key: cord-0949513-2hpqrjul
authors: Zamir, Muhammad; Nadeem, Fawad; Alqudah, Manar A.; Abdeljawad, Thabet
title: Future implications of COVID-19 through Mathematical modeling
date: 2021-12-25
journal: Results Phys
DOI: 10.1016/j.rinp.2021.105097
sha: bc77d652502d6a1fbc9ebca69186ab31ac178476
doc_id: 949513
cord_uid: 2hpqrjul

COVID-19 is a pandemic respiratory illness. The disease spreads from human to human and is caused by a novel coronavirus SARS-CoV-2. In this study, we formulate a mathematical model of COVID-19 and discuss the disease free state and endemic equilibrium of the model. Based on the sensitivity indexes of the parameters, control strategies are designed. The strategies reduce the densities of the infected classes but don’t satisfy the criteria/threshold condition of the global stability of disease free equilibrium. On the other hand, the endemic equilibrium of the disease is globally asymptotically stable. Therefore it is concluded that the disease cannot be eradicated with present resources and the human population needs to learn how to live with corona. For validation of the results, numerical simulations are obtained using fourth order Runge–Kutta method.

At the end of 2020, the most dangerous communicable viral disease COVID-19 appeared in China. The disease's main source was the virus SARS-CoV-2. Once the disease entered the human community, its alarming transmission rate was un-ignorable, therefore the experts of all areas and disciplines focused to stop the further spreading of the disease. Different strategies have been adopted but humanity is still at risk of the disease. Close human contact is the main root of the disease transmission and this ultimately leads to a disaster in business and education.

The SARS-CoV-2 released from the mouth or nose of the COVID-positive individuals may directly hit another susceptible human if it is within range of 5-6 feet. Otherwise, the virus contaminates the available surfaces. As such these surfaces play the role of the virus carrier [1] . There are currently 257,007,274 confirmed cases and 5,156,403 deaths from the coronavirus COVID-19 outbreak as of November 21, 2021, worldwide [2] .

The disease fatality rate is not high because almost 75% of the COVID-positive individuals recover without treatment. Since these individuals don't develop any symptoms of the disease, therefore they are not treated [3] . In 20-25% cases tiredness, severe headache, losing taste and smell, dry cough or high fever appears on the victim, from two to fourteen days after the attack of virus [4, 5] . Most of the cases of COVID-19 are mild and recover in two weeks, however, in critical cases, the recovery may take 21 to 42 days [6] . The genetic features, history, and clinical features of the disease can be found in [9, 10, 11, 12, 13, 16] .

Round the world, all well-wishers of humanity are very much concerned about the future forecast of COVID-19. If the disease maintains its present status, humanity may face starvation and illiteracy. To forecast the future of the disease different mathematical models including [17, 18, 19, 20] have been presented recently. The studies focussed on different dynamics of the disease. For a new update about modeling in fractional calculus, we refer to [21, 22, 23, 24, 25, 26, 27, 15] .

Motivating from the work presented in [28] , where four control variables were used for the optimal control of COVID-19. We, in this study, formulate the mathematical model of the disease and discuss both the endemic equilibrium and disease free equilibrium of the model. Control strategies are designed based on sensitivity indexes of the transmission parameters. The results show that the conditions required for the globally asymptotically stable eradication of the disease do not hold. Furthermore, the endemic equilibrium of the disease is globally stable. Based on these results the study concludes that with the present resources the disease cannot be defeated, therefore humanity needs to learn how to live with corona. Our paper is organized as: section 1 gives the brief introduction of COVID-19 and the contributions of different researchers. In section 2 different stages of the disease and its mathematical modeling is discussed. Section 3 is concerned with analysis of the model and addresses J o u r n a l P r e -p r o o f Journal Pre-proof invariant region,well-posedness and reproduction number. Section 4 addresses threshold condition for global stability of disease free equilibrium. Section 5 focusses sensitivity indexes of the parameters, control strategies and the results of the strategies obtained with help of numerical simulations. In section 6, we address global stability of endemic equilibrium of the model and the conclusion.

To formulate the mathematical model of COVID-19, we consider different stages of the disease and accordingly divide the human population and surfaces in different compartments as shown below:

W i denotes surfaces/stuff contaminated with SARS-CoV-2.

S denotes the class of susceptible human population.

R denotes the Recovered class of human population. W s denote the surfaces or stuff that the human class can touch on daily basis. The following model (1) represents different stages of the disease:

The flowchart of COVID-19 is shown in figure (1) below. 

