The deadly coronavirus continues to spread across the globe, and mathematical models can be used to show suspected, recovered, and deceased coronavirus patients, as well as how many people have been tested. Researchers still do not know definitively whether surviving a COVID-19 infection means you gain long-lasting immunity and, if so, for how long? In order to understand, we think that this study may lead to better guessing the spread of this pandemic in future. We develop a mathematical model to present the dynamical behavior of COVID-19 infection by incorporating isolation class. First, the formulation of model is proposed; then, positivity of the model is discussed. The local stability and global stability of proposed model are presented, which depended on the basic reproductive. For the numerical solution of the proposed model, the nonstandard finite difference (NSFD) scheme and Runge-Kutta fourth order method are used. Finally, some graphical results are presented. Our findings show that human to human contact is the potential cause of outbreaks of COVID-19. Therefore, isolation of the infected human overall can reduce the risk of future COVID-19 spread.
Mathematical models are useful to understand the behavior of an infection when it enters a community and investigate under which conditions it will be wiped out or continued. Currently, COVID-19 is of great concern to researches, governments, and all people because of the high rate of the infection spread and the significant number of deaths that occurred. In December 2019, coronavirus first reported in Wuhan, China, is an infectious disease caused by a newly discovered coronavirus. The virus that causes COVID-19 is mainly transmitted through droplets generated when an infected person coughs, sneezes, or exhales. These droplets are too heavy to hang in the air and quickly fall on floors or surfaces. Coronavirus-confirmed cases reached nearly four million in 187 countries, and approximately 295,000 people have lost their lives due to this virus.
According to figures collated by Johns Hopkins University, the largest cases occurred at the US. Noting that more than 77,000 deaths happened, it also has the world’s highest death toll (see the Figure
BBC News up to 8 May [
Researchers have been tracking the spread of the virus, have mobilized to speed innovative diagnostics, and are working on a number of vaccines to protect against COVID-19. Cao et al. [
Parameters and description.
Symbols | Description |
---|---|
Susceptible population | |
Exposed population | |
Infected population | |
Isolated population | |
Recovered population | |
Rate at which susceptible population moves to infected and exposed class | |
Rate at which exposed population moves to infected one | |
Presents the rate at which exposed people take onside as isolated | |
Shows the rate at which infected people were added to isolated individual | |
Rate at which isolated persons recovered | |
Natural death rate plus disease-related death rate |
Unfortunately, the number of coronavirus victims is expected to be much higher than that predicted on February 10, 2020, since 12289 new cases (not previously included in official counts) have been added two days later. Further research should focus on updating the predictions with the use of up-to-date data and using more complicated mathematical models. Currently, there are no licensed vaccines or therapeutic agents for coronavirus prevention or treatment though research studies into potential antivirals and vaccine candidates are underway in a number of countries. Vaccine testing, development, and distribution are typically a much longer process than drug development, and it is not likely that a vaccine for COVID-19 will be ready before 2021 at the earliest. The virus can easily spread in dense places. Social distancing or low contact rate refers to measures that are taken to increase the physical space between people to slow down the spread of the virus. Batista [
From the above discussion, it was concluded that human to human contact is the potential cause of outbreaks of COVID-19. Therefore, isolation of the infected human overall can reduce the risk of future COVID-19 spread. In order to do this, we divided the total population into five compartments: susceptible, exposed, infected, isolated, and recovered from the disease. This study will lead to the mathematical model formulation in which the interaction of the exposed population and infected population occurred to the susceptible populations. The infected individuals, the individuals showing no symptoms apparently but have the disease in weak form inside their bodies, must be sent to isolated class in different rates. The local stability and global stability of model is discussed, by using the approach of the basic reproductive. For the numerical solution of the proposed model, the nonstandard finite difference (NSFD) scheme and Runge-Kutta fourth order method are used. Finally, some graphical results are presented. Our findings show that human to human contact is the potential cause of outbreaks of COVID-19.
The paper is organised as follows: Section
In this section, we develop the mathematical model by taking into account the above assumptions.
As the first four equations are independent of
For system (
In the remaining sections, we will discuss the local and global stability of the proposed model with initial conditions. First, a result is observed for the positivity and boundedness of the solution of system (
Under the initial conditions (
By the initial conditions (
The existence of unique positive equilibrium and stability of system (
Consider the following matrices for finding the basic reproductive number
Now Jacobian of
The dominant eigenvalue of
The system (
For local stability at
There exists a unique positive virus equilibrium point
By letting the right hand sides of all equations of system (
From the value of
If
For the proof of this theorem, first, we construct the Lyapunov function
Differentiating equation (
The NSFD method is used for the numerical solution of the proposed model (
Now, using the NSFD method for numerical solution of system (
We assume the parameters of the system (
Figure
Plots present the susceptible, exposed, infected, isolated, and recovered population when
Plots present the susceptible, exposed, infected, isolated and recovered population when
In this work, we presented that isolation of the infected human overall can reduce the risk of future COVID-19 spread. Our model shows that the coronavirus spreads through contact and describes how fast something changes by counting the number of people who are infected and the likelihood of new infections. Those new infections are what induce the epidemic. For this reason, we think that this research may lead to better guessing of the spread of this pandemic in the future. This paper is devoted to implement the coronavirus mathematical model containing isolation class. The reproductive number-related stability is discussed, which showed the impact of interaction of infected people to susceptible population and proved graphically and analytically that if we control this contact rate, the control of the current disease is possible, otherwise. State and territory governments have different restrictions in place for public gatherings. Therefore, citizens need to follow the directions from time to time to minimize the health risk.
The authors confirm that the data supporting the findings of this study are available within the article.
The authors declare that there is no conflict of interest regarding the publication of this paper.
The authors equally contributed in preparing this manuscript.
This project was funded by the Deanship of Scientific Research at King Abdulaziz University, Jeddah. The authors, therefore, gratefully acknowledge DSR for the technical and financial support.