Mathematical model virus

The paper is organised as follows: Section 2 is related to the model formulation keeping in mind the assumptions that exposed and infected individuals are making contacts with susceptible individuals at the same rate. Section 3 is concerned with the local stability and existence of positive equilibrium solution. Some numerical simulations are executed to illustrate the analytical results in Section 4. Finally, conclusions are presented in Section 5. In this section, we develop the mathematical model by taking into account the above assumptions.

As the first four equations are independent of , so omit without generality the last equation for and the modified system 1 becomes. For system 2 , let , , , , and , and rescale the system 2 to get the normalized form with the initial conditions. 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 3. Lemma 1. Under the initial conditions 7 , all the solutions of system 3 remain nonnegative for. By the initial conditions 7 , it was discovered that. The existence of unique positive equilibrium and stability of system 3 depends on the basic reproductive number on free equilibrium point FEP , which is determined with the help of the next generation matrix method [ 8 ].

The free coronavirus equilibrium point is. Consider the following matrices for finding the basic reproductive number :. Now Jacobian of and at are. The dominant eigenvalue of represents , which is. Theorem 2. The system 3 is locally stable related to virus-free equilibrium point , and unstable if. For local stability at , the Jacobian of system 3 is which follows the eigenvalues , , , and , if.

So the system 3 is locally stable if and unstable if. Theorem 3. There exists a unique positive virus equilibrium point for system 3 , if. By letting the right hand sides of all equations of system 3 to zero, as implies that From the value of , it is obvious that all the values of are positive if. Theorem 4. If , then the system 3 is globally stable. For the proof of this theorem, first, we construct the Lyapunov function as Differentiating equation 15 with respect to time and keeping the reality in mind that and , we obtained Therefore, if , then , which implies that the system 3 is globally stable for.

The NSFD method is used for the numerical solution of the proposed model 3. Basically, NSFD is an iterative method in which we get closer to solution through iteration [ 9 , 10 ]. Now, using the NSFD method for numerical solution of system 3 , it follows that. We assume the parameters of the system 3 shown in Table 1 [ 4 ].

Figure 2 shows the solutions for , , , , and obtained by NSFD, RK4, and ode45 for , which show that it is unstable and will never become stable because of contact rates of infected people to susceptible people.

So according to the act of Governments to keeping people within specified bounds may be their home stay at home and save your lives , offices, etc. For , when the contact rate becomes smaller, then, the current infectious disease may be controlled see Figure 3.

In this work, we presented that isolation of the infected human overall can reduce the risk of future COVID 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 birth rate and death rate of bats were defined as n B and m B. The birth rate and death rate of hosts were defined as n H and m H.

The people were divided into five compartments: susceptible people S P , exposed people E P , symptomatic infected people I P , asymptomatic infected people A P , and removed people R P including recovered and death people. The birth rate and death rate of people were defined as n P and m P. Therefore, we added the further assumptions as follows:.

Based on our previous studies on simulating importation [ 13 , 14 ], we set the initial value of W as following impulse function:. In the function, n , t 0 and t i refer to imported volume of the SARS-CoV-2 to the market, start time of the simulation, and the interval of the importation.

During the outbreak period, the natural birth rate and death rate in the population was in a relative low level. However, people would commonly travel into and out from Wuhan City mainly due to the Chinese New Year holiday.

Therefore, n P and m P refer to the rate of people traveling into Wuhan City and traveling out from Wuhan City, respectively.

In the model, people and viruses have different dimensions. Based on our previous research [ 15 ], we therefore used the following sets to perform the normalization:.

In the normalization, parameter c refers to the relative shedding coefficient of A P compared to I P. The normalized RP model is changed as follows:. Commonly, R 0 was defined as the expected number of secondary infections that result from introducing a single infected individual into an otherwise susceptible population [ 13 , 16 , 17 ]. In this study, R 0 was deduced from the RP model by the next generation matrix approach [ 18 ].

The mean incubation period was 5. We set the same value 5. The duration from illness onset to first medical visit for the 45 patients with illness onset before January 1 was estimated to have a mean of 5. In our model, we set the infectious period of the cases as 5. Since there was no data on the proportion of asymptomatic infection of the virus, we simulated the baseline value of proportion of 0.

Since there was no evidence about the transmissibility of asymptomatic infection, we assumed that the transmissibility of asymptomatic infection was 0. We assumed that the relative shedding rate of A P compared to I P was 0. Since 14 January, , Wuhan City has strengthened the body temperature detection of passengers leaving Wuhan at airports, railway stations, long-distance bus stations and passenger terminals. As of January 17, a total of nearly 0. In Wuhan, there are about 2. We assumed that there was 0.

This means that the 2. Therefore, we set the moving volume of 0. Since the population of Wuhan was about 11 million at the end of [ 25 ], the rate of people traveling out from Wuhan City would be 0.

However, we assumed that the normal population mobility before January 1 was 0. Therefore, we set the rate of people moving into and moving out from Wuhan City as 0.

The parameters b P and b W were estimated by fitting the model with the collected data. Based on the equations of RP model, we can get the disease free equilibrium point as:. By the next generation matrix approach, we can get the next generation matrix and R 0 for the RP model:. In this study, we developed RP transmission model, which considering the routes from reservoir to person and from person to person of SARS-CoV-2 respectively.

