Mathematical Epidemiology Model Analysis on the Dynamics of COVID-19 Pandemic

In the present work, Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) mathematical model for COVID-19 Pandemic is formulated and analyzed. The positivity, boundedness, and existence of the solutions of the model are proved. The Disease-free equilibrium point and endemic equilibrium points are identified. Local Stability of diseasefree Equilibrium point is checked with the help of Next generation matrix. Global stability of endemic equilibrium point is proved using the Concept of Liapunove function. The basic reproduction number for Novel Corona virus pandemic is computed as R0 = (αβΛ) ⁄ [(δ + μ) (β + δ + μ) (γ + δ + μ)] which depend on six different parameters. It is observed that if basic reproduction number is less than one, then number of cases decrease over time and eventually the disease dies out, and if the basic reproduction number is equals to one, then number of cases are stable. On the other hand, if the basic reproduction number is greater than one, then the number of cases increase over time gets worth. Sensitivity analysis of the basic reproduction number is done with respect to each parameter. It is observed that only some parameters Λ, α, β have high impact on the basic reproduction number. Consequently, with real data on the parameter it is helpful to predict the disease persistence or decline in the present situation. Lastly, numerical simulations are given using DEDiscover software to illustrate analytical results.

Severe acute respiratory syndrome corona virus 2 (SARS-Cov-2), commonly known as Novel Corona virus (nCoV), is a single, positive-stranded, RNA virus that belongs to Nidoviral type, which are responsible for the Current covid-19 Pandemic [4]. A species that host corona virus is known to be bat. Recent research shows that SARS-CoV-2 virus and bat corona virus have 96% identical genetic sequences [1,2].
The novel corona virus (nCoV) or COVID-19 may show signs of fever, cough, breathing difficulties, organ failures or even death of whole society. [1,5]. It can be transmitted from person to person even before any actual signs appeared, which is difficult to prevent and control [5]. Researchers all round the world have been trying to know how the virus spreads and find out effective ways control the outbreak. Compared the reproduction number R 0 of severe acute respiratory syndrome of H1N1 virus (1.25), SARS (2.2-3.6), the R 0 of COVID-19 indicates awful potential transmission as 2.2, 3.8 and 2.68 by different researcher in the world. WHO published an estimated R 0 of COVIDh-19 is (1.4-2.5). The larger basic reproduction number R 0 the greater power of transmission rate of disease and the smaller the basic reproduction number R 0 the lower the transmission rate of the disease [1,4,6] According to WHO report, the virus that causes COVID-19 is mainly transmitted through droplets generated when an infected person coughs, sneezes, or speaks. These droplets are too heavy to hang in the air. They quickly fall on floors or surfaces. You can be infected by breathing in the virus if you are within 1 meter of a person who has COVID-19, or by touching a contaminated surface and then touching your eyes, nose or mouth before washing your hands. There is no specific medicine to prevent or treat corona virus disease . People may need supportive care to help them breathe. If you have mild symptoms, stay at home until you have recovered. You can relieve your symptoms if you:(i) rest and sleep, (ii) keep warm, (iii) drink plenty of liquids, and (iv) use a room humidifier or take a hot shower to help ease a sore throat and cough. People with COVID-19 develop signs and symptoms, including mild respiratory symptoms and fever, on an average of 5-6 days after infection (mean incubation period 5-6 days, range 1-14 days).
According to WHO report, most people infected with the COVID-19 virus will experience mild to moderate respiratory illness and recover without requiring special treatment. Older people, and those with underlying medical problems like cardiovascular disease, diabetes, chronic respiratory disease, and cancer are more likely to develop serious illness.
Novel Therefore it is urgent to study and provide more scientific information for a better understanding of the novel corona virus (nCoV) or Covid-19. Thus susceptible-Exposed-infectious-recoveredsusceptible (SEIRS) model is adopted to estimate the dynamics and the potential spread based on the current data of cases, to calculate the basic reproduction number R 0 under different scenarios of the epidemics and to draw preliminary conclusions about the effectiveness of public health measures like hygiene, Masks.
The main purpose of this article is to formulate and to made Mathematical model analysis that describes the disease transmission dynamics of COVID-19 based on different literature reviews. The paper will create better understanding of the current corona virus pandemic. In [2,4] SEIR epidemic model of a data-driven analysis is done, and some parametric estimation is computed based on curve fitting and numerical methods. In [2] SIRS model of COVID-19 is constructed in the case of Indian country. This paper is organized as follows: In section 2, Mathematical model formulation Model assumptions, description of variables and parameters, Model diagram and Model equations are presented. In section 3, Mathematical Analysis of Model: positivity, Boundedness, and existence of solution, Equilibrium points are Discussed. In section 4, Stability Analysis of Equilibrium points; Next Generation matrix, Local Stability of disease free equilibrium point (LSDFEP), Global Stability of endemic equilibrium point (GSEEP), Basic Reproduction number will be presented. In Section 5, Simulation Study of our model equations are performed with initial conditions given for the variables and some values are assigned for the parameters. The results and discussion are given in section 6. In Section 7, Conclusions and Recommendations are drawn depend on the stability analysis and simulation study.

