Mathematical Modeling of COVID-19 Pandemic with Treatment

Subject Areas : International Journal of Mathematical Modelling & Computations

Abayneh Bezabih ^{1 *}

1 - Hawassa Meryjoy site near to silasie church

Received: 2020-09-22 Accepted : 2021-04-30 Published : 2021-06-20

Keywords: Simulation, Stability Analysis, model, COVID-19, Reproduction Number, Pandemic, Next Generation matrix,

Abstract :

In this paper, mathematical model of COVID-19 Pandemic is discussed. The positivity, boundedness, and existence of the solutions of the model equations are proved. The Disease-free & endemic equilibrium points are identified. Stability Analysis of the model is done with the concept of Next generation matrix. we investigated that DFEP of the model E_0 is locally asymptotically stable if α≤β+δ+μ & unstable if α>β+δ+μ . It is shown that if reproduction number is less than one, then COVID-19 cases will be reduced in the community. However, if reproduction number is greater than one, then covid-19 continue to persist in the Community. Lastly, numerical simulations are done with DEDiscover 2.6.4. software. It is observed that with Constant treatment, increase or decrease contact rate among persons leads great variation on the basic reproduction number which is directly implies that infection rate plays a vital role on decline or persistence of COVID-19 pandemic.

Full-Text:



Mathematical Modeling of COVID-19 Pandemic with Treatment 

 

 

Abstract :In this paper, mathematical model of COVID-19 Pandemic is discussed. The positivity, boundedness, and existence of the solutions of the model equations are proved. The Disease-free & endemic equilibrium points are identified. Stability Analysis of the model is done with the concept of Next generation matrix. we investigated that DFEP of the model   is locally asymptotically stable if  & unstable if  . It is shown that if reproduction number is less than one, then COVID-19 cases will be reduced in the community. However, if reproduction number is greater than one, then covid-19 continue to persist in the Community. Lastly, numerical simulations are done with DEDiscover 2.6.4. software. It is observed that with Constant treatment, increase or decrease contact rate among persons leads great variation on the basic reproduction number which is directly implies that infection rate plays a vital role on decline or persistence of COVID-19 pandemic.

 

 

Keywords: COVID-19 , Pandemic , Model, Stability Analysis, Next Generation matrix, Reproduction number, Simulation .

 

 

                                            

Index to information contained in this paper

1.       Introduction 2.       Model formulation and Assumptions 3.       Mathematical Analysis of the Model 4.       Equilibrium Points 5.       Stability Analysis of Equilibrium Points 6.       Basic Reproduction Number 7.       Numerical Simulation 8.       Conclusion and recommendation

 

 

 

 

 

 

 

 

1.       Introduction

Mathematical modeling is an important tool to understand and analyze real world problems, for instance modeling infectious disease transmission dynamics in human and animals. Infectious disease is one of the most important factors in creating illness and even death in hundreds thousands people all over the world. In particular COVID‐19 (Corona virus Disease 2019) is a family of RNA Beta virus in Nidoviral order. This beta virus is medium size Viruses enveloping a positive -stranded RNA which Contain very large viral RNA genome. Corona prefix Comes from Latin word for Crown named for "crown-like" appearance of virus. This virus originated in Wuhan Huanan seafood wholesale market that contain aquatic products, and some wild animals.[1-3]

         The novel corona virus 2019 (nCoV-2019) or COVID‐19 is Started at Wuhan, Hubei province of china in 2019.It is the seventh Corona virus found to cause illness in humans. Some researchers believed that the virus transmitted from either snakes to humans or from bats to humans. There is animal market in Wuhan that seems to be center of this out break and it is suggested that there was exposure to live and dead animals. Right now, peoples are suspecting bats are sources of COVID-19 .[1,3,5] Corona virus infect birds and mammals. Bats are hosts to the large number of viral genotype of corona virus. epidemic occur when viruses transmit from one species to another species. The species that hosts the severe acute respiratory syndrome corona virus 2(SARS‐CoV‐2) is probably bat, containing 96% identical at the whole‐genome sequence level.[1,2,5]

