key: cord-0302258-57iulmae
authors: Zamir, Muhammad; Shah, Kamal; Nadeem, Fawad; Bajuri, Mohd Yazid; Ahmadian, Ali; Salahshour, Soheil; Ferrara, Massimiliano
title: Threshold conditions for global stability of disease free state of COVID-19
date: 2021-02-28
journal: Results in Physics
DOI: 10.1016/j.rinp.2020.103784
sha: 9c005893c1bf4ba88d7b290a8df1099ec1cc5f58
doc_id: 302258
cord_uid: 57iulmae

This article focus the elimination and control of the infection caused by COVID-19. Mathematical model of the disease is formulated. With help of sensitivity analysis of the reproduction number the most sensitive parameters regarding transmission of infection are found. Consequently strategies for the control of infection are proposed. Threshold condition for global stability of the disease free state is investigated. Finally, using Matlab numerical simulations are produced for validation of theocratical results.

The pandemic of viral infection, causing COVID-19, initiated in China. COVID-19 is highly infectious disease. The causative agent of the disease disease is a virus called severe acute respiratory syndrome coronavirus 2, or SARS-CoV-2. Very rapid viral transmission occur in human population, whenever they are in close contact. The transmission probability is high when the contact range is less than 2 meters. In such a circumstances the virus spreads by respiratory droplets released when an infected individual coughs, sneezes or talks. These droplets can be inhaled or directly reach the mouth or nose of a nearby person with medium of air. However if a susceptible human is not close range, the shredded virus contaminate the nearby surfaces. These contaminated surfaces help the viral transmission in community, however this isn't considered to be a main way it spreads through [1] .

Coronaviruses represents a big family. This family causes different types of infections. The infection ranges from common cold/flue to the most severe infection like severe acute respiratory syndrome and MERS; middle east respiratory syndrome [2] . The novel COVID-19 first emerged in December, 2019, Wuhan, China, in the form of severe cases of pneumonia and respiratory problems. The correct etiology of the infection could not be traced that time. WHO reported the virus as a novel coronavirus (2019-nCoV). The disease was named as severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2). The virus was first identified from a single individual. Subsequently the virus was verified in sixteen more cases [4, 5] .

It is expected that the virus might be bat origin [3] , and the infection transmission might be initiated from a seafood market (Huanan Seafood Wholesale Market) of China [6] . Currently 7,597,304 cases of the disease as been confirmed and 423,844 deaths has occurred as of June 12, 2020, world wide [7] .

About 75% of the victims of COVID-19 don't develop symptoms of the disease and recovered naturally [8] . 20% of the exposed individuals develop symptoms. The most common symptoms of COVID-19 are tiredness, fever and dry cough. Some patients may have aches and pains, runny nose, nasal congestion, Muscle aches, Chills, Loss of taste or smell or both, Headache, Chest pain and sore throat. Other less common symptoms have been reported, such as rash, nausea, vomiting and diarrhea. These symptoms are usually mild and starts slowly and gradually. Most symptomatic individuals (about 80%) recover from the disease without needing special treatment. In children and young adults, COVID-19 is generally minor. However, for some people it can cause serious illness. This type of severe attack of the virus may cause death. In some cases the attack may result SARS (severe acute respiratory syndrome) or pneumonia. The symptoms of infection appears in 2-14 days [9, 10] . The recovery time of mild cases is approximately 2 weeks and in severe/critical cases the recovery may take 3-6 weeks [11] . The individuals who developed severe form of disease, the medium time to dyspnea ranges from 5-8 days. The average time to acute respiratory distress syndrome (ARDS) varies from 8 days to 12 days. The average time to intensive care at vent bol (ventilated class) ranges from 10-12 days [27, 26, 28, 29] .The recovered individuals of disease can have antibodies for at least two weeks, long-term data are still lacking [12] .

The coronavirus (2019-nCoV) is genetically related to the coronavirus that caused the SARS-2003, however the diseases they caused are quite different [13] . The genetic features and some clinical findings of the infection have been reported recently [14] [15] [16] . International air travel contributed the international spread of the infection. The infection has got global attention regarding its elimination and control [17] .

The whole world is highly concerned with drastic future forecast of the disease. The scientists and researchers, therefore focus the development of mathematical model. The model not only helps estimating dynamics of the transmission of the virus but other important forecasts. Recent mathematical modeling includes [18] [19] [20] [21] [22] [23] . These models mainly focused the transmission/spreading of coronavirus or basic reproduction number of coronavirus, (R 0 ). The authors followed intrinsic growth rate and the serial intervals. Wu et al. in their study focus the forecasting and Newscasting of the novel coronavirus both nationally and internationally. The authors used Markov Chain Monte Carlo methods in their study [24] .

