Mathematical Modelling of COVID-19 Pandemic with Demographic Effects

In this paper, an asymptomatic infection transmission Susceptible-Exposed-Infectious-Recovered (SEIR) model with demographic effects is used to understand the dynamics of the COVID-19 pandemics. We calculate the basic reproduction number ( 𝑅 0 ) and prove the global stability of the model by solving the differential equations of the model using the disease-free equilibrium (DFE) and endemic equilibrium (EE) equations, respectively. We showed that when 𝑅 0 < 1 or 𝑅 0 ≤ 1 and 𝑅 0 > 1 or 𝑅 0 ≥ 1 the DFE and EE asymptotic stability exist theoretically and numerically respectively. We also demonstrate the detrimental impact of the direct and asymptomatic infections for the COVID-19 pandemic.


Introduction
The COVID-19 is a novel flu infection belong to the Coronaviruses family that causes illness ranging from a common cold to severe illness in humans like the Middle East Respiratory Syndrome (MERS) and Severe Acute Respiratory Syndrome (SARS) in adults and children [1,2]. The COVID-19 started in the city of Wuhan, Hubei Province, China, in 2019 and has spread to all parts of the world, affecting 213 countries and territories [3]. It is the third coronavirus species to infect human populations in the past two decades [4][5][6]. As of 10 June 2020, there have been global confirmed cases of 7,145,539, and 408,025 resulted in deaths [3]. Symptoms of the virus are fever, cough, shortness of breath, fatigue, body aches, headache, the loss of taste or smell, sore throat, congestion or runny nose, nausea or vomiting, and diarrhea [7]. Close contact and respiratory droplets within 6 feet (1.8 m) approximately are the most common primary ways of transmission [8].
However, people who contracted the virus can take between 2 to 14 days before signs and symptoms manifested [7], and within this periods they can infect others [9,10]. That is an asymptomatic individuals can infect an uninfected person. Although most of those infected get cured, there is currently no vaccine or specific antiviral therapy to prevent contacting the virus. In a mild case, usual flu treatments like antibiotic drugs are used, and in severe cases, supportive treatment like a breathing machine is given to protect vital functions of the organs. The virus infected all ages of humans, but the higher risk is more on adult individuals with severe illness relating to respiratory diseases, organ diseases, and blood diseases [7].
The basic reproduction number ( 0 ) is a critical threshold quantity associated with viral transmissibility, and it has been used to understand the transmission of the COVID-19.
Epidemiological 0 is an equation used to describe the contagiousness of the pathogen. It determined the number of people on average that would be infected from a case introduced into a population. The initial COVID-19 pandemic 0 , according to the World Health Organization (WHO) was estimated to be between ranges of 1.4 to 2.5 [6]. That is, one infected person will infect an average of 2 persons in his/her lifetime. Also, in other studies of the epidemic, Zhao et al. [11] estimated the average 0 for COVID-19, from 3.3 to 5.5, and Read et al. [12], estimated to range between 3.6 and 4.0.
Stability analysis which has a direct relationship with 0 is also another way to understand infectious disease. It is believed that when 0 is above unity, the disease will persist, and the stability is endemic, and when 0 less than unity, the disease will die out, and the stability is disease-free. The analysis is done by partition the state of individuals in the population into different compartments. For instance, since COVID-19 have an incubation period, the population can be divided into those who are susceptible (S) to the virus, those who are exposed (E) to the virus, those who are infected (I) with the virus and those recovered (R) from the illness. The SEIR is interpreted using differential equations, where calculus and simple algebraic methods are used to study the dynamic of the disease.
However, during these approaches the 0 is calculated directly from the differential equations model at the state when the disease is free from the population.
In this study, a deterministic four compartments SEIR model with an asymptomatic infection transmission is used to inspect the stability of the COVID-19 pandemic using differential equations techniques. This is done by formulating four nonlinear differential equations and provides a theoretical and numerical analysis of the model. Our results show that, theoretically, the disease-free and endemic equilibria of the model locally and globally asymptotically stable and the direct and symptomatic infection transmissions are detrimental for the COVID-19 pandemic.

Model framework
In this section, we describe an epidemic transmission SEIR model with demographic changes. The model is used in epidemiology to compute the amount of susceptible, exposed (infected), infectious, recovered people in a population (N). This model is used under the following assumptions: ✓ The population is constant but large.
✓ The only way a person can leave the susceptible state (S) is to become infected either from the exposed (E) or infectious (I) state.
✓ The only way a person can leave the E state is to show signs and symptoms of the illness or die of natural death.
✓ The only way a person can leave the I state is to recover from the disease or die from natural death or die as a result of the disease.
✓ A person who recovered (R) from the illness received permanent immunity.
✓ Age, sex, social status, and race do not affect the probability of being infected.
✓ The member of the population has the same contacts with one another equally.
✓ All births are into the susceptible state, and it is assumed that the birth and natural death rates are equal.
The transmission is measure at ( + )⁄ , where is the direct transmission rate, and is the probability of getting infected when an uninfected individual comes into contact with an individual from state E. We assume natural birth and death rate to be measure at an equal rate and the disease induced death rate measure at . The rate for an individual to move from state E to state I is measured at rate , and the rate of recovery is measure at . Figure 1 represents the SEIR model which is described using the system of nonlinear ordinary differential equations Where ( ) = , ( ) = , ( ) = and ( ) = , denote the number of susceptible, exposed, infectious, and remove individuals at time , respectively, and = + + + . System (1) is subjected to the initial condition For simplicity system (1) is reduced to a proportional framework given as where = ⁄ , = , ⁄ = ⁄ , and = ⁄ . By considering the total population

