key: cord-0986090-ypv7brcj
authors: Shakhany, Mohammad Qaleh; Salimifard, Khodakaram
title: Predicting the Dynamical Behavior of COVID-19 Epidemic and the Effect of Control Strategies
date: 2021-03-11
journal: Chaos Solitons Fractals
DOI: 10.1016/j.chaos.2021.110823
sha: e0a5918bdd399fe9dbab65737c8b3a0fa34adbf3
doc_id: 986090
cord_uid: ypv7brcj

This paper uses transformed subsystem of ordinary differential equation [Formula: see text] model, with vital dynamics of birth and death rates, and temporary immunity (of infectious individuals or vaccinated susceptible) to evaluate the disease-free [Formula: see text] and endemic [Formula: see text] equilibrium points, using the Jacobian matrix eigenvalues [Formula: see text] of both disease-free equilibrium [Formula: see text] and endemic equilibrium [Formula: see text] for COVID-19 infectious disease to show S, E, I, and R ratios to the population in time-series. In order to obtain the disease-free equilibrium point, globally asymptotically stable ([Formula: see text]), the effect of control strategies has been added to the model (in order to decrease transmission rate [Formula: see text] and reinforce susceptible to recovered flow), to determine how much they are effective, in a mass immunization program. The effect of transmission rates [Formula: see text] (from S to E) and [Formula: see text] (from R to S) varies, and when vaccination effect [Formula: see text] , is added to the model, disease-free equilibrium [Formula: see text] is globally asymptotically stable, and the endemic equilibrium point [Formula: see text] , is locally unstable. The initial conditions for the decrease in transmission rates of [Formula: see text] and [Formula: see text] reached the corresponding disease-free equilibrium [Formula: see text] locally unstable, and globally asymptotically stable for endemic equilibrium [Formula: see text]. The initial conditions for the decrease in transmission rate [Formula: see text] and [Formula: see text] and increase in [Formula: see text] reached the corresponding disease-free equilibrium [Formula: see text] globally asymptotically stable, and locally unstable in endemic equilibrium [Formula: see text].

In the early stages of the COVID-19 pandemic, epidemiological scientists tried to understand the dynamics of disease and control measures effectiveness (Kucharski et al., 2020; Rahimi & Talebi Bezmin Abadi, 2020) in order to reach the strategies which reduce the infectious disease spreading speed, through some activities that effect on direct contacts, such as quarantine and travel restrictions, social distancing, hygiene measures (Mushayabasa et al., 2020) , school closure, physical distancing (Davies et al., 2020) , and also, activities that reduce risk of infection, such as, shielding of people aged 70 years old or older, self-isolation of symptomatic cases and wearing masks (Davies et al., 2020; Rader et al., 2021) to prevent an overload of local medical services (Ilnytskyi, 2020) . But lately vaccination as a tool for immunization came in to the system, and broaden the strategies to prevention and immunization types (De la Sen et al., 2012; Le et al., 2020) .

Unidentified and unprecedented dimensions of the unfolding crisis of COVID-19 have been a great challenge to forecast and also to solve (Panovska-griffiths, 2020), but because of the novelty, the accuracy, and also the accessibility to data about this phenomenon is another challenge (Id & Id, 2020) . Over the past decades, planning to combat the spreading of infectious diseases through mathematical models was a field of studies to implementing control strategies (Hethcote H.W, 1984) , In most of these mathematical models, the population is divided into different compartments, where each one of them represents the epidemics stage state. Derivatives are used to mathematically express the transmission rate of each compartment to another one.

The system of the ordinary differential equations is a tool for describing the population changes in continuous time series in the assorted compartment of the population for the derivatives of the transmission rate (Trawicki, 2017) .