However, these models don't discuss the origin(bat) and the route of transmission/spreading (seafood market). Chen et al. extended this work. The authors in their work presented a comprehensive model of novel coronavirus [25] . Some more detailed information about the said infection can be found in [26, 27, 29, 30] . In this study, we focus the effect of different intervention on the transmission of the disease with help of sensitivity analysis of the parameters of the model. We combine different interventions in particular ratio and formulate a strategy. The effect of different strategies on disease control is shown graphically to facilitate strategy selection for the agencies fighting against COVID-19. Lastly we find a threshold condition for global stability of the disease free state in the community.

The total human population is divided in nine sub-classes: susceptible humans class (S), the quarantine humans class (Q), exposed humans class (E), symptomatic infected humans class (I 1 ), asymptomatic infected humans class (I 2 ), isolated human class (I s ), mild human class (I M ), critical human class (I c ) and recovered humans class (R). W I ; denote the class of contaminated surfaces/stuff shedded/stained with coronavirus.

The susceptible human population can catch infection from infected humans (both symptomatic and asymptomatic), exposed humans and also from the stuff stained/contaminated with coronavirus at different rates. All those susceptible individuals are quarantined who got contact with infectious human in the last 14 days. The quarantined individuals are passed through laboratory tests and accordingly move to susceptible class or exposed class. The exposed individuals after completing latency or incubation period move to symptomatic infectious class I 1 or asymptomatic infectious class I 2 . The symptomatic infectious individuals are isolated in isolation class I S . After completing transition period at I S some of the individuals move to mild class I M and the rest move to critical class I C . Most of the individuals at mild class recover and about 49% of the infected individuals at critical class die due to disease. The symptomatic and asymptomatic infectious individuals contaminate both the stuff and environment close to them at the rate called shedding coefficient. The class of such stuff is denoted by W I .

The following system (1) of coupled nonlinear differential equations represents the model of COVID-19:

The following table (1) contains the values of the different parameters used in the model (1) .

In this section properties of the model including Disease-Free-Equilibrium, Invariant region and the Basic Reproduction Number are addressed.

The state variables and parameters used in the model are always nonnegative because the model is concerned with the living population. Adding all the compartments related to human population we have:

From Eq. (2) we have rt Transition period in IS 14-21 days [34] r1

The ratio of exposed moving to symptomatic

Disease induced death ratio of critical class 0.49% [35] ε

The life time of virus on stuff 1-10 days [25] eX Expiry period of contaminated stuff 30 days assume k1

Ratio of quarantine moving to IS 66% assume k2

Ratio of quarantine moving to S 33% assume k4

Recovery ratio of mild class 99% [36] k5

Recovery rate of critical class 47% [37] k3

Ratio of asymptomatic moving to IC 4% [38, 39] δ Ratio of isolated moving to IC 15% [40] β Immunity loosing period 15 days [41] N⩽Λ − μN.

Solving the above equation we have:

On the basis of above discussion we claim the following result:

Proof. Ω; the region of the proposed model, defined by

is positively invariant domain, and the model is epidemiologically and mathematically well posed [42] and all the trajectories are forward bounded. □

The number of secondary infections caused by a single primary infection in completely susceptible population is called reproduction number denoted by R 0 . The reproduction number is find by next generation matrix [44, 43] as:

The column in matrix f denotes the individuals who get infected.

The column of matrix V denotes the individuals that enter the infected class or leave the infected class, excluding those coming from susceptible class.

The dominant Eigenvalue of ( − FV − 1 ) and hence R 0 is:

.

In this part, the threshold condition for global asymptotic stability of Disease Free State of the system (1) is studied. The following theorem would be used in the upcoming results, stated here for convenience: Theorem 3.1. ( [45] ) Let the given model be presented as

Then the DFS (Disease free state) is GAS (globally asymptotically stable) if the following holds. However if B 2 ∈Ñ then for any Ỹ ∈ Ω such that B 2 =B 2 (Ỹ ),

(a 5 ): The spectral radius of (B 2 ) is less then or equal to zero.

To prove the global stability of the disease free equilibrium let Y= 

⎧ ⎨ ⎩Ṡ = Λ − ( β 1 ( I 1 + c 2 I 2 + c 4 E ) + β 2 W I ) S − μS + k 2 Q + βṘ R = k 4 I M + k 5 I C + ( 1 − k 3 ) r n I 2 − ( μ + β ) R (3)

Proof. We re-write the above system as:

At the Domain G, the above system reduces to the form. ⎧ ⎨ ⎩Ṡ = Λ − μṠ R = − μR (4) Here

All the entries j (i,i) of the matrix C s are − ve, Hence the said system is GAS at Disease Free equilibrium (DFE). . □

The sub system of infected population is:

Theorem 3.3. In the system (5), B I is irreducible and metzler ∀Y ∈ Ω. Further more there exist some B I so as

Also

ϱ is modulus of stability and denotes the dominant real part of the eigenvalues of B I .