Mathematical Model Formulation
In the present study SEIRS model of COVID-19 is Constructed as follows. The total populations are divided into four classes: (i) Susceptible class denoted by Contains population which are capable of becoming infected (ii) Exposed class denoted by consists of populations being infected but not infectious and waiting for a short period time is called latency period.(iii) Infected class denoted by consists of population which are infected with COVID-19 and are also infectious and (iv) Recovered class denoted by consists of recovered class from infectious disease COVID-19. Mathematical SEIRS model of COVID-19 is formulated based on the following hypotheses so as to predict the past or future dynamics of COVID-19 progression and transmission Dynamics in the world.
(i) The size of total population is assumed to be constant, = + + + (ii) Both the number of births and death are may not be equal and populations are well mixed. (iii) Susceptible class are recruited into the compartment at a constant rate Λ (iv) The Exposed class has short incubation period and not yet infective but moved to infective class at rate (v) Susceptible class are infected when they come into contact with COVID-19 patient and the disease transmitted according to bilinear interaction rate where, = which is force of infection. (vi) Recovered class revert to the susceptible class after losing their immunity at a rate (vii) All types of population suffer natural mortality at a rate . (viii) All types of population suffer die due to  Pandemic at a rate (ix) All parameters in the model are positive.  Having the above assumptions, variables, and parameters the model diagram can be given as in Figure 1  Based on the assumptions, the notations of variables, parameters, and Model diagram, system of ordinary differential equations are formulated as follows: with initial conditions, (0) > 0, (0) ≥ 0, (0) ≥ 0, (0) ≥ 0 , ( ) = ( ) which is force of infection

Mathematical Model Analysis
In this section mathematical model analysis part is Presented. The analysis consists of the following features:

Positivity, Boundedness, and Existence of Solution
In order to show that the model is biologically valid, it is required to prove that the solutions of the system of ordinary differential equations (1)-(4) are positive and bounded for all time [7].
Theorem 1 (Positivity) Solutions of the model equations (1)-(4) together with the initial conditions , can be expressed without loss of generality, after eliminating the positive terms ( Λ + ) which are appearing on the right hand side, as an inequality as ⁄ ≥ −( + + ) . Using variables separable method and on applying integration, the solution of the foregoing differentially inequality can be obtained as ( ) ≥ $ %(&'"("))* . Recall that an exponential function is always non-negative irrespective of the sign of the exponent, Hence, it can be concluded that ( ) ≥ 0.
Positivity ( ): The model equation (2) arranged as ⁄ = − − − can be expressed without loss of generality, after eliminating the positive term ( ) which are appearing on the right hand side, as an inequality as ⁄ ≥ −( + + ) . Using variables separable method and on applying integration, the solution of the foregoing differentially inequality can be obtained as ( ) ≥ $ %(,"("))* . Recall that an exponential function is always non-negative irrespective of the sign of the exponent. Hence, it can be concluded that ( ) ≥ 0.
Positivity of ( ) : The model equation (3) arranged ⁄ = − − − can be expressed without loss of generality, after eliminating the positive term ( ) which are appearing on the right hand side, as an inequality as ⁄ ≥ −( + + ) . Using variables separable method and on applying integration, the solution of the foregoing differentially inequality can be obtained as ( ) ≥ $ %(-"("))* . Recall that an exponential function is always ( ) non-negative irrespective of the sign of the exponent. Hence, it can be concluded that ( ) ≥ 0.
Positivity of ( ) : The model equation (4) arranged ⁄ = − − − can be expressed without loss of generality, after eliminating the positive term ( ) which are appearing on the right hand side, as an inequality as ⁄ ≥ −( + + ) . Using variables separable method and on applying integration, the solution of the foregoing differentially inequality can be obtained as ( ) ≥ $ %(."("))* . Recall that an exponential function is always non-negative irrespective of the sign of the exponent. Hence, it can be concluded that ( ) ≥ 0. Thus, the model variable Proof: Recall that each population size is bounded if and only if the total population size is bounded. Hence, in the present case it is sufficient to prove that the total population size = ( ) + ( ) + ( ) + ( ) is bounded for all . It can be shown that all feasible solutions are uniformly bounded in a proper subset Ω ∈ ℝ " # where the feasible region Ω is given by Ω = 0( , , , ) ∈ ℝ " # ; N ≤ (Λ (δ + μ) ⁄ ) 6 It is clear that the derivative of total population with respect to time t is given by ⁄ = 7 ⁄ 8 + 7 ⁄ 8 + 7 ⁄ 8 + 7 ⁄ 8 . Then summation of all the four model equations (1)-(4) as follows: . This can be written as dN(t) dt is an upper bound of ( ). Hence, feasible solution of the system of model equations (1) Table 3 as follows: Table 3. Partial derivatives of functions with respect of model variables. To understand the dynamics of the system, it is necessary to identify equilibrium points of the model Equation.
An equilibrium solution is a steady state solution of the model equations (1)-(4) in the sense that if the system begins at such a state, it will remain there for all times. In other words, the population sizes remain unchanged and thus the rate of change for each population vanishes. Equilibrium points of the model are found, categorized, stability analysis is done and the results have been presented in the following sub-sections.