From the basic SIR compartmental model introduced by Kermack and McKendrick (Kermac & McKendrick, 1927) and the later ones such as SIS (Zhou & Liu, 2003) , SEIR with permeant immunity (Biswas et al., 2014; D'Onofrio, 2002; C. Sun & Hsieh, 2010) , and SEIRS with temporary immunity for other infectious diseases (Bai, 2015; Gao et al., 2006 Gao et al., , 2007 Jana et al., 2017; Mateus et al., 2018; Meng et al., 2007; Teng, 2008) , and also for COVID-19 (Ilnytskyi, 2020; Oluyori et al., 2020) and SEAIR which includes asymptomatic compartment (Barman & Mishra, 2021; Basnarkov, 2020; Boulant et al., 2020) , one of COVID-19 characteristic and unusual feature is the abundance of asymptomatic cases A, with mild (or without any) symptoms which are unaware of being a host of a virus and may actively spread it around, and enlarge the epidemic size (Chatterjee et al., 2020; T. Sun & Weng, 2020) . The real situation of these cases is unknown, but the role of them through mathematical modeling could be estimated (Basnarkov, 2020; J. Sun et al., 2020) .

In compartmental models, population N is divided into classes (compartments). In the susceptible S compartment, all individuals are susceptible S, if they have contact with the disease (Hethcotet, 1976) . Unidentified infected individuals E, who is in latent period (times between exposure and infected, without symptoms and infectious ability) or incubation period (times between exposure and onset of clinical symptoms, they do not have any clinical symptoms but they do have infectious ability to infect the others) of the disease which they are exposed (Ilnytskyi, 2020) , in identified infected compartment I, all infectious individuals have clinical symptoms and they are able to infect susceptible individuals if they contact, and recovered compartment R, individuals have been removed from identified infected class, and they are immune from disease (Trawicki, 2017) which could be permanently SEIR (Li & Muldowney, 1995) or temporarily SEIRS, which turns back to susceptible class S, after a specific time (Gao et al., 2007) .

Compartmental models such as SIR, SEIR and recently introduced SEAIR are in the consensus of specialists and mathematicians to use and expand (Basnarkov, 2020) . In order to respond to the questions about the future behavior of susceptible S, exposed E, asymptomatic A, infected I, and recovered R individuals, determining the proportion of each compartment to the population in continuous-time is required (De la Sen et al., 2012) . As mentioned above both exposed E, and asymptomatic A compartments, can be classified as unidentified infected in E (in this paper E, stands for unidentified infected, i.e. both exposed and asymptomatic classes). There are different factors such as time-delays, vital dynamics of the population, transmission rates, the infectionage structure of population, quarantine, and isolation, spatial structure along with treatment and vertical transmission which determines the number of individuals, who currently exist, or enter the classes in future (Trawicki, 2017) , but since, still there is not any periodical data available for COVID-19, and as Armstrong 1992 said, all the forecasting methods have a chance to mistake (Armstrong, 1992) . In the case of COVD-19, as was the bird flu and SARS misreported to the extent the epidemic (Panovska-griffiths, 2020), there might the correct number of unidentified infected cases be times more than, the officially identified infected reports (Contreras et al., 2020; Wu et al., 2020) . Such challenges make all the forecasting trials facing with wrong conclusions, or at least misleading results that must be regarded cautiously (Ghafari et al., 2020; Taghrir et al., 2020) .

Applying different scenarios to lower the dissemination of infection, during latent and incubation periods and also incorporate time-decaying effects due to loss of acquired immunity (in both of vaccinated and removed individuals), awareness about physical distancing, wearing masks and non-pharmaceutical interventions (López & Rodó, 2020) leads to conclude that, (1) abundance of unidentified infected (exposed and asymptomatic) cases, (2) absence of effective medication,

(3) virus mutation, (4) vaccination efficacy to maintain acquired immunity, (5) the effectiveness of direct contact and risk of infectious, on transmission rate from S to E, and (6) temporary immunity duration. While, 1, 2, and 3 are ambiguous, 4, 5 and 6 calls for some discussion.

During the early stages of COVID-19 outbreak, it was widely accepted amongst researchers that, each cases who recovers from disease has "immunity passport" and can do social activities without getting infected again, however later studies (immunodiagnostic tests) showed that this immunity lasts only 3-4 months, and reinfection cases has reported (Edridge et al., 2020; Tillett et al., 2021) .