We used the models to fit the reported data in Wuhan City, China from published literature [ 3 ]. The different values might be due to the different methods. The methods which Li et al. Our previous study showed that several methods could be used to calculate the R 0 based on the epidemic growth rate of the epidemic curve and the serial interval, and different methods might result in different values of R 0 [ 26 ].

This means that the transmission route was mainly from person to person rather than from reservoir to person in the early stage of the transmission in Wuhan City. However, this result was based on the limited data from a published literature, and it might not show the real situation at the early stage of the transmission. To contain the transmission of the virus, it is important to decrease R 0.

According to the equation of R 0 deduced from the simplified RP model, R 0 is related to many parameters. All these interventions could decrease the effective reproduction number and finally be helpful to control the transmission. Since there are too many parameters in our model, several limitations exist in this study.

Firstly, we did not use the detailed data of the SARS-CoV-2 to perform the estimation instead of using the data from literatures [ 3 ]. Secondly, the parameters of population mobility were not from an accurate dataset. This assumption might lead to the simulation been under- or over-estimated. All these limitations will lead to the uncertainty of our results. Therefore, the accuracy and the validity of the estimation would be better if the models fit the first-hand data on the population mobility and the data on the natural history, the epidemiological characteristics, and the transmission mechanism of the virus.

Since the objective of this study was to provide a mathematical model for calculating the transmissibility of SARS-CoV-2, the R 0 was estimated based on limited data which published in a literature.

More data were needed to estimate the transmissibility accurately. World Health Organization. World Health Organization, cited January 19, A pneumonia outbreak associated with a new coronavirus of probable bat origin. Early transmission dynamics in Wuhan, China, of novel coronavirus-infected pneumonia. N Engl J Med. Clinical features of patients infected with novel coronavirus in Wuhan, China. Table 1 gives a brief overview of the contributions and limitations of mathematical modeling on informing influenza pandemic preparedness and response that we have discussed in this review, which cover only a subset of all mathematical modeling studies on this topic.

We envision that mathematical modeling will remain an important tool for infectious disease control in the future. Some contributions and limitations of mathematical modeling on influenza pandemic preparedness and response. Remaining uncertainty over the public health and economic burden posed by the pandemic. Remaining uncertainty over the explanation for seasonality of annual influenza epidemics.

National Center for Biotechnology Information , U. Exp Biol Med Maywood. Author manuscript; available in PMC Aug 1. Author information Copyright and License information Disclaimer. Copyright notice. The publisher's final edited version of this article is available at Exp Biol Med Maywood. See other articles in PMC that cite the published article. Summary Influenza pandemics have occurred throughout history and were associated with substantial excess mortality and morbidity. Background Influenza pandemics have occurred throughout history.

Mathematical models The application of mathematical models to describe infectious disease dynamics is a systematic way of translating assumptions and data regarding disease transmission into quantitative estimates of how an epidemic evolves through time and space. Open in a separate window. Figure 1. Figure 2. The nonlinear dependence of the final attack rate and critical coverage on R 0 Knowing R 0 , we can use the Susceptible-Infectious-Recovered model to estimate the final attack rate as well as the level of vaccination coverage required to prevent an epidemic left.

Use of models to guide pandemic preparedness Estimation of epidemiologic parameters Combining historical epidemic data e. Predicting the speed of global spread and the effectiveness of travel restrictions Mathematical models have been used to study the global spread of infectious agents as early as the s. Assessing the effectiveness of containment and mitigation strategies The need for influenza pandemic preparedness in the past decade has been a strong driving force for the development of large-scale agent-based epidemic simulations[ 11 , 13 — 15 , 20 , 21 ] which represent a substantial advance in infectious disease modeling.

Figure 3. A schematic of transmission network for agent-based simulations In general, a network is consisted of nodes the circles and edges the links between circles. Optimizing antiviral strategies Maintaining a stockpile of antiviral drugs is a major component of many influenza preparedness plans.

Optimizing vaccination strategies Vaccine is the long-term solution for reducing morbidity and mortality associated with a novel influenza strain. Assessing the logistical requirement of interventions Mathematical models of epidemics can be easily extended to take into account operational constraints in order to assess logistical feasibility of interventions.

Discussion While influenza pandemics occur infrequently, the possibility of many millions of deaths worldwide in a severe pandemic means that they present a significant threat to public health. Table 1 Some contributions and limitations of mathematical modeling on influenza pandemic preparedness and response.

Contributions Limitations Transmissibility and mean generation time of influenza Rate of global spread and limited effectiveness of travel restrictions Potential effectiveness of containment and mitigation strategies Optimal use of antivirals and control strategies for drug-induced antiviral resistance Optimal pandemic and pre-pandemic vaccination strategies Logistical requirement of mitigation and treatment strategies Strategies for strengthening influenza surveillance Accurate severity estimates early in the pandemic Accurate epidemic forecasts during the early pandemic stages Remaining uncertainty over the public health and economic burden posed by the pandemic Remaining uncertainty over the explanation for seasonality of annual influenza epidemics No consensus on the effectiveness and feasibility of school closure.

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