is related to prevalence (I + qA) with a linear relation as

In order to make relation simple we substitute β = β 0 /(N − Q). β is per capita transmission rate.

The population is disease free until I(0) number of infected enter into the population. At the time t = 0, R(0) = 0, S(0) = N, Q(0) = 0, A(0) = 0 and E(0) = 0. The feasible region of the system will be

From the system (1), S (0) < 0 for all t > 0 implies that S(t) is positive, monotone and bounded by N . Hence, the final size of epidemic is

In the same way, R is positive, bounded and monotonic which implies that

The trivial equilibrium point for the model is (Λ/µ, 0, 0, 0, 0, 0). For sufficiently large time, number of infected exponentially decay

for t → ∞. This implies that

Probability of individuals who recovered from the disease, indicate the cumula-90 tive distribution function F (t):

To find the probability density function P (t) from cumulative distribution, we must use fundamental theorem of calculus:

First moment about the origin gives the expectancy of time spend in infectious class:

The incubation period is thus, the inverse of sum of transmission rates α, ν, µ.

In this subsection, we sketch a graph of the model. Let us take total population be N = 1000. The birth rate of the population is assumed to Λ = −0.2% 

In Epidemiology, we often interested to know the long term behaviour of system. We assume that population is not open thus the quantity we deal in modelling are finite one. A graph between S and I with time as parameter is call orbits or trajectories and graph is often called Phase plane (fig 2) . If we look the system in long term, the system gets steady state equilibrium and at this point

The system possesses two singular points (equilibria points). One, at dis- reproduction number is greater than one. 

where E * is

The reproduction number generally defined as number of secondary infections appear from one infected individual. It provides a threshold condition for 115 the stability of the system. Finding reproduction number through Jacobian approach using linearization of the system often does not work for complex system.

We use next generation approach, known as Van den Driessche and Watmough approach [26] , here.

In this approach, we first decompose system (1) into infected compartment:

and non-infected compartment:

We arrange infected compartment such that 120 d dt

The matrices F and V chosen such that

and

Note that various combinations of F and V are possible, however, the functions defined in (16) and (17) satisfy the following conditions:

• New infections in the populations are secondary infections, that is,

• There is no immigration from susceptible population, that is, V = 0,

• Total output from infected compartments is positive.

Moreover, feasibility condition on F allows only non-negative output and Matrices of partial derivatives of F and V are respectively,

and

Next generation matrix is defined as K = F V −1 . Van den Driessche and Watmough approach suggests the reproduction number R 0 = spectral radius of K.

with

Reproduction number can be defined as

The reproduction number R 0 can be visualised as sum of two reproduction numbers. The term In fig (3) , infected population has been plotted for different values of reproduction number. Fig (3) shows higher the value of reproduction number (from unity), the greater peak of infected population. The proposed model has five transmission rates, namely, β, η, θ, ρ, α and three recovery rates δ, ν, γ apart from Λ, µ and two probabilities p, q. One of the primary mandate of the epidemiological model is to provide an effective tool to control the transmission. Sensitivity analysis suggests that which transmission 165 rate we should control sensitively in order to prevent the transmission. To perform global sensitivity analysis we need to grow the framework to incorporate all parameter involved in the system. This is a computationally hard issue, especially for a multi-dimensional model [27] . We focus here on local sensitivity analysis first.

Suppose we need a local sensitivity analysis of infected with respect to the 

Other variables Z E , Z I , Z Q , Z A , Z R are defined in the same fashion by replacing S to respectively E, I, Q, A, R. Our interest is to simulate Z I and for this, we need to solve the following system of differential equations

Z E = S(I + qA) + βZ S (I + qA) + βS(Z I + qZ A ) − (η + θ + µ)Z Q (28) (fig 7) .

We observe that the increase upto the value 0.01 in θ lowers the peak of infection leaving the position of peak unaltered.

Elasticity of a static quantity with respect to a parameter measures the per-190 centage change in the quantity with respect to percentage change in parameter. From equation (24) , elasticity of reproduction number with respect to β is

Furthermore, elasticity of reproduction number with respect to η

Expression (34) shows reciprocal relation of η, θ and µ. The elasticity of reproduction number with respect to ρ is the following:

Note that equation (26) is independent from ρ. This means R 0 and R a 0 have same elasticity. Elasticity of reproduction number with respect to θ and α are

and

This investigation reveals that increase in ρ, that is, identification of more infected from asymptotic class helps to reduce the reproduction number. The same is with η and θ. Global stability of the system can be determined using Lyapunov's function.

Lyapunov's stability theorem states that a globally positive definite and radially unbounded Lyapunov's function whose derivative is negative on entire feasible region except a point x * , possesses the globally stable equilibrium at x * . Con-210 struction of suitable Lyapunov's function is tricky. Krasovkii-LaSalle theorem helps to establish such function for an autonomous system.

Suppose (S * , E * , I * , Q * , A * , R * ) is an endemic equilibrium point stated in equation (7-12). We define

It is clear from the definition that function defined in equation (38) are positive definite and radially unbounded. The values of k 2 , k 4 and k 5 are assumed to be

In order to establish global asymptotic stability, S < S * and k 1 must be chosen such that

It is worth mentioning here that this global asymptotic stability shall be disease 215 free. 

Prophylaxis are off-medical control strategies in order to prevent the spread of the disease. Such measures include social distancing, mask wearing, handwash, sanitization and lockdown. Prophylaxis reduces the transmission rates. Lockdown reduces person to person contact in susceptible and exposed group which leads to reduction of β and η. This we observe reduction of reproduction number from equation (24).

If we see the relationship between reproduction number and isolation rate α, we get a graph (fig 8) . Smaller value of β provides the steeper decline in 225 reproduction number. The reproduction number is a decreasing function of the isolation rate α. The critical isolation rate that gives R 0 = 1, is given by

The equation (42) provides an information that one can control the reproduction number by applying the prophylaxis in a proactive manner. Variation of infected with respect to β and α is shown in fig. (9) which is a decreasing, 230 concave up function of α.

In the proposed model, we have considered isolation and quarantine as a separate compartment. Isolation has been studied in different disease model and found that it destabilize the dynamics and lead to oscillations [28, 29] and thus, suggested as potential intrinsic mechanism to combat the disease. 4. Discussion:

Proposed model considers six stages of populations: susceptible (S), exposed (E), asymptotic (A), infected (I), quarantine/isolation (Q) and recovered (R).

The model discriminates between infected and asymptotic people depending upon whether infected people from virus do not show symptom or otherwise. As 250 per study, transmission of COVID-19 disease through asymptotic population is an evident feature. The quarantine and isolation in the population are carrying out from both stages: from exposed and infected. In the model, we consider that dead do not transmit the disease and a person who recovered from the disease will not undergo again through infection.

However, reproduction number is an effective tool to control the disease.

From (fig 3) , it is clear that larger the value of R 0 invite the disaster in a large amount quickly.

The sensitivity analysis of the model advocates to reduce the value of η in order to control the disease in an efficient way. Transmission rate β also controls 260 the infection but in less sensitive manner than η. Changing ρ and θ do not affect in the number of infected in a large scale.

If we enhance the isolation rate α as given by equation (42), the disease will disappear from population. Thus it provides a threshold value of transmission which prevent the spreading of the disease using isolation only. 

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The authors hereby declare their intensions to publish the attached manuscript in Chaos, Solitons and fractals

The authors thank the referee for his concrete suggestions which resulted in a better organization of this article.

All authors make equal contributions