Local SEIR model applied to quantify outbreak dynamics in United States, china (Peirlinck et al., 2020) and Italy (Gatto et al., 2020) , to shorten the transmission window of infectious cases (asymptomatic/symptomatic) through isolation, movement restrictions and awareness of the population (Choi & Ki, 2020; Juanjuan Zhang et al., 2020) to mitigate the direct contact of individuals (Block et al., 2020) and found that the optimal strategy is to release approximately half the population 2-4 weeks from the end of an initial infection peak, then wait another 3-4 months to allow for second peak before releasing everyone else. In this case the classical concept of recovered and immunized cases within 3-4 months is valid, but after 3-4 months, temporarily immunized population (recovered or vaccinated) loss their immunity, and they will be susceptible again to infect. Compartmental SEIRS, covers SEIR model, containing four classes of susceptible S, unidentified infected E, identified infected I, and recovered R individuals, considering the temporary immunity and turning back of recovered, to susceptible.

The epidemiological control strategies objective is preventing susceptible individuals to infect, and establishing immunization program to immunize them (De la Sen et al., 2012; Jiao & Shen, 2020) . In the case of COVID-19, As all the past pandemics occurred around the globe, the most hope for prevention and controlling, is vaccination (Le et al., 2020) , but technological limitations and also the high speed mutation of Coronavirus has hampered designing the vaccine formula for biological researchers (Jeyanathan et al., 2020) and also for modelers to design and simulate the COVID-19 susceptible S, exposed E, infected I and recovered R time series behavior regarding to vaccine accessibility. Inventing the vaccine is one problem, but accessibility and inject it to the total susceptible individuals because of social and economic limitations, might make a great troubles for any mass immunization programs to support all the population that they need it (Glatman-freedman & Nichols, 2012) .

A vaccine-based control strategy, guarantee the proportion of infectious patients in the total population in a positive and boundedness SEIR model converges to a desired value (Jiao & Shen, 2020) , In another word in a mass immunization program, anyone who receives the vaccine directly proceeds to R from S compartment. Proper vaccine providing to the public, reduces the basic reproduction number value to less than unity (De la Sen et al., 2012; Jeyanathan et al., 2020; Jiao & Shen, 2020) . Anderson and May (Anderson & May, 1985) studied vaccination as an extension to the SEIR and SEIRS. Wang et al, 2019 showed that optimal vaccination strategy as an optimal control strategy, can control the spread of epidemic despite of "seasonality varying incidence, monotonic successfully immune rate and monotonic increasing vaccine yield" as a three time-varying constraint factors (X. Wang et al., 2019) .

The basic reproduction number is generally compared with unity to assess the spread of infectious diseases to the population. If it is greater than unity ( ), means that an epidemic has occurred, and each infectious individual generates more than one new case, and if its less than unity ( ) means that the epidemic likely fades out (Khan et al., 2014) . There is two different approaches about equal unity , the first one's believe that, it leads to an epidemic (Bahrampour, 2005; Becker & Bahrampour, 1997; Nikbakht et al., 2018; Safi & Garba, 2012) , while the others have accept it as a true that If ≤ 1, on average, the number of new infections produced by one infectious individual over the mean course of the infectious disease, implies the infectious disease dies out eventually and also, the disease-free equilibrium is globally asymptotically stable, and the disease always dies out, and endemic equilibrium is locally unstable. otherwise (means ), there exists a unique endemic equilibrium which is globally asymptotically stable, and the disease-free equilibrium is locally unstable (Badole et al., 2018; Trawicki, 2017; Juan Zhang & Ma, 2003) .

Effectiveness of control strategies, such as lowering down the direct contact and risk of infectious in order to decrease transmission rate , and temporarily immunized individuals (vaccinated and previously infected), are considerations must be seen before any decision making about mass immunization programs. In this paper we try to response the importance of control strategies through rescaled model to stabilize globally asymptotically disease-free equilibrium ̅ . The remainder of this paper has been organized into the following sections.

