Predicting the dynamical behavior of COVID-19 epidemic and the effect of control strategies

Abstract

This paper uses transformed subsystem of ordinary differential equation $seirs$model, with vital dynamics of birth and death rates, and temporary immunity (of infectious individuals or vaccinated susceptible) to evaluate the disease-free $DFE{\overline{X}}_{DFE},$and endemic $EE{\overline{X}}_{EE}$ equilibrium points, using the Jacobian matrix eigenvalues ${\lambda }_i$ of both disease-free equilibrium ${\overline{X}}_{DFE},$ and endemic equilibrium ${\overline{X}}_{EE}$ 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 ($R_0\le 1$), the effect of control strategies has been added to the model (in order to decrease transmission rate$\beta ,$ and reinforce susceptible to recovered flow), to determine how much they are effective, in a mass immunization program. The effect of transmission rates $\beta$ (from S to E) and $\alpha$ (from R to S) varies, and when vaccination effect$\rho$, is added to the model, disease-free equilibrium ${\overline{X}}_{DFE}$ is globally asymptotically stable, and the endemic equilibrium point${\overline{X}}_{EE}$, is locally unstable. The initial conditions for the decrease in transmission rates of $\beta$ and $\alpha ,$ reached the corresponding disease-free equilibrium ${\overline{X}}_{DFE}$ locally unstable, and globally asymptotically stable for endemic equilibrium${\overline{X}}_{EE}$. The initial conditions for the decrease in transmission rate$s\beta$and$\alpha ,$ and increase in $\rho ,$ reached the corresponding disease-free equilibrium ${\overline{X}}_{DFE}$ globally asymptotically stable, and locally unstable in endemic equilibrium${\overline{X}}_{EE}$.

1. Introduction

In the early stages of the COVID-19 pandemic, epidemiological scientists tried to understand the dynamics of disease and control measures effectiveness [42,59] 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 [51], school closure, physical distancing [20], 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 [20,58] to prevent an overload of local medical services [36]. But lately vaccination as a tool for immunization came in to the system, and broaden the strategies to prevention and immunization types [22,43].

Unidentified and unprecedented dimensions of the unfolding crisis of COVID-19 have been a great challenge to forecast and also to solve [56], but because of the novelty, the accuracy, and also the accessibility to data about this phenomenon is another challenge [35]. Over the past decades, planning to combat the spreading of infectious diseases through mathematical models was a field of studies to implementing control strategies [33], 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 [69].

From the basic SIR compartmental model introduced by Kermack and McKendrick [40] and the later ones such as SIS [77], SEIR with permeant immunity [11,19]; C. [62], and SEIRS with temporary immunity for other infectious diseases [7,27,28,37,47,49,67], and also for COVID-19 [36,54] and SEAIR which includes asymptomatic compartment [8,9,14], 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 [16]; T. [64]. The real situation of these cases is unknown, but the role of them through mathematical modeling could be estimated [9], [63].

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 [34]. 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 [36], 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 [69] which could be permanently SEIR [44] or temporarily SEIRS, which turns back to susceptible class S, after a specific time [28].

Compartmental models such as SIR, SEIR and recently introduced SEAIR are in the consensus of specialists and mathematicians to use and expand [9]. 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 [22]. 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 infection-age 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 [69], but since, still there is not any periodical data available for COVID-19, and as [4] said, all the forecasting methods have a chance to mistake [4]. In the case of COVD-19, as was the bird flu and SARS misreported to the extent the epidemic [56], there might the correct number of unidentified infected cases be times more than, the officially identified infected reports [18,74]. Such challenges make all the forecasting trials facing with wrong conclusions, or at least misleading results that must be regarded cautiously [30,65].

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 [45] 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 [25,68].