Positivity of the solution
Assume that system (4) has a global solution corresponding to non-negative initial conditions. Then the solution is non-negative at all times. The statement is confirmed by the following Lemma. Hence the positively invariant for the system (4) is

The equilibria of the model
There are two equilibrium points for the system (4), i.e., the disease-free equilibrium (DFE), the state when the disease is absence, and the endemic equilibrium (EE), which is the state when the disease continues to persist in the population. To understand the stability of the model we need an expression to estimate the basic reproduction number ( 0 ).
Adding the first two equations of (6) and substitute for 0 we get Because at the disease-free state no one have the infection then, 0 = 0. We can see that 0 = ( 0 , 0 , 0 ) = (1,0,0).

Stability analysis of the Disease-free equilibrium points
Theorem 1. If 0 < 1 and < ( + ) + (2 + + + ) the DFE is locally asymptotically stable in Ω. The proof of Theorem 1 is complete.

Proof.
To prove the global asymptotic stability (GAS) of the DFE, we construct the following Substituting 0 we get Thus, for ⁄ ≤ 0, 0 = 0, ℎ 0 ≤ 1. According to the Lasalle invariance principle [13], the DFE point is GAS.

Stability analysis of the endemic equilibrium
Theorem 2: If 0 > 1 the endemic equilibrium is locally asymptotically stable.

Proof.
To prove the LAS of the endemic equilibrium we consider the Jacobian matrix associated with * , that is * = ( Substituting for * , * and * we get * = From 0 we get Since > 0, > 0, > 0, and − > 0, according to the Routh-Hurwitz criterion, the endemic equilibrium of system (4) is LAS.

Theorem 3.
If 0 ≥ 1 the endemic equilibrium point is globally asymptotically stable in Ω.

Numerical simulations
In this section, we illustrate the DFE and EE theorems numerically using the integration technique in R-software. The model parameter values are obtained from COVID-19 literature, and we focus our analysis in a small settlement approximately 1000 population. Using data of 10 June 2020, we estimate the global case fatality rate as the ratio of total deaths and total confirmed cases ( = 408025 7145539 ⁄ = 0.057) [3], the incubation period has a mean average of 5.2 days and the recovery period is 5.8 days [14],i.e., ( = 1 5.2 ⁄ = 0.192, = 1 5.8 ⁄ = 0.172). The birth and death rate is assumed to be ( = 0.00005), the virus asymptotic infection proportion = 0.5 [15]. Using these parameters values together with = 0.533 as in [16], we can see that our 0 = 3.71, which is equivalent to its estimate in [12,[16][17][18].
Firstly, we investigate the DFE by assuming = 0.0533; we observe that when 0 = 0.371 in Figure 2(a), Theorem 1 and Theorem 3 are satisfied for the DFE to asymptotically stable. Also, in   Figure 2. That is, the positive effect when the magnitude of reduce is noticeable when the curves in Figure 2(b) become similar to that of Figure 2(a). It is observed in Figure 3 that as decreases the endemic trajectories patterns is similar to the DFE curves in Figure 2(a), indicating that the asymptomatic infection is detrimental to the COVID-19 pandemic.

Figure 3. The SEIR model with asymptomatic infection transmission effects
Similarly, we further investigate the effect of the direct transmission by regulating the magnitude of , and we also observe the same curves pattern as Figure 3. That is, when is lower in magnitude, lesser susceptible individuals become infected as the curve tends to increase in proportion, indicating that direct transmission can enhance the persistence of the COVID-19 pandemic.

Conclusion
In this paper, we formulate an asymptomatic infection SEIR model to investigate the stability analysis of the COVID-19 pandemic with demographic effects. We use simple algebraic procedures to describe the dynamics of the model theoretically. We showed that the model has two equilibrium states, which are disease-free and endemic equilibrium. The stability analyses show that the two equilibria states are locally and globally asymptotically stable, which are confirmed numerically using epidemiological data of COVID-19 pandemic. Also, we show numerically that the COVID-19 pandemic can be put to rest if both the direct and asymptomatic infections are control.

Declarations
Availability of data materials. Authors can confirm that all relevant data source are included in the article.
Competing interests. The authors declare that they have no competing interests, Funding: Not applicable.
Authors' contributions. AAK designed, analysed and interpreted the results of this article; LNM analysed and interpreted the results; GB substantively revised the article. All authors read and approved the final manuscript.