This section discusses three properties of the model; Invariant region, Disease-Free-Equilibrium (DFE) and the Basic Reproduction Number. 

The transition period at

The ratio of recovery of critical class 51%

The of ratio asymptomatic moving to vent bol 2% [31, 32] δ

The of ratio exposed moving to asymptomatic 75%

The transmission rate of infection from

The transmission rate of infection from stuff

The shedding coeffieient of

The shedding coeffieient of I 2 on W 0.5 [20] γ

The multiple of the transmissibility 0.5 [20] of

The ratio of symptomatic moving to vent bol 5%

[33] β 3 The Immunity loosing rate of recovered indivivduals 0 

Let N be the total population of humans and Z be the total density of stuff in human use. Adding all the classes of human population together and the compartments in stuff together, we haveṄ andŻ as under:Ṅ

Solving equations (2) and (3), we obtain the following results:

Above two inequalities shows that both the trajectories representing human population and the density of stuff, are forward bounded. Thus the following result is proved:

The following region Φ,

is +ve invariant domain. Furthermore the trajectories of human population and stuff density are bounded above. Hence the proposed model is well posed.

Communicable diseases spread generally by the contact of infected and non-infected individuals. How many susceptible individuals, an infectious individual do infect, is called R 0 or the disease reproduction rate and is find by next generation matrix [39, 36] . Let X I = (E, I 1 , I v , I 2 , W i ) T and X = (W s , R, S) T Following [45] , we find f (x) and v(x) and their derivatives at the disease-free equilibrium point as:

R 0 = the dominant eigenvalue of F V −1 and is given by;

J o u r n a l P r e -p r o o f Journal Pre-proof R 0 denote the transmission of the disease both from human to human and contaminated surfaces to human. The transmission from human to human is denoted by R H 0 and from contaminated surfaces to humans by R W 0 .

So,

In this section, we derive threshold condition for global stability for DFE (disease free equilibrium) of the system (1), using theorem 3.1 of [37] .

Let P = (S, W s , E, I 1 , I 2 , I v , W i , R) T be the class of state variables of the proposed model, P I = (W i , E, I 2 , I 1 , I v ) T be the class of infected population and contaminated surfaces, and P s = (S, W s , R, ) T be the class of susceptible and recovered human population, and noncontaminated stuff/surfaces. Theorem 4.1. Given the sub-system:

System (7) is GAS at the domain ∆, where ∆ = {P ∈ Φ; P I = 0, P s = 0, }.

Proof. : If P I = 0, the system;

reduces to the form: 

Here all the entries M (r,n) are −ve for r=n. Therefore the Disease Free equilibrium ( Γ h µ , 0, 0, 0, 0, 0, A ex+ε , 0) is stable globally.

The sub-system:Ṗ

is simply written as:Ṗ

Theorem 4.2. Given the sub-system (10). The matrix G I , as defined in (11), is metzler and irreducible, for all P ∈ Φ.

Proof. : Consider the system (11):Ṗ

For m=n the entries G (m,n) are −ve and non-negative for f or m = n. Hence G I (P) is metzler and irreducible for all P ∈ Φ.

Theorem 4.3. There always exist G I , for matrix G I as defined in equation (11) , such that:

Also

being dominant eigenvalue of G I . Where G I is the upper bound matrix of G I .

Proof. : Since S ≤ S 0 and W s ≤ W s 0 .

Then obviously the matrix G I (P), given below, is the upper bond of the matrix G I (P).

Simplifying the above matrix and putting S 0 = Γ h µ and W s 0 = A e X , we have:

The above upper bound matrix G I (P) is obtained if S = N. And the model attains the status of S = N at DFE point. Hence G I (P) is the upper bond matrix obtained at disease free equilibrium. The jacobian of the system (10) is

Matrix J I at disease free equilibrium is given by:

Clearly G I (P) is equal to the block of the Jacobian J I at the DFE. Which proves equations (13) and (12) .

To prove equation (14), we state the following theorem: 

Where is given by:

.

And

Proof. : Let M, N, O and P be the decomposition of the matrix G I , so that;

We need to show that G I is stable. For this we show that P − OM −1 N and M are stable.

Since for i=j all the entries M (i,j) are negative. Therefore all eigenvalues of M are −ve and the entries M (i,j) 0 for i = j. Hence M is metzler stable.