Local SEIR model applied to quantify outbreak dynamics in United States, china [57] and Italy [29], to shorten the transmission window of infectious cases (asymptomatic/symptomatic) through isolation, movement restrictions and awareness of the population [17]; Juanjuan [76] to mitigate the direct contact of individuals [12] 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 [22,39]. In the case of COVID-19, As all the past pandemics occurred around the globe, the most hope for prevention and controlling, is vaccination [43], but technological limitations and also the high speed mutation of Coronavirus has hampered designing the vaccine formula for biological researchers [38] 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 [31].

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 [39], 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 $R_0$ to less than unity [22,38,39]. Anderson and May [3] studied vaccination as an extension to the SEIR and SEIRS. [71] 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. [71].

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 $\left(R_0>1\right)$, means that an epidemic has occurred, and each infectious individual generates more than one new case, and if its less than unity$(R_0<1),$ means that the epidemic likely fades out [41]. There is two different approaches about $R_0$ equal unity$R_0=1$, the first one's believe that, it leads to an epidemic [6,10,53,60], while the others have accept it as a true that If $R_0$ ≤ 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  $R_0>1)$, there exists a unique endemic equilibrium which is globally asymptotically stable, and the disease-free equilibrium is locally unstable [5], [69], [75].

Effectiveness of control strategies, such as lowering down the direct contact and risk of infectious in order to decrease transmission rate$\beta$, 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 $seirs$ model to stabilize globally asymptotically disease-free equilibrium${\overline{X}}_{DFE}$. The remainder of this paper has been organized into the following sections. The proposed ordinary differential equation $seirs$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${\overline{X}}_{DFE}$ and endemic equilibrium${\overline{X}}_{EE}$ are explained in Section 3. Section 4 is devoted to experimental results and the paper is concluded in Section 5.

2. Methodology

The mathematical epidemiological $SEIRS$ 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$N$ . Fig. 1 , shows positive and boundedness (for more details about this key factors see Ref. [21]) $SEIRS$ model consisting of susceptible S, unidentified infected E, identified infected I, and recovered R compartments, which they are component individuals of population$N$. In other words,$N=S+E+I+R$.

Fig. 1.

SEIRS Model with vital dynamic (birth and death rates), temporary immunity and vaccination.

From that in compartmental models, each compartment shows the condition of individuals who are in different stages of the disease, and the SEIR consists of the total population, we can conclude that each one of S, E, I and R divided by N, shows the ratio of the individuals to the total population who are in one compartment. Thus we have Eq. (1).

$$s=S/N;e=E/N;i=I/N;r=R/N;s+e+i+r=1$$ (1)

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

Table 1.

Compartments of SEIR model with its definitions.

Parameter	Name	Units	Definition
$s=S/N$	Ratio of susceptible individuals to total population	Ratio of individuals	The ratio of the population who are susceptible to getting infected if they exposed it.
$e=E/N$	Ratio of unidentified infected individuals to total population	Ratio of individuals	The ratio of the population who are exposed to the infection, but they have not any clinical symptoms.
$i=I/N$	Ratio of identified infected individuals to total population	Ratio of individuals	The ratio of the population who are infectious and they have clinical symptoms.
$r=R/N$	Ratio of recovered individuals to total population	Ratio of individuals	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 $b$and death $d$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.

Table 2.

Description of $\mathit{seirs}$model parameters.

Parameter	Name	Unit	Meaning
$b$	Birth rate	$\frac{birth}{\frac{Population}{yearlydays}}$	Yearly new born birth rate. The number of births per 1000 people per year (360 days)
$d$	Death rate	$\frac{death}{\frac{Population}{yearlydays}}$	Yearly S, E, I and R death rate. The number of deaths per 1000 people per year (360 days)
$\alpha$	Transmission rate (from R to S)	$\frac{1}{days}$	Rate at which individuals who are in compartment R, enter to compartment S, because of losing immunity.
$\beta$	Transmission rate (from S to I)	$\frac{1}{days}$*direct contact per person per day*risk of infection	Rate at which individuals who are in compartment S, enter to compartment E, because of exposure to infectious individual.
$\sigma$	Transmission rate (from E to I)	$\frac{1}{days}$	Rate at which individuals who are in compartment E, enter to compartment I, because of emerging infection symptoms.
$\gamma$	Transmission rate (from I to R) Temporary immunity of infected	$\frac{1}{days}$	Rate at which individuals who are in compartment I, enter to compartment R, because of temporary immunity from disease.
$\rho$	Vaccination rate (From R to S) Temporary immunity of vaccinated	$\frac{1}{days}$*efficacy rate*the probability of availability	Rate at which S individuals, vaccinate and enter to R compartment.