         severe acute respiratory syndrome (SARS) is a beta corona virus that occur in Guangdong province of china 2003. Initially the virus transmitted from bats to civets to humans . Due to SARS illness, more than 8000 total cases,774 deaths, and approximately 9.6% fatality rate are recorded. Middle East respiratory Syndrome(MERS) is beta Corona virus that Started at Saudi Arabia in 2012. This virus transmitted from Camels to humans through eating Camels meat, drinking camels milk , exposure to camels. In this epidemic more than 2400 Cases,858 deaths, and around 34.4% of fatality rate are recorded. Novel corona virus 2019 may include signs of fever, cough, shortness of breath and general breathing difficulties, organ failures or even death, posing a severe threat to the whole society. It can be transmitted from person to person even before any actual signs appeared, which is difficult to prevent and control[4,5,7]. Researchers all around the world have been trying to know how the virus spreads and find out the effective ways to put this outbreak quickly under control. Compared the reproduction number R0 of SARS 2.2 to 3.6 , the R0 of COVID‐19 shows awful transmission as 2.2, 3.8 and 2.6 by different research in the world. WHO published an estimated R0 of COVID-19 is 1.4 to 2.5 . The larger the R0 the higher power the transmission rate[1-5].

         There is no specific medicine to prevent or treat corona virus disease (COVID-19). 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[6,7,16] 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 are more likely to develop serious illness[7,8,16]. Novel Corona virus (2019-nCoV), Situation Report - 1, Initially the Disease spread in china, republic of Korea, and Thailand and It was 282 number of confirmed cases reported globally up to 21 January 2020. Corona virus disease 2019 (COVID-19) Situation Report – 94, shows that there are 2,695,418 peoples are infected and 188,804 peoples are died and 739,871 peoples are recovered from Corona virus disease(Covid-19) pandemic up to 23 April 2020 in which this paper is organized.[16]

         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‐infectious-Treated‐recovered (SITR) model is adopted to understand the transmission dynamics or potential spread of COVID-19 based on the current data. The basic reproduction number R0  of the COVID-19 pandemic will be computed for different infection rates and conclusions drawn depending on the value of the reproduction number .

        The paper Contain the following sections: In section 2, Mathematical model formulation: Model assumptions, description of variables and parameters, Model diagram and Model equations are presented. In section 3,Mathematical Model Analysis: positivity, Boundedness, and existence of solution, Equilibrium points are Discussed. In section 4, Stability Analysis of Equilibrium points ; Next Generation matrix, Local Stability of DFEP, Global Stability of endemic equilibrium point, 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. results and discussion are presented in section 6. In Section 7, Conclusions and Recommendations are drawn depend on the stability analysis and simulation study.

2.        Model formulation and Assumptions

In this paper mathematical model of COVID-19 with treatment will be discussed. The total populations are divided into four compartments: (i) Susceptible Compartment denoted by  consists persons which are capable of becoming infected (ii) Infected compartment denoted by  consists of persons which are infected with COVID-19 and are also infectious (iii) Treatment compartment denoted by  consists of persons being treated and (iv) Recovered compartment denoted by  consists of recovered persons from COVID-19. A system of differential equation is formulated based on the following assumptions.

(i)       Suppose total populations is constant .

(ii)       The number of births and death may not be equal, the population is well mixed , immigration and migration is not considered in the model.

(iii)       Susceptible persons are recruited into the compartment  at constant rate

(iv)        Susceptible persons will be infected, if they come into contact to infective & transmitted at rate

(v)       The infected persons join treatment compartment and treated at rate .

(vi)        The treated persons join recovery compartment at rate  .

(vii)       Recovered persons revert to the susceptible person after losing their immunity at rate.

(viii)       All types of persons suffer natural mortality at rate.

(ix)       Infected persons die due to COVID-19 Pandemic at rate .

(x)       Assume that all parameters are positive.

Table 1 Notations and description of model variables

 

Table 2 Notations and description of model parameters

Variables	Descriptions
	Population size of susceptible person
	Population size of infected & infectious person
	Population size of under treatment person
	Population size of recovered person

 

 Having the above assumptions the model diagram shown in Figure 1.

 

Based on the assumptions and Model diagram , the following system of differential equations are obtained .

                                                         (1)

                               (2)

                                                         (3)

                                                                          (4)

with initial conditions. 