Disease Free Equilibrium Point
Disease free equilibrium points are steady state solutions where there is no disease in the population. In the absence of the disease this implies that ( ) = ( ) = 0 and the right hand side of the model is equal to zero.

Endemic Equilibrium Point
The endemic equilibrium point * 0 * , * , * , * 6 in the feasible region is a steady state solution where the disease persists in the population. The endemic equilibrium point is obtained by setting rates of changes of variables with respect to time in model equations (1)-(4) to zero. That is, setting ⁄ = ⁄ = ⁄ = ⁄ = 0 the model equations can be written as the system of non linear equations Where, O = + , P = + + , Q = + + , = + + solving equations (7) and (8) will give expression for and in terms of variable as follows: = ( ⁄ ) = ( ⁄ )( Q ⁄ ) , This expression could be re-written as Now substitute (9) and (10) into (6) so as to solve which results ( Q ⁄ ) − P = 0 This can be arranged 7 ( Q ⁄ ) − P8 = 0 However, does not vanish, since the disease is assumed endemic and it is a computation of non zero equilibrium point of the system. Thus the only meaningful solution ( Q ⁄ ) − P = 0, then solution is given by the expression * = 7PQ8 7 8 ⁄ Then substituting equations (9), (10) and (11)  Finally, substitution of * in (10) and (11)

Basic Reproductive Number
The basic reproduction number represent the average number of new infections generated by each infected person [7][8][9]. The higher value of R the speedy the disease transmission rate and the Smaller values of R the slower the disease transmission rate [12]. There are three options for the values of R (i) R < 1 means the number of new cases will decrease over time and eventually the outbreak will end on its own.(ii) R = 1means the cases are stable. (iii) R > 1means the outbreak is self-sustaining unless effective control measures are implemented. According to WHO Determining factors of basic reproduction number R : (i) infectious periodhow long the infection is contagious, for instance flu typically up to 8 days and in children up to 2 weeks. The longer an infection is contagious for the higher the reproduction number.(ii) Contact rate-how many people an infected person comes into contact with R will be lower if a person stays at home, higher if they are out and about. (iii) Mode of transmission (shedding potential): rapid speed transmission, if the disease transmitted by airborne, flu or measles, no physical contact or fomite necessary, and Slower transmission, the disease transmitted by body fluids, Ebola, Hepatitis B, C, or HIV. [1,5,9] In the discussion of disease transmission if 20% of infected individuals are responsible for 80% of transmissions those spreaders is called Super-spreaders and if 80% of infected individuals are responsible for 20% of transmissions are called mini-spreaders. [1,5] To

Stability Analysis of the Disease Free Equilibrium
In absence of the infectious disease, the model equations have a unique disease free steady state U .
It is shown that DFEP of model (1)-(4) is given by U = 07Λ ( + ) ⁄ 8, 0, 0, 06. Now local stability analysis of DFEP is presented in the following theorem and proved with the help of next generation matrix. Where, O = + , P = + + , Q = + + , = + + Now the Next generation matrix of functions (;, X, ℎ, Z) with respect to ( , , , ) is given by Therefore the Next generation matrix [ of model at the disease free equilibrium U reduces to Then the eigen values of [( U ) are computed from characteristic equation Therefore, it is concluded that the LSDFEP U of the system of differential equations (1)-(4) is locally asymptotically stable due to all eigen value is negative.