Based on original model SEIRS model, parameters in Tables 1 and 2, and positive consents in Table 3 , Pan system of non-linear ordinary differential equation [55] could be transformed into $seirs$ mathematical model in Eq. (2), for analyses and evaluation disease-free equilibrium, and endemic equilibrium stability and also determining critical points (bifurcations) [1].

$$\begin{array}{ccc}\hfill ds/dt& =& A-\alpha e-\alpha i-\beta si-Bs;de/dt=\beta si-Ce;\hfill \\ \hfill di/dt& =& \sigma e-Di;dr/dt=\gamma i+\rho s-Fr\hfill \end{array}$$ (2)

Table 3.

Positive constants definition in $\mathit{seirs}$model.

Positive constant	Value
A	$b+\alpha$
B	$b+\rho +\alpha$
C	$b+\sigma$
D	$b+\gamma$
F	$b+\alpha$

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.

3. Stability

Stability is a tool for analyzing and determining disease-free equilibrium${\overline{X}}_{DFE},$ and endemic equilibrium points. Stability of disease-free equilibrium ${\overline{X}}_{DFE},$Shows that there is no epidemic in a system and all of population $N,$ are susceptible to infect, but they are not exposed or infectious yet, or they are recovered from disease. While, stability of endemic equilibrium ${\overline{X}}_{EE},$ shows that the ratios of unidentified$e$, and identified infected$i$ cases, are growing. Mathematically, Eqs. (2) and (3) shows $D{\overline{X}}_{DFE}$ and ${\overline{X}}_{EE},$ respectively.

$${\overline{X}}_{DFE}=\left(s,e,i\right)=\left(s^{*},e,i\right)$$ (3)

$${\overline{X}}_{EE}=\left(s,e,i\right)=\left(s^{*},e^{*},i^{*}\right)$$ (4)

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

$$\frac{ds}{dt}=0;\frac{de}{dt}=0;\frac{di}{dt}=0$$ (5)

In a Jacobian matrix $J,$and solving it for $s,e,i$in Eq. (2), Such that $J\left(X\right)=J\left(s,e,i\right)$ and evaluate equilibrium points to decide on the stability, which is directly determined from the eigenvalues ${\lambda }_i$ of$\left|J\left(X\right)-\lambda I\right|=0$.

If all three eigenvalues${\lambda }_i$ of $J,$ evaluated at disease-free equilibrium point, contains negative real parts, ${\overline{X}}_{DFE}$ is globally asymptotically stable, and ${\overline{X}}_{EE}$ is locally unstable. If at least, one eigenvalue evaluated at disease free equilibrium point, contains non-negative real part, ${\overline{X}}_{DFE}$ is locally unstable and ${\overline{X}}_{EE}$ is globally asymptotically stable (Juan [75]).

By substituting Eq. (2), into ${\lambda }_i$ of Eq. (3), the ${\overline{X}}_{DFE}$ in Eq. (3), will be computed as Eq. (6).

$$\frac{ds}{dt}\left|{\overline{X}}_{DFE}=0=\left(A-\alpha e-\alpha i-\beta si-Bs\right)\right|{\overline{X}}_{DFE}$$ (6)

Eq. (4) will be applied to generate $s=\frac{A}{B}$ or

$${\overline{X}}_{DFE}=\left(s,e,i\right)=\left(s^{*},e,i\right)=\left(\frac{A}{B},0,0\right)$$ (7)