 

3.       Mathematical Analysis of the Model  

3.1        Primary Results[Positivity , boundedness, and existence of solution of the model]

 To show the formulated model is meaningful and valid, it is necessary to prove that the solutions of the system of differential equations (1) – (4) are positive , bounded for all time. This properties of the model can be considered as primary results.

Theorem 1 [Positivity] Solutions of the mathematical model (1) – (4) are always positive with initial conditions  .That is the model variables  and are positive for all  and will remain in.

Proof: Positivity is shown separately for each variables  and.

 Positivity of:  Equation (1) given by  can be expressed without loss of generality, eliminating the positive terms  , as an inequality as  . Using separable of variable 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.

 positivity of: Equation (2) 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. Hence, it can be concluded that.

 Positivity of:  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.  

 Positivity of:  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.

Thus variables  and   represent persons at different compartment which are positive quantities and will remain in  for all time.[9,10,13,16]

Theorem 2 [Boundedness] The positive solutions of mathematical models (1) – (4) are bounded. That is variables  &  are bounded for all.

Proof:  each population size of variable is bounded if and only if the total population size N is bounded. Hence, it is sufficient to show that total population size   is bounded for all. This can be depicted that all feasible solutions are uniformly bounded in a proper subset  where the feasible region  is given by.

It is trivial that   Then adding all four equations (1) – (4) results which implies  ). after eliminating negative term    results inequality . Equivalently this inequality can be expressed as   solving this differential equation yields  general solution. But the term  represents initial values of variable  at . Thus, particular solution can be written as. Furthermore,  as . Thus, the total population size of  takes off from the value  at  and ends up with bounded value  as time goes to infinity. hence  is bounded as . Hence, feasible solution of the system of differential equations (1) – (4) remains in the region  which is positively invariant set. Thus, the given mathematical model is meaningful and mathematically well posed in the domain of region. hence, variables  &  are bounded for all. [9,10,13,16]

 

Theorem 3 [Existence] Solutions of differential equations (1) – (4) exist in with the initial conditions . That is variables  &   exist for all  and remain in.

Proof: Define the differential equation (1) – (4) as follows:

 

 

 

 

 

 

Suppose   represent a region. Then According to Derrick & Groosman theorem, the differential equations (1) – (4) have a unique solution if  are continuous & bounded in a region for     Here  , Continuity and the boundedness can be verified as follows:

 

Table 3. Partial derivatives of model with respect to variables

Parameters	Descriptions
	Recruitment rate of susceptible person. new susceptible person will Recruited & enter into susceptible compartment at this rate.
	Infection rate. susceptible persons become infected with covid-19 and transfer from compartment  to at this rate
	Treatment rate .With this rate infective persons at this rate and moves from compartment  to for treatment.
	Recovery rate(gain immunity rate) . With this rate treated class moves from compartment  to
	Loss immunity(re-infection rate). recovered persons re-infected with this rate and moves from compartment  to
	Mortality rate due to infection of Covid-19. Infected persons of compartment and infected persons under treatment of Compartment   are die.
	Natural death rate. With this rate all class of Compartment suffer natural death rate.

For:       .

For:

For:

For:           Thus, all the partial derivatives  exist, continuous and bounded in. Hence the solution of differential equations (1) – (4) exists and unique by Derrick & Groosman theorem.[9,10,14,16] 4.       Equilibrium points of the model 4.1.       Disease-free equilibrium points of the model  Disease-free equilibrium points are solutions of model equations where  due to no disease in the population and the right hand side of differential equation (1)-(4) is equal to zero. Thus     which results . hence, the disease-free equilibrium point of differential equation (1) – (4) is given by       4.2.        Positive or Endemic Equilibrium of the model   The endemic equilibrium point  is a positive equilibrium point solutions, where the disease persists in the population. The endemic equilibrium point is obtained by taking rates of changes of variables with respect to time is zero. hence  ,then differential equations(1)-(4) can be written as the system differential equations ;   Where . Then Equation (6)  arranged as which results or. but does not vanish due to the disease is assumed to exist in the system. Then the only meaningful solution of the system is solving for S yields                                                         (9) In Similarly fashion, solving equations (7) & (8) results expression for  &  in terms of variable  .                                                                                              (10)                                                       (11) Then plug equations (9),(10) & (11) into (5) results;                                  (12) plug the value of total population size  on equation (12) and solving for I, will yield expression for . hence the endemic equilibrium points are                                                          (14) Now substitute in (10) & (11) will result expressions for and interms of parameters.                                                         (15)                                                          (16) Hence, the positive equilibrium point is given by 5.       Stability Analysis of Equilibrium points of the model 5.1.                      Local Stability of Disease-free equilibrium point (LSDFEP) of the model   In this section the Local Stability of Disease-free equilibrium point (LSDFEP) of the model is established and proved as follows in theorem 4. Theorem 4 The differential equations (1) – (4) is locally asymptotically stable at DFEP  if  & unstable if              Proof : Define the differential equations (1) – (4) as follows;           , Where Now, Next generation matrix of the functions   with respect to variables  is        Therefore, the Next generation matrix evaluated at disease-free equilibrium point(DFEP)   yields   Then eigen values of the next generation matrix  will be computed from the roots of characteristic equation  . That is  0  0     Thus, the four eigen values of the next generation matrix are       plug value of  and  results,                          Hence disease-free equilibrium point (DFEP)  of the system of differential equations (1) – (4) is locally asymptotically stable if    & unstable if     which means the DFEP  of the system of differential equation is locally asymptotically stable if  & unstable if      5.2.                      Global Stability Analysis of Endemic Equilibrium Point(GSEEP) of the model The Global stability analysis of positive or endemic equilibrium point of the model    is established and proved as follows in theorem 5. Theorem 5 The endemic equilibrium point   is globally asymptotically stable. Proof: To prove the theorem take appropriate liapunove function[9,10,13,15,16].    Suppose that               (17) Then plug the differential equations (1) - (4) into (17)                                                                                          Take the variables   out of each bracket as follows        Take negative sign out from each bracket ,then observe resulting equations to complete the proof.       Thus, it is possible to conclude that , for non negative integers   and hence endemic equilibrium point is globally stable. 6.       Basic Reproduction Number The basic reproduction number denoted by  and defined as the expected number of people getting secondary infection among the whole susceptible population . This number shows a potential for spread of disease within a given population. When each infected individual produces on average less than one new infected individual so that the disease is expected to die out. On the other hand if  , then each individual produces more than one new infected individual so that the disease is expected to continue spreading in the population[11,12,14].                  The basic reproductive number  can be determined using the next generation matrix. In this method,   is defined as the largest eigen value of the next generation matrix. Constructing the next generation matrix involves classifying all compartments of differential equation(1)-(4) in to two classes: infected and non-infected class. Thus, basic reproduction number could not be found from structure of model equation alone but rather from definition of infected & uninfected compartments.         Assume that there are 𝑛 compartments in the model of which the first 𝑚 compartments are with infected individuals [3].  From the system (1) – (4) the first three equations are considered and decomposed into two groups; contains newly infected cases , V contains the remaining terms, and Let  be a column vector and the differential equations of the first three compartments are rewritten as           let. Here (i)  denote newly infected cases which arrive into the infected compartment; (ii) denotes newly infected cases arrived into the treated compartment, and (iii)  denotes newly infected case from susceptible compartment.  