Proof. Let us re-write sub system (5) as:

Since the off diagonal entries are non-negative and the diagonal entries are negative. Therefore the matrix B I (Y ) is irreducible and metzler for all Y ∈ Ω. Next let B I be the upper bond of the matrix B I (Y ). Then,

Since S⩽S 0 . Therefore B I (Y ) is the upper bond of B I (Y ). This maximum is uniquely realized in Ω if S = N = S 0 . This corresponds to the DFE.

Also the matrix B I (Y ) is equal to J 2 . Where J 2 represent the Jacobian of the infected sub-system (3) at the DFE. Thus the assumption a 4 of theorem (3.1) holds. This proves (7) and (6). □ Next we prove a 5 or (8).

where Proof. We use the following decomposition of the matrix B I .

p 3 = (r n + μ), p 4 = (r t + μ), p 5 = (k 4 + μ) p 6 = (k 5 + D 2 + μ), Then the stability B I depends on the stability of G.

We know from Routh-Hurwitz [46] that in our case: ϱ(B I )⩽0 only if

Let us call the R.H.S of this equation as ξ. Then we have shown that the assumption a 5 or (8) is satisfied for ξ < 1. □

In the above discussion we have proved all the assumptions of the theorem (3.1).

On the basis of above findings we claim the following theorem:

Theorem 3.5. : The DFE of Given system would be globally asymptotically stable If the parameters used in the given model satisfy ξ < 1, where ξis as defined above.

Different parameters used in the model effect the transmission of the disease differently. The role of the parameter K t in the phenomenon Z t is called sensitivity of Z t w.r. t K t and is given by [47, 31] .

The sensitivity indices of parameters are given in the table (2) .

The parameter is directly proportional to initial transmission rate R 0 if has got positive sensitivity index, like the sensitivity index of human birth rate, Λ is +1 and is inversely proportional to R 0 if the index is negative. The parameter has more effective role in the transmission of the disease if its sensitivity index is high. However some parameters, though have got high sensitivity index but are beyond human control, like the natural mortality rate of human population or the disease incubation period.

We intervene 6 parameters, k I ; the clinical detection of disease, β 1 ; the disease transmission probability from human to human, β 2 ; the disease transmission probability from contaminated surfaces, ξ 1 ; the shedding coefficient of symptomatic infectious individuals, ξ 2 ; the shedding coefficient of asymptomatic infectious individuals, ε the life time of virus on the surface.

Non pharmaceutical interventions are,

• Face mask; This intervention addresses β 1 , the transmission from human to human and the shedding coefficients ξ 1 and ξ 2 . • Wash hands; This intervention addresses both β 1 , the transmission from human to human and β 2 , the transmission from contaminated surfaces to human. The control strategies given in table 2.2 present the magnitudes of these interventions.

We use RK-4 method for six set of values (called strategy) to generate numerical simulation, using matlab. Unit used is per day. The behavior of the disease is forecasted for 3000 days. For better visibility of disease's behavior some figures are split time wise, like Figs. ()()()1-3 shows the behavior of quarantine class, in the initial period, over all period and final period of 1500 days to 3000 days respectively.

In this work, a mathematical model of COVID-19 was formulated. The focus of the study is the elimination of the disease and the global stability of the disease free state so obtained. For this the sensitivity of R 0 was calculated and on the basis of sensitivity analysis, six parameter were chosen for intervention as shown in table (2.2) . Three control strategies were designed for the mentioned six interventions. On the basis of the results obtained from numerical simulations, we recommend strategy 3rd. Fig. (1) shows that as a result of strategy 3, the density curve of quarantined class is flattened as compared to that of strategy 1 and strategy 2. In fact this result is desired to reduce the burden of quarrantina centers. Fig. (3) shows that the recommended strategy overcome the quarantined class in 2600 days completely.

Similarly the recommended strategy flattens the density curve of exposed class and succeed to eliminate it in 2200 days, as shown in Figs. (5) and (4) . The density of symptomatic infectious class reduce to zero in 1400 days (See Fig. 6 ).

The behavior of asymptomatic infectious class is presented in Figs. (8) and (7). Since these infected individual do not develop signs and symptoms, so the control of this class takes about 1800 days (See Figs. 9-12) .

The densities of the isolated and mild classes reduces to zero in about 2200 days, as shown in figures. The behavior of critical class is presented by Fig. (14) , Figs. (15) and (13) . The recommended strategy 3 reduces the density of this class in 2600 days as shown in Fig. (13) . The effective use of sanitizer reduces the density of contaminated stuff in very short time, as shown in Figs. (16) and (17) .

It is concluded that the disease can be eliminated with help of relax non phrenetical interventions (like smart lock down of Pakistan). However it may take from six to seven years in its complete eradication. The study also derive threshold condition for global stability of disease free state. The condition agrees that there will be no outbreak, once the disease was eliminated. However it is suggested not to focus just the 5 . The graph represents the comparison of the strategies regarding exposed human population for the initial period. Fig. 4 . The graph represents the comparison of the strategies regarding exposed human population. 

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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.