By substituting ${\overline{X}}_{DFE}=\left(\frac{A}{B},0,0\right)$in $J\left(X\right)$ we have $J\left({\overline{X}}_{DFE}\right),$ with eigenvalues ${\lambda }_i,$then we have $|J\left({\overline{X}}_{DFE}\right)-\lambda I|=0,$which will be expanded through determinant and the eigenvalues${\lambda }_i$ determined from the cubic polynomial as Eq. (8).

$$\begin{array}{ccc}\hfill P\left(\lambda \right)& =& {\lambda }^3+\left(B+C+D\right){\lambda }^2+\left(BC+BD+CD+\sigma \beta \frac{A}{B}\right)\lambda \hfill \\ & & +\left(BCD+\sigma \beta A\right)=0\hfill \end{array}$$ (8)

All three eigenvalues${\lambda }_i$are dependent to $\sigma$ and $\beta$parameters, and also $A,B,C,D$ constants. If

$$a_1=\left(B+C+D\right)$$ (9)

$$a_2=\left(BC+BD+CD+\sigma \beta \frac{A}{B}\right)$$ (10)

$$a_3=\left(BCD+\sigma \beta A\right)$$ (11)

Based on Routh-Hurwitz criterion, if all these three conditions$,a_1>0,a_3>0$and$a_1{a}_2>a_3$ are satisfied, the system is globally asymptotically stable at disease-free equilibrium, and if not, the system is locally unstable at disease- free equilibrium [2]. From that, all three conditions in Eq. (3), have been satisfies,$DFE{\overline{X}}_{DFE}$ is globally asymptotically stable. From the transformed subsystem in Eq. (2), the $EE{\overline{X}}_{EE}$ will be computed by Eq. (12).

$$\frac{di}{dt}=0=\sigma e-Di$$ (12)

Where

$$\frac{e}{i}=\frac{b}{\sigma }$$ (13)

And

$$\frac{de}{dt}=0=\beta si-Ce$$ (14)

Or

$$s=\frac{C}{\beta }\frac{e}{i}$$ (15)

To deliver the first coordinate of $EE{\overline{X}}_{EE},$ as

$$s=\frac{CD}{\beta \sigma };\frac{ds}{dt}=0=A-\alpha e-\alpha i-\beta si-Bs$$ (16)

Which is equal to

$$s\left(\beta i+B\right)=A-\alpha e-\alpha i$$ (17)

we have

$$i\left(\frac{CD}{\sigma }+\alpha \right)=A-\alpha e-\frac{BCD}{\beta \sigma }$$ (18)

After distributing and collecting, which will be more simplified with $i=\frac{\sigma }{D}e$ and by distributing and combining factors, the second coordinate of $EE{\overline{X}}_{EE}$ will be as Eq. (19).

$$e=\left(\frac{D}{CD+\alpha \sigma +\alpha D}\right)\left(\frac{\beta \sigma A-BCD}{\beta \sigma }\right)$$ (19)

Through substitution of Eq. (19), into $i=\frac{\sigma }{D}e,$ the third coordinate of $EE{\overline{X}}_{EE}$ is Eq. (20).

$$i=\frac{\beta \sigma A-BCD}{\beta (CD+\alpha \sigma +\alpha D})$$ (20)

With Eqs. (16), (19), and (20), the $EE{\overline{X}}_{EE}$ in Eq. (4), is Eq. (21).

$$\begin{array}{ccc}\hfill {\overline{X}}_{EE}& =& \left(s,e,i\right)=\left(s^{*},e^{*},i^{*}\right)(\frac{CD}{\beta \sigma },\left(\frac{D}{CD+\alpha \sigma +\alpha D}\right)\hfill \\ & & \times \left(\frac{\beta \sigma A-BCD}{\beta \sigma }\right),\frac{\beta \sigma A-BCD}{\beta (CD+\alpha \sigma +\alpha D})\hfill \end{array}$$ (21)

Which only makes physical sense of$\beta \sigma A-BCD>0$.