let. Here; , where parameters    .         The next step is the computation of the square matrices  and  of order  , where  is the number of infected classes, defined by  and  with  , such that  is non-negative,  is a non-singular matrix and  is the disease free equilibrium point DFE. If  is non-negative and  non-singular, then  is non-negative and thus  is also non-negative. The matrix   is called the next generation matrix for the model. The basic reproduction number  where  denotes the spectral radius of matrix  and the spectral radius is the biggest non-negative eigenvalue of the next generation matrix. The Jacobian matrices for and  at  can be Computed as      and  . The respective Jacobian Matrix of  and  at the disease free equilibrium point  takes the form      and  . Clearly  is non-zero. Thus Matrix is invertible and its inverse matrix exists. So after computation of its inverse matrix  results;   The product of  matrices  and can be given by    Recall that the basic reproduction number(threshold value) is the largest eigen value in spectral radius matrix . Thus, eigen values of spectral radius Matrix  are determined from the roots of characteristic polynomial equation   which implies   Hence basic reproduction number is    7.       Numerical Simulation  In this section, the numerical simulation of model equations (1) – (4) is done , using the software DEDiscover 2.6.4. For Simulation purpose, Model equations are arranged and a set of meaningful values are assigned to the model parameters. These sets of parametric values are given in Tables 3 dS/dt=Lambda+Rho*R-(Alpha*S*I)/N-Mu*S // susceptible class dI/dt=(Alpha*S*I)/N-(Beta+Delta+Mu)*I // Infected class dT/dt=Alpha*I-(Gamma+Delta+Mu)*T // Population Class under Treatment dR/dt=Gamma*T-(Delta+Mu)*R // Recovered class Table 3 Parameter values   Using the parameter values given in Table 2 and the initial conditions   in the model equations (1) – (4) a simulation study is conducted and the results are given in Figure 2 - 4   Figure 2: Time series plot for COVID-19 pandemic at  with From Fig. 2, it is observed that the number of infected individuals looks like exponential curve which tell us Covid-19 expanded rapidly in the community. This phenomenon supplemented by the basic reproduction number . Thus COVID-19 pandemic will persist in the community for some period of time. The remaining population gradually decrease in number as shown above.   Figure 3: Time series plot for COVID-19 pandemic at  with   Figure 4: Time series plot for covid-19 pandemic at  with From Fig. 3 and Fig. 4, one can observe that the number of infected individual look like logistic curve, meaning the disease continue to spread in the community for some period of time. These phenomenon stable or die out in some time in the future which is supplemented by the basic reproduction number   in fig 3 From the time series plot in fig4,  that shows the disease dies out or under control. 8.       Conclusion and recommendation In this paper, a mathematical model of COVID-19 pandemic with treatment given to the infected individuals has been developed based on reasonable assumptions. Moreover, positivity , boundedness ,and existence of solution of the model is verified or proved to clarify the model is meaningful and mathematically well posed. Existence of equilibrium points of the model were identified. Local Stability analysis of disease-free equilibrium point(DFEP) is proved with the concept of next generation matrix and global stability analysis of endemic equilibrium point is done by taking appropriate liapunove function.         In this study, the basic reproduction numbers were computed as   for different infection rate   respectively with constant treatment. For those basic reproduction numbers  , the numbers of cases of COVID-19 pandemic increase rapidly and will persist in the community. For those basic reproduction numbers  , COVID-19 pandemic become slow or weak in spreading and gradually the disease may dies out.The smaller infection rate leads to smaller reproduction number. Hence it is recommend to reduce infection rates by Masks , gloves , wash and sanitize hands . Moreover, both Health and government officials have to create awareness and apply proper interventions about COVID-19 pandemic and its transmission so as to reduce COVID-19 pandemic.     References [1]       Yaqing Fang, Yiting Nie, Marshare Penny. Transmission dynamics of the COVID‐19 outbreak and effectiveness of government interventions: A data‐driven analysis. Journal of medical virology, Wiley. march 2020 [2]       Chayu Yang and Jin Wang,March2020. A mathematical model for the novel corona virus epidemic in Wuhan, China. Mathematical biosciences and engineering 17(3): march (2020) 2708–2724. [3]       Alexander Okhuese Victor. mathematical predictions for covid-19 as a global pandemic. Research Gate. march 2020 [4]       Jyoti Bhola, Vandana Revathi Venkateswaran and Monika Koul . Corona Epidemic in Indian context: Predictive Mathematical Modeling, medRxiv preprint doi: https://doi.org/10.1101/2020.04.03.20047175 [5]       Binti Hamzah FA, Lau C, Nazri H, Ligot DV, Lee G, Tan CL, et al. Corona Tracker: World-wide COVID-19 Outbreak Data Analysis and Prediction. Bull World Health Organ. E-pub: 19 March 2020. doi: http://dx.doi.org/10.2471/BLT.20.255695 [6]       website: https://www.who.int › health-topics › corona virus [7]       website: https://www.who.int › docs › default-source › who-china-joint [8]         [9]         [10]         [11]         [12]         [13]         [14]         [15]         [16]