Global Stability Analysis of Endemic Equilibrium Point
The Global stability Analysis of endemic equilibrium point * ( * , * , * , * ) is stated in Theorem 5 and proved by taking appropriate liapunove function. [7,8] Theorem Thus it is possible to set e < , e = , e > , e > are non negative integers such that dL dt ⁄ ≤ 0 and endemic equilibrium point is globally stable.

Numerical Simulation
In this section, the numerical simulation of model equations (1) [5] Using the parameter values given in Table 2 with different initial conditions in model equations (1)-(4) a simulation is done and the results are given in Figure 2-5.
From Figure 2, it is observed that when the number of exposed individual increases then the number of infected people will increase that leads to the decrease on the susceptible and recovered populations. Figure 3, shows all those class of population changing for some time and eventually the populations will be constant. Thus in any situations there will exist those class of populations and continue in stable state after some time in future.
From Figure 4, recovered and infected populations are inversely proportional. That is if the population gets greater the number of infection, then will be less the number of recovery may be due to lack of resources or management, and Similarly, from Figure 5, exposed and infected populations are directly proportional. If there is more infection in the community, there will be more exposed individual and the less infected community, the less the exposed community.

Sensitivity Analysis
Sensitivity analysis tells us how important each parameter is to disease transmission. It is used to discover parameters that have a high impact on U and should be targeted by intervention strategies. Sensitivity indices allow us to measure the relative change in a variable when a parameter changes. The normalized forward sensitivity index of a variable with respect to a parameter is the ratio of the relative change in the variable to the relative change in the parameter. If the result is negative, then the relationship between the parameters and U is inversely proportional. In this case, we will take the modulus of the sensitivity index so that we can deduce the size of the effect of changing that parameter. On the other hand, a positive sensitivity index means an increase in the value of a parameter.
A highly sensitive parameter should be carefully estimated, because a small variation in that parameter will lead to large quantitative changes. An insensitive parameter, on the other hand, does not require as much effort to estimate, since a small variation in that parameter will not produce large changes to the quantity of interest [9,11,13] The explicit expression of U is given by . Since U depends only on six parameters, we derive an analytical expression for its sensitivity to each parameter using the normalized forward sensitivity index as follows:  Table 5, It is to be noted that the parameters Λ, , are positive and hence play a vital role in controlling the stability aspects of the system, and the remaining parameters are negative and hence have not big influence on the system.

Result and Discussion
A mathematical SEIRS model of COVID-19 was Conducted and the sensitivity indices of the basic reproduction number U is Computed to determine the relative importance of the model parameters in the disease transmission. This information leads us to identify the influence of each model parameter in the basic reproduction number. Consequently, it is helpful to know and predict the disease progress, persistent or decline in the past, present and future.
This mathematical model analysis may provide critical information for decision makers and public health officials, who may have to deal with infectious disease of COVID-19 pandemic. I hope that the paper will have a great impact on citizens of the whole world to concentrate on prevention and control of COVID-19 pandemic. Such contribution is interesting regarding to COVID-19, which causes a large disruption in the lives of sufferers and has enormous socioeconomic costs occurred in the world.
Almost all latest research recommend the following protective measures: hygiene, masks, physical distancing, staying indoors, boosting immune system, raising awareness through networking (like face book, and twitter), boosting moral of front-line workers such as (medical practitioners, nursing staff, cleaning staff, health care centers) who are interacting directly with the patients.

Conclusion
In this Paper, SEIRS mathematical model describing the dynamics of COVID-19 is formulated and analyzed. The model is developed based on biologically reasonable assumptions. The mathematical analysis has shown that if basic reproduction number is less than one, then number of cases decrease over time and eventually the disease die out, and if the basic reproduction number is equals to one, then cases are stable. On the other hand, if the basic reproduction number is greater than one then the number of cases increase over time gets worth, and the disease continue to spread more rapidly.
Moreover, existence, positivity and boundedness of the solution of the model is shown to clarify the model is biologically meaningful and mathematically well posed. Stability analysis of the model is checked with help of next generation matrix and global stability are proved by taking appropriate liapunove function.