Since all $A,B,C,D$are constants, and $\alpha ,\beta ,\sigma$parameters are positive values in Eq. (20), by manipulating $\beta \sigma A-BCD>0,$ the epidemic condition $R_0,$is given as:

$$R_0=\beta \sigma A-BCD>0$$ (22)

By substituting Eq. (4), in $J\left(X\right)$ we have $J\left({\overline{X}}_{EE}\right)$ with eigenvalues${\lambda }_i$, then we have

$$\left|J\left({\overline{X}}_{EE}\right)-{\lambda }_i\right|=0$$ (23)

Which will be expanded through determinant, and the eigenvalues${\lambda }_i$, are determined from the cubic polynomial as Eq. (24).

$$\begin{array}{ccc}\hfill P\left(\lambda \right)& =& {\lambda }^3+\left(B+C+D+\beta i^{*}\right){\lambda }^2+(BC+BD+CD+\beta C{i}^{*}+\beta D{i}^{*}\hfill \\ & & -\sigma \beta s^{*})\lambda +\left(BCD+\beta CD{i}^{*}-\sigma \beta B{s}^{*}\right)=0\hfill \end{array}$$ (24)

Where eigenvalues${\lambda }_i,$are dependent to $\sigma$and $\beta$ parameters, $A,B,C,D$constants, and first and third coordinates of $EE{\overline{X}}_{EE}$, namely $s^{*}$, and $i^{*},$ in Eq. (21).

In a similar manner to the eigenvalues ${\lambda }_i,$for the cubic polynomial in Eq. (8), the eigenvalues${\lambda }_i,$for the cubic polynomial in Eq. (24), are even more difficult to compute without any specific values for $\sigma ,\beta$ parameters and $A,B,C,D$ constants. The Routh-Hurwitz criterion with conditions$a_1>0,a_3>0$, and$a_1{a}_2>a_3$, is again applied to show the cubic polynomial in Eq. (24), to determine the parameters and constants global stability of the $EE{\overline{X}}_{EE}$ in Eq. (21), with the coefficients.

$$a_1=B+C+D+\frac{\beta \sigma A-BCD}{CD+\alpha \sigma +\alpha D}$$ (25)

$$a_2=BC+BD+C\left(\frac{\beta \sigma A-BCD}{CD+\alpha \sigma +\alpha D}\right)+D\left(\frac{\beta \sigma A-BCD}{CD+\alpha \sigma +\alpha D}\right)$$ (26)

$$a_3=CD\left(\frac{\beta \sigma A-BCD}{CD+\alpha \sigma +\alpha D}\right)$$ (27)

From that, in the first condition$a_1>0$, all $A,B,C,D$ constants and $\alpha ,\sigma and\beta$parameters, are all bigger than zero, In the second condition $a_3>0$, we have $\beta \sigma A-BCD>0$, and for the third condition $a_1{a}_2>a_3$, all the eigenvalues${\lambda }_i$, in Eq. (24), have negative real parts, we conclude that the $EE{\overline{X}}_{EE}$ in Eq. (21), is globally asymptotically stable with $\beta \sigma A-BCD>0$.

4. Experimental Results

The proposed transformed subsystem of ordinary differential equation in Eq. (2), was evaluated to consider the effect of $\beta ,\alpha$ and $\rho$ on the globally asymptotically stability and locally unstable conditions of disease-free $DFE{\overline{X}}_{DFE}$ and endemic ${\overline{X}}_{EE}$equilibriums of COVID-19. Since the values of $\sigma$ and $\gamma$parameters cannot be manipulated by human activities as an effective control strategy, they are exempted for their values being changed. Therefore, it is assumed that they do not varies.