The proposed ordinary differential equation epidemiological model based on vital dynamics, temporary immunity and the effect of vaccination to immunize susceptible individuals temporarily are discussed in Section 2, stability (globally and locally) of disease-free equilibrium ̅ and endemic equilibrium ̅ are explained in Section 3. Section 4 is devoted to experimental results and the paper is concluded in Section 5.

The mathematical epidemiological model with vital dynamics (birth and death rates), the temporary immunity of recovered and vaccinated individuals, provides a cursory description of spreading Coronavirus amongst the population . (1)

The four compartments of model have been described further detail in Table 1 . 

The ratio of the population who are susceptible to getting infected if they exposed it. Ratio of unidentified infected individuals to total population

The ratio of the population who are exposed to the infection, but they have not any clinical symptoms.

The ratio of the population who are infectious and they have clinical symptoms.

The ratio of the population who are recovered from the infection and they are temporarily immune from the infection.

In SEIRS model, with unequal ratios of birth and death rates, and temporarily immunized individuals which they could be vaccinated individuals or previously infected to COVID-19, all the population are susceptible to infect at first. By direct contact of susceptible with the infectious individual by the risk of infectious, the susceptible individual leaves its compartment and enter to unidentified infected E compartment after specific time. Then by passing from latent and incubation periods, clinical symptoms of the unidentified infected cases, starts to emerge and enters them to the identified infected I compartment. Unidentified and identified infected individuals might recover from COVID-19 and enter to the recovered R compartment without any vaccine to cure, temporarily or permanently. As any other infectious diseases recovering from COVID-19 with or without vaccination is temporarily (It means that, they are temporarily immune from infection and potentially transition back to susceptible compartment). Table 2 summarizes the parameters interpretation embedded in SEIRS model. Based on original model SEIRS model, parameters in Table 1 and Table 2 , and positive consents in Table 3 , Pan system of non-linear ordinary differential equation (Pan et al., 2011) could be transformed into mathematical model in Eq.

(2), for analyses and evaluation disease-free equilibrium, and endemic equilibrium stability and also determining critical points (bifurcations)

Positive constants A, B, C, D and F values has been shown in Table 3 . In the next step, stability mathematics of disease-free equilibrium and endemic equilibrium has been presented.

Stability is a tool for analyzing and determining disease-free equilibrium ̅ and endemic equilibrium points. Stability of disease-free equilibrium ̅ Shows that there is no epidemic in a system and all of population are susceptible to infect, but they are not exposed or infectious yet, or they are recovered from disease. While, stability of endemic equilibrium ̅ shows that the ratios of unidentified , and identified infected cases, are growing. Mathematically, Eq. (2) and Eq. (3) shows ̅ and ̅ respectively.

Equilibrium points are computed by setting Eq. (5).

; ;

In a Jacobian matrix and solving it for in Eq.

(2), Such that ( ) ( ) and evaluate equilibrium points to decide on the stability, which is directly determined from the eigenvalues of | ( ) | .

If all three eigenvalues of evaluated at disease-free equilibrium point, contains negative real parts, ̅ is globally asymptotically stable, and ̅ is locally unstable. If at least, one eigenvalue evaluated at disease free equilibrium point, contains non-negative real part, ̅ is locally unstable and ̅ is globally asymptotically stable (Juan Zhang & Ma, 2003) .

(3), will be computed as Eq. (6).

Eq. (4) will be applied to generate or

By substituting ̅ ( ) in ( ) we have ( ̅ ) with eigenvalues then we have | ( ̅ ) | which will be expanded through determinant and the eigenvalues determined from the cubic polynomial as Eq. (8).

All three eigenvalues are dependent to and parameters, and also constants. If

Based on Routh-Hurwitz criterion, if all these three conditions and are satisfied, the system is globally asymptotically stable at disease-free equilibrium, and if not, the system is locally unstable at disease-free equilibrium (Al-azzawi, 2012 (2), the ̅ will be computed by Eq. (12).

Where (13) And (14) Or (15) To deliver the first coordinate of ̅ as ;

Which is equal to

we have

After distributing and collecting, which will be more simplified with and by distributing and combining factors, the second coordinate of ̅ will be as Eq. (19).