To show that G I is stable, we need to prove that D = P − OM −1 N is stable. Applying Routh-Hurwitz criteria [42] , we obtain the following inequality:

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

⇒ D is stable only if < 1 M and D are stable and thus we have proved that

In the above discussion we have proved all the five assumptions of global stability of disease free equilibrium, so we claim the following theorem:

Theorem 4.5. : Disease free equilibrium of COVID-19 is globally stable subject to the condition < 1, where

In theorem (4.5), it is concluded that the disease free state obtained with the help of different interventions, is globally stable subject to the condition < 1. We, therefore, first need to decide the interventions and control strategies. In this section we discuss The sensitivity/role of different parameters in the transmission of COVID-19, and decide intervention.

Formulation of control strategies based on decisive parameters.

The results of control strategies with help of numerical simulations.

The change observed in the dependent variable R by changing the value of independent parameter β is called sensitivity of R for parameter β, denoted by Υ β R and is given by:

The result obtained from above equation is called the sensitivity index of β. For further detail on sensitivity the reader is refer to [41, 28, 36] .

Using the above formula we have obtained the following table (2) The magnitude of the sensitivity index of a parameter represents the role of the parameter in the transmission of COVID. Highest the magnitude of sensitivity index of parameter, greatest the role of the parameter in disease transmission. However, some parameters have high sensitivity indexes yet we do not consider these parameters for intervention or decisive parameters. Because these parameters can not be addressed so simply, for example, human birth rate or mortality rate.

The sensitivity index of non of the parameters shown in the table (2) is zero. So all these parameters have got a role in the transmission of COVID-19. The value of the parameter is directly proportional to the rate of disease transmission, R 0 , if the sign of its sensitivity index is +ve and inversely proportional if the sign is −ve. For example, as shown in the table, k; the transition period of an individual at the stage I 1 , has got sensitivity index of -0.99. Since the index is negative, hence an increase in the value of the parameter will cause a decrease in the transmission rate R 0 . We select parameters κ, β 1 , β 2 , θ 1 , θ 2 , η 1 , ε, δ 1 , γ, e x fit for intervention or the decisive parameters.

We intervene in the above defined decisive parameters according to the available resources and formulate control strategies. Accordingly, the following 3 strategies are designed, as shown in the table (3) . Due to the high transmission rate of COVID-19, the increase in densities of infected classes is too rapid to be accommodated by the hospitals. For global stability of the disease free state the following two conditions must hold:

1: The densities of the infected classes must reduce to zero.

2: The value of must be less than 1, for the selected values of decisive parameters.

All the above figures show that the densities of the targeted infected classes have reduced to zero. Thus satisfying condition 1 of global stability. But the second condition of global stability, < 1, is not satisfied as shown in table (3). This shows that the disease-free state so obtained is not globally stable and hence the disease cannot be permanently eliminated with help of present tools of elimination. Since the disease is highly communicable it therefore appears again in the disease free zones. As such the disease attains the endemic mood. In the following section, we show that the endemic equilibrium is globally asymptotically stable. 

To check the global stability of endemic equilibrium we use the concept used in [43] .

Theorem 6.1. If R 0 > 1, the endemic equilibrium (S * , E * , I * 1 , I * 2 , I * v , R * , W * s , W * i ) is global asymptotically stable.

Proof. Consider the following sub-system of (1)   Ṡ = Γ h + β 3 R − (β 1 (I 1 + γI 2 )S + β 2 W i S) − µṠ E = (β 1 (I 1 + γI 2 )S + β 2 W i S) − η 1 E − µĖ I 1 = (1 − δ)η 1 E − (k + µ)I 1 (16) The jacobian matrix of system (16) is

and its second additive compound matrix is

Where m = (β 1 (I 1 + γI 2 ) + β 2 W + 2µ + η 1 ) n = β 1 (I 1 + γI 2 ) + β 2 W + 2µ + k

Which implies q ≤ −2µ < 0. Thus, based on theorem 3.5 of [43] , the endemic equilibrium is globally asymptotically stable.

In this work, we discussed different dynamics of the COVID-19 pandemic. The sensitivity test of the reproduction number shows that non of the parameters and particularly easily addressable parameters, have got a dominant role in disease transmission. Therefore the control of this disease is comparably tough but not disappointing, however, permanent eradication of the disease with present resources does not seem effective. What would be the long last effect of vaccination is still awaited. Our result shows that a disease-free state can be achieved but the state is not globally stable. This means that reinfection and new outbreaks of the disease in the community may happen time and again. The endemic equilibrium of the disease is globally stable. As a conclusion, it is recommended that the public shall be given awareness about 'how to live with corona using non-pharmaceutical approach'. For future work it is recommended to address the effect of vaccination on the global stability of disease free state.

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