The birth and death number are 20.75 and 7.752 per 1000 person per year respectively [61]. In case of vaccination there is three main subjects that should be considered, (i) interval time between susceptibility and temporarily immune, (ii), efficacy rate to produce antibody protection, and (iii), distribution and availability. Based on nature's reports, it takes about 2 weeks or more, in few days to be immunized from the day of receiving the vaccine, and the efficacy rate of widely accepted ones (for example: Sputnik V [24], Modena [32], Pfizer–BioNTech [72]) is more than 90% to produce antibodies [52], and there is a positive correlation between vaccination coverage and socioeconomic factors [26] which in turn makes it difficult to cover all susceptible individuals in poorer countries such as Iran, in a mass immunity program. This fact will be clearer, when all susceptible individuals are not immunized yet, and temporarily immunized cases losses their immunity and turn back into susceptible state from the other side. The real-world data in terms of availability for Iran population is not provided yet, thus scenario based simulation for this part of $\rho$ variable, has been provided.

It is evident, that transmission rate $\beta ,$reduces when $\rho$ is added to the system and it actively does its role, because transmission through direct contact rate decreases [50], but at this stage we suppose that, the highest risk of infectious in a close contact is approximately 10.2% [46] from 9.191 direct contact for each person per day [66] (because of social and physical distancing and also awareness, this number is low, while it can be much more than of that, such that it can be 22 direct contact, per person per a day in a normal relations of life [23]), and it takes 2 days, to transmission from S to E, and also symptoms may appear 2-14 (with the average of 5-6 days) days after exposure to the virus (transmission from E to I) [15] . It usually takes 10-20 days to recover from COVID-19 after onset symptoms (transmission from I to R) [70]. As stated earlier after 3-4 months, reinfection cases has reported (transmission from R to S) [25,68].

Table 4a presents a typical set of numerical values of the model parameters for Iran's real condition, in birth and death rates, which is estimated daily based on statistical center of Iran [61], and evaluated parameters value for COVID-19 pandemic.

Table 4a.

Numerical values of models parameter.

Parameter	Units Value	Value
$b$	$\frac{\frac{20.75}{1000}}{360}$	0.00005
$d$	$\frac{\frac{7.752}{1000}}{360}$	0.00002
$\alpha$	$\frac{1}{90}$	0.011
$\beta$	$\frac{1}{2}$*9191*0.102	0.468741
$\sigma$	$\frac{1}{5.5}$	0.1818
$\gamma$	$\frac{1}{6.5}$	0.1538
$\rho$	$\frac{1}{14}*0.9*$ variant	-

Table 4b shows $A,B,C,D,F$ constants and also the value of $R_0$ in a simulated condition of COVID-19 in Iran, without any control strategies (Note that; this evaluation is done supposing that, people do not care about wearing masks, and physical distance, and also there is no vaccination).

Table 4b.

Positive constants.

Parameter	Value
A	0.0110
B	0.0110
C	0.1818
D	0.1539
F	0.0110
$R_0$	3.0451

Based on numerical values of the model's parameter shown in Table 4a, and positive $A,B,C,D,$ and $F$constant values shown in Table 4b, COVID-19 basic reproduction number is$R_0=3.0451,$which is greater than unity, and it shows that the disease-free equilibrium ${\overline{X}}_{DFE}$ is locally unstable and endemic equilibrium $EE{\overline{X}}_{EE}$is globally asymptotically stable.

The rescaled $seirs$ 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 $i$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 $\beta ,$and $\alpha ,$ to see how much they are effective with no vaccination$\rho ,$ and in the second one, vaccination effect on the stability has been tested. For both situations, the stability of disease-free equilibrium ${\overline{X}}_{DFE}$ in Eq. (7), and endemic equilibrium ${\overline{X}}_{EE}$in Eq. (21) has been evaluated using eigenvalues${\lambda }_i$ of Jacobian matrix$J\left(X\right)$. The proportions of $s,e,i$and $r$have been evaluated as the total population$N$ 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 [73].