Through substitution of Eq. (19), into the third coordinate of ̅ is Eq. (20).

With Eq. (16), Eq. (19), and Eq. (20), the ̅ in Eq. (4), is Eq. (21).

Which only makes physical sense of .

Since all are constants, and parameters are positive values in Eq. (20), by manipulating the epidemic condition is given as:

By substituting Eq. (4), in ( ) we have ( ̅ ) with eigenvalues , then we have

Which will be expanded through determinant, and the eigenvalues , are determined from the cubic polynomial as Eq. (24).

Where eigenvalues are dependent to and parameters, constants, and first and third coordinates of ̅ , namely , and in Eq. (21).

In a similar manner to the eigenvalues for the cubic polynomial in Eq. (8) 

The proposed transformed subsystem of ordinary differential equation in Eq. (2) The real-world data in terms of availability for Iran population is not provided yet, thus scenario based simulation for this part of variable, has been provided.

It is evident, that transmission rate reduces when is added to the system and it actively does its role, because transmission through direct contact rate decreases (Mossong et al., 2008) , but at this stage we suppose that, the highest risk of infectious in a close contact is approximately 10.2% (Luo et al., 2020) Based on numerical values of the model's parameter shown in Table 4a , and positive and constant values shown in Table 4b , COVID-19 basic reproduction number is which is greater than unity, and it shows that the disease-free equilibrium ̅ is locally unstable and endemic equilibrium ̅ is globally asymptotically stable. The rescaled model (based on Eq.

(2)), with numerical values of model parameters (presented in Table 4a ), and constants (presented in Table 4b ) is simulated, in order to differentiate between, control strategies to prevent Coronavirus dissemination. It has been assumed, that each person who receives the vaccine is the same as an individual who recovers from identified infected class, and is temporarily immune for 90 days. Thus, two different scenarios are possible. In the first one, we have focused on reducing transmissions rates and to see how much they are effective with no vaccination and in the second one, vaccination effect on the stability has been tested. For both situations, the stability of disease-free equilibrium ̅ in Eq. (7), and endemic equilibrium ̅ in Eq. (21) has been evaluated using eigenvalues of Jacobian matrix ( ). The proportions of and have been evaluated as the total population ratios of Iran's, based on the average ratios of daily unidentified and identified infected, mortality and morbidity and recovered individuals, in 357 days from February 10, 2020, to February 7, 2021 which it's data is available in (who, 2021).

Decreasing the numbers of susceptible individuals, depends on values of death rate , and transmission rates , and the value of is constant and it depends on other factors, and also COVID-19 infectious disease mortality and morbidity. The value of transmission rate alters with human activities, such as physical distance and wearing masks (Rader et al., 2021) , which the first one effects on direct contact, and the second one effects on risk of infectious.

To recognize the real epidemic's condition of Iran, in order to have a context to analyze the control strategies effectiveness, suppose that there is not any strategies to reduce the transmission rates and or increases the transmission rate . Based on model's parameter numerical values, in Table 4a and constant's value in Table 4b , with for in Eq.

(2), the basic reproduction number estimated at 3.0451, which is bigger than unity. It implies that, there is a growing infectious, and disease-free equilibrium ̅ is locally unstable. Table 5 shows the disease-free equilibrium ̅ , endemic equilibrium ̅ , eigenvalues of Jacobian matrices ( ̅ ), and ( ̅ ), along the stabilities. 