4.1. No-Control strategy

Decreasing the numbers of susceptible individuals, depends on values of death rate$d$, and transmission rates$\beta$, $\rho$ and $\alpha .$ the value of $d,$ is constant and it depends on other factors, and also COVID-19 infectious disease mortality and morbidity. The value of transmission rate $\beta ,$ alters with human activities, such as physical distance and wearing masks [58], 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$\beta$ and $\alpha ,$ or increases the transmission rate$\rho$. Based on model's parameter numerical values, in Table 4a and constant's value in Table 4b, with $\beta =0.468741,\alpha =0.011.\rho =0.0,$ for $seirs$ in Eq. (2), the basic reproduction number $R_0$estimated at 3.0451, which is bigger than unity. It implies that, there is a growing infectious, and disease-free equilibrium${\overline{X}}_{DFE}$is locally unstable. Table 5 shows the disease-free equilibrium${\overline{X}}_{DFE}$, endemic equilibrium${\overline{X}}_{EE}$, eigenvalues${\lambda }_i$ of Jacobian matrices$J\left({\overline{X}}_{DFE}\right)$, and$J\left({\overline{X}}_{DFE}\right)$, along the stabilities.

Table 5.

Parameters, eigenvalues and stability of COVID-19 ($\mathit{\beta }=0.468741,\mathit{\alpha }=0.011,\mathit{\rho }=0.00)$.

Point	$s$	$e$	$i$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	Stability
$DFE$	1	0	0	-0.0111	-0.4601	0.1244	Unstable
$EE$	0.3284	0.0361	0.0426	-0.0129 -0.4601	-0.0129 + 0.0411i	-0.3410 + 0.0000i	Stable

From that all 3 eigenvalues${\mathit{\lambda }}_1,{\mathit{\lambda }}_2$ and${\mathit{\lambda }}_3$of $EE{\overline{X}}_{EE},$ have negative real parts for $\beta =0.468741\alpha =0.011.\rho =0.0$ of $s,e$ and $i,$ endemic equilibrium ${\overline{X}}_{EE}$ is globally asymptotically stable (there is no non-negative${\lambda }_i$), but because of at least, one non-negative real part eigenvalue${\mathit{\lambda }}_3$, disease-free equilibrium ${\overline{X}}_{DFE}$is locally unstable. Fig. 2 , illustrates the simulation of Iran's COVID-19 initial conditions of rescaled $s,e,i$ and $i$ variables data, in time series from the epidemic starting point using numerical values of model's parameter in Table 4a.

Fig. 2.

Proportional population in $\mathbf{s},\mathbf{e},\mathbf{i}$ and $\mathbf{r},$ with $\mathit{\beta }=0.468741,\mathit{\alpha }=0.011,\mathit{\rho }=0.0$.

From that the proportion of susceptible$s,$is decreasing, and unidentified infected $e,$ identified infected $i,$ and recovered $r$ from disease are increasing in 69 days after the epidemic outbreak, implies on the $R_0$bigger than unity, and endemic equilibrium point${\overline{X}}_{EE}$ is globally asymptotically stable. Conditions of $s,e,i$ and $r$ continues approximately in next 5 days (approximately day 73), and eventually unidentified infected $e,$ and identified infected $i$ cases start's to decay. Susceptible $s,$and recovered $r,$ reach there steady-states values in the same weeks number, where unidentified infected $e,$and identified infected $i,$are maximum. After approximately 102 days from starting point, the majority of proportional population are recovered individuals $r=0.73,$ and the proportions of susceptible $s,$ unidentified infected $e,$ and identified infected $i,$ cases is 0.183, 0.03 and 0.054, respectively.

4.1.1. Transmission rate $\beta$

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 $\beta$ [48]. Suppose that wearing masks reduces 0.3 risk of infectious (transmission rate will be $\beta =0.328),$and with restrictive measures direct contact of individuals decline to 5 contact per a day per each person (the transmission rate $\beta$ 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 $\beta$ with decrease in direct contact, on ordinary differential equation of $seirs$ model on the$DFE{\overline{X}}_{DFE}$,$EE{\overline{X}}_{EE}$ and eigenvalues ${\mathit{\lambda }}_{\mathit{i}}$ of Jacobian matrices $J\left({\overline{X}}_{DFE}\right)$, and $J\left({\overline{X}}_{DFE}\right)$ along stability.