Most of human activities such as quarantine, social and physical distance, movement restrictions, washing hands, closing school and universities, wearing masks etc. implies on trying to decline transmission rate (Mccallum et al., 2001) . Suppose that wearing masks reduces 0.3 risk of infectious (transmission rate will be ) and with restrictive measures direct contact of individuals decline to 5 contact per a day per each person (the transmission rate will be 0.255). Table 6a shows the effect of wearing masks with 0.3 effectiveness to decrease risk of infectious, and Table 6b shows the effect of reducing with decrease in direct contact, on ordinary differential equation of model on the ̅ , ̅ and eigenvalues of Jacobian matrices ( ̅ ), and ( ̅ ) along stability. From the eigenvalues with non-negative real part in 0, the diseases free equilibrium ̅ of is locally unstable (at least one has non-negative real part), but since there is not any non-negative real part eigenvalue the endemic equilibrium ̅ is globally asymptotically stable. From the eigenvalues with non-negative real part in 0, the diseases-free equilibrium ̅ of model is locally unstable (at least one has nonnegative real part), and ̅ is globally asymptotically stable, because there is not any nonnegative real part eigenvalues for endemic equilibrium. Error! Reference source not found.a shows the simulation result of decreases in risk of infectious by 0.3 through wearing masks, and Figure 3b , illustrates direct contact reduces to 5 (as the result of physical distance), for 360 days, while the other parameters are constant. 

One strategy to decrease the transmission rate that can be do, is applying both wearing masks and social distance simultaneously. Suppose that, as mentioned in 4.1.1., because of restrictive measures (5 direct contact per each person per day), and wearing masks (0.3 decrease in risk of infectious) the transmission rate 0.1785, while other parameters are constant. Table 7a shows numerical results of 0 effect, on the ordinary differential equation of model, disease-free equilibrium ̅ , endemic equilibrium ̅ and eigenvalues of a Jacobian ( ̅ ) and ( ̅ ) along stability. From the eigenvalue with non-negative real part in 0 the diseases-free equilibrium ̅ of is locally unstable (at least one has non-negative real part) and endemic equilibrium point ̅ is globally asymptotically stable.

There are possibilities that could be presumed, such as transmission rate of decreases (Blyuss & Kyrychko, 2010) . Suppose that, 90 days of transmission from to compartment, increases to 120 days ( 0.0083), while transmission rate is as previously mentioned in 4.1. Table 7b shows increase in temporary immunity duration to 120 days ( 0) effect on the ordinary differential equation of model disease-free equilibrium ̅ , endemic equilibrium ̅ and eigenvalues of a Jacobian ( ̅ ) and ( ̅ ) along stability. 

Combining control strategies such as restrictive measures to decrease direct contact, wearing masks to reduce risk of infectious (in order to decrease transmission rate ) and considering increase in temporary immunity duration, is able to slowing down the speed of the infectious disease spreading (basic reproduction number is still greater than unity ), and vaccination as an effective control strategy to transmit susceptible s to temporarily immunized r compartment is able to stabilize globally asymptotically disease-free equilibrium.

Disease-free equilibrium ̅ , endemic equilibrium ̅ , eigenvalues of Jacobian matrix ( ̅ ) and ( ̅ ) along stability evaluated are shown in Table 8a , for combined control strategies without vaccination, and Table 8b considering the vaccination effect on the ordinary differential equation of model. Beside strategies to decrease transmission rate and increase the duration of temporary immunity (transmission rate ) which the population remain susceptible and only the speed of spreading decreases, vaccination is able to reduce the ratio of susceptible individual s to population, with the difference that it transmit the susceptible s to recovered r directly, and also decreases the cost of other strategies such as physical distance and wearing masks. by applying combining strategies ( ) as has been shown in Table 8b , the disease-free equilibrium ̅ has no non-negative real part eigenvalues along its stability and is globally asymptotically stability, with From the negative real part eigenvalues the disease-free equilibrium ̅ is globally asymptotically stable, because there is not any non-negative real part eigenvalues for and endemic equilibrium ̅ of is locally unstable ( has non-negative real part). These results shows that, even with even with regulating rules about physical distance, wearing masks and even increase in temporary immunity duration, there it is need to vaccinate 0.217 ( ) of the susceptible individuals to stabilize the disease-free equilibrium ̅ globally asymptotically. Decrease in transmission rate and increase in temporary immunity duration shown in Figure   5a , reduces the speed of infectious spread, but does not stable the disease-free equilibrium ̅ globally asymptotically.

Considering the vaccination effect on the ordinary differential equation model stabilizes the disease-free equilibrium ̅ globally asymptotically and makes the endemic equilibrium ̅ locally unstable. In other words, the basic reproduction number is less than unity and we do not have epidemics outbreak. By applying the combination of these control strategies susceptible individuals s, declines in first 50 days, and it will be added to the compartment who have temporary immunity and it will remain for the next 310 days.

In this paper transformed subsystem of ordinary differential equation has been used, to focus on the simulation of vital dynamics (birth and death rates) with different rates and transmission rates and . From the experimental results, various scenarios has been simulated to examine stability (locally and globally) of disease-free equilibrium ̅ and endemic equilibrium points ̅ when is less than, equal or greater than unity.

Experimental results shows that, by applying different control strategies such as restrictive measures in direct contact, and wearing masks to decrease the risk of infectious (decreasing the transmission rate ) and also increase in temporary immunity duration (without vaccination ), at least one eigenvalue of Jacobian matrix of disease-free equilibrium ̅ has non-negative real part which implies that ̅ is locally unstable, and endemic equilibrium ̅ is globally asymptotically stable.

Vaccination as a tool of transmitting susceptible individuals s, to recovered compartment r, can have an effective role on reducing susceptible individuals s, and transmit it to the population who have temporary immunity to make less than or equal unity ( ) Vaccination of the greater proportion of susceptible individuals is able to compensate all delinquencies in other control strategies, but it is evident that a combination of control strategies (decreasing increasing temporary immunity duration and vaccination) in poorer countries such as Iran, makes less than unity, and all Jacobian matrix ( ̅ ) eigenvalues of disease-free equilibrium with negative real part, which implies that disease free equilibrium ̅ is globally asymptotically stable, and endemic equilibrium ̅ is locally unstable.

Briefly findings of this research could be stated as the initial conditions for decrease in transmission rates of and reached the corresponding disease-free equilibrium ̅ locally unstable, and globally asymptotically stable for endemic equilibrium ̅ . The initial conditions for decrease in transmission rate and and increase in reached the corresponding disease-free equilibrium ̅ globally asymptotically stable, and locally unstable in endemic equilibrium ̅ .

For future research, two kinds of researches can be done. Firstly, investigating a mathematical model to find the critical point (bifurcation) where is equal unity, what would be the proportions of because supposing that the COVID-19 vaccine does exist, it is impossible to imagine that everyone can take it because of economic limitations for example. Secondly, the model with vital dynamics, temporary immunity and vaccination could be modified to incorporate quarantine, isolation, age-structure, and any other factors that can effect on Coronavirus epidemic to obtain more realistic simulation results. 

Using ∆ -Discriminate Method to Determine the Stability and Bifurcation of Chen Chaotic System

Stability and bifurcation of pan chaotic system by using Routh -Hurwitz and Gardan methods

Vaccination and herd immunity to infectious diseases

Error measures for generalizing about forecasting methods : Empirical comparisons

Global Dynamics of an SEIR Epidemic model with saturated incidence under treatment

On Multi-type Branching Process in determining reproduction number ander a markovian mode movement

Threshold dynamics of a time-delayed SEIRS model with pulse vaccination

A time-delay SEAIR model for COVID-19 spread. 1-6

SEAIR Epidemic spreading model of COVID-19

Preventing epidemics with age-specific vaccination schedules

A SEIR model for control of infectious diseases with constraints

Social network-based distancing strategies to flatten the COVID-19

Stability and Bifurcations in an Epidemic Model with Varying Immunity Period Stability and Bifurcations in an Epidemic Model

SEAIR Framework Accounting for a Personalized Risk Prediction Score: Application to the Covid-19 Epidemic

Symptoms & Emergency Warning Signs

Studying the progress of COVID-19 outbreak in India using SIRD model

Estimating the reproductive number and the outbreak size of COVID-19 in Korea

Statistically-based methodology for revealing real contagion trends and correcting delay-induced errors in the assessment of COVID-19 pandemic

Stability properties of pulse vaccination strategy in SEIR epidemic model

Effects of nonpharmaceutical interventions on COVID-19 cases, deaths, and demand for hospital services in the UK: a modelling study

On the equilibrium points, boundedness and positivity of a SVEIRS epidemic model under constant regular vaccination

On vaccination controls for the SEIR epidemic model

Mixing patterns between age groups in social networks

Safety and efficacy of an rAd26 and rAd5 vector-based heterologous prime-boost COVID-19 vaccine : an interim analysis of a randomised

Seasonal coronavirus protective immunity is short-lasting

Forecasted trends in vaccination coverage and correlations with socioeconomic factors : a global time-series analysis over 30 years

Analysis of a delayed epidemic model with pulse vaccination and saturation incidence

Impulsive vaccination of an SEIRS model with time delay and varying total population size

Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures

under-reporting of prevalence and deaths Title : Ongoing outbreak of COVID-19 in Iran : challenges and signs of concern with underreporting of prevalence and deaths

The effect of social determinants on immunization programs

Neurologic Features in Severe SARS-CoV-2 Infection

A Thousand and One Epidemic Models

Qualitative Analyses of Communicable Disease Models *

Forecasting the novel coronavirus COVID-19

SEIRS epidemiology model for the COVID-19 pandemy in the extreme case of no acquired immunity

Global Dynamics of a SEIRS Epidemic Model with Saturated Disease Transmission Rate and Vaccination Control

Immunological considerations for COVID-19 vaccine strategies

Dynamics Analysis and Vaccination-Based Sliding Mode Control of a More Generalized SEIR Epidemic Model

A Contribution to the Mathematical Theory o f Epidemics

Estimating the basic reproduction number for single-strain dengue fever epidemics

Early dynamics of transmission and control of COVID-19: a mathematical modelling study. The Lancet Infectious Diseases

The COVID-19 vaccine development landscape

Global Stability for the SEIR Model in Epidemiology

The end of social confinement and COVID-19 re-emergence risk

Modes of contact and risk of transmission in COVID-19 among close contacts

Optimal control of nonautonomous seirs models with vaccination and treatment

How should pathogen transmission be modelled ?

Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination

Social contacts and mixing patterns relevant to the spread of infectious diseases

On the role of governmental action and individual reaction on COVID-19 dynamics in South Africa: A mathematical modelling study

COVID research updates: One man's COVID therapy drives worrisome viral mutations

Estimation of the basic reproduction number and vaccination coverage of influenza in the United States (2017-18)

Backward and Hopf bifurcation analysis of an SEIRS COVID-19 epidemic model with saturated incidence and saturated treatment response

Chaos synchronization between two different fractionalorder hyperchaotic systems

Can mathematical modelling solve the current Covid-19 crisis ?

Outbreak dynamics of COVID-19 in China and the United States

Mask-wearing and control of SARS-CoV-2 transmission in the USA: a cross-sectional study. The Lancet Digital Health

Practical Strategies Against the Novel Coronavirus and COVID-19-the Imminent Global Threat

Global Stability Analysis of SEIR Model with Holling Type II Incidence Function

International Conference on Population and Development (ICPD) -2014 Report. Statistical Center of Iran

Global analysis of an SEIR model with varying population size and vaccination

COVID-19: Epidemiology, Evolution, and Cross-Disciplinary Perspectives

Estimating the effects of asymptomatic and imported patients on COVID-19 epidemic using mathematical modeling

Genomic evidence for reinfection with SARS-CoV-2: a case study

Deterministic Seirs Epidemic Model for Modeling Vital Dynamics

Duration and key determinants of infectious virus shedding in hospitalized patients with coronavirus disease-2019 (COVID-19)

Optimal vaccination strategy of a constrained time-varying SEIR epidemic model

mRNA vaccine-elicited antibodies to SARS-CoV-2 and circulating variants

In Iran (Islamic Republic of)

Substantial underestimation of SARS-CoV-2 infection in the United States

Global dynamics of an SEIR epidemic model with saturating contact rate

Evolving epidemiology and transmission dynamics of coronavirus disease 2019 outside Hubei province, China: a descriptive and modelling study. The Lancet Infectious Diseases

Stability of periodic solutions for an SIS model with pulse vaccination

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This research has been financially supported by the National Institute of Genetic Engineering and Biotechnology of Iran.