Table 6a.

Parameters, eigenvalues and stability of COVID-19 ($\mathit{\beta }=0.328\mathit{\alpha }=0.011.\mathit{\rho }=0.0$0).

Point	$s$	$e$	$i$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	stability
$DFE$	1	0	0	-0.0111	-0.4125	0.0767	Unstable
$EE$	0.4693	0.0285	0.0337	-0.0096 - 0.0307i	-0.0096 + 0.0307i	-0.3386 + 0.0000i	Stable

Table 6b.

Parameters, eigenvalues and local stability of COVID-19 ($\mathit{\beta }=0.255\mathit{\alpha }=0.011.\mathit{\rho }=0.0$0)

Point	$s$	$e$	$i$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	stability
$DFE$	1	0	0	-0.0111	-0.3836	0.0479	Unstable
$EE$	0.6037	0.0213	0.0251	-0.0079 - 0.0232i	-0.0079 + 0.0232i	-0.3374 +0.0000i	Stable

From the eigenvalues${\lambda }_i$with non-negative real part${\lambda }_3$ in $\beta =0.328.\alpha =0.011.\rho =0.0$0, the diseases free equilibrium${\overline{X}}_{DFE}$ of $seirs$ is locally unstable (at least one $\lambda$ has non-negative real part), but since there is not any non-negative real part eigenvalue ${\lambda }_i,$ the endemic equilibrium ${\overline{X}}_{DFE}$ is globally asymptotically stable.

From the eigenvalues ${\mathit{\lambda }}_{\mathit{i}},$ with non-negative real part ${\mathit{\lambda }}_3,$ in $\mathit{\beta }=0.255,\mathbf{\alpha }=0.011,\mathbf{\rho }=0.0$ 0, the diseases-free equilibrium ${\overline{\mathit{X}}}_{\mathit{DFE}}$ of $\mathit{seirs}$ model is locally unstable (at least one $\mathit{\lambda }$ has non-negative real part), and ${\overline{\mathit{X}}}_{\mathit{EE}}$ is globally asymptotically stable, because there is not any non-negative real part eigenvalues ${\mathit{\lambda }}_{\mathit{i}},$ for endemic equilibrium. Fig. 3 a shows the simulation result of decreases in risk of infectious by 0.3 through wearing masks, and Fig. 3 b, illustrates direct contact reduces to 5 (as the result of physical distance), for 360 days, while the other parameters are constant.

Fig. 3.

(a). Proportional population in $\mathbf{s},\mathbf{e},\mathbf{i}$ and $\mathbf{r}$ with $\mathit{\beta }=0.328,\mathit{\alpha }=0.011,\mathit{\rho }=0.0$, (b). Proportional population in $\mathbf{s},\mathbf{e},\mathbf{i}$ and $\mathbf{r}$ with $\mathit{\beta }=0.255,\mathit{\alpha }=0.011,\mathit{\rho }=0.0$.

Both of wearing masks and physical distance have the same effect (but with different rates), on transmission rate$\beta ,$ in one direction, with basic reproduction number for both of them greater than unity, $R_0=$ 2.13 for wearing masks and $R_0=$ 1.6566 for decrease in direct contact. Such that, with decrease in transmission rate $\beta$ only with wearing masks, the proportional conditions of$s,e,i$ and $r$ continues approximately in 115 days from starting point to where susceptible $s$ and recovered$r$ cross each other, and unidentified infected $e$ and identified infected $i$ are in their maximum levels on that day, and after that both of them start to decline. The ratios of susceptible s and recovered r, are in maximum and minimum levels in days 150 and 132 with 0.6 and 0.31 values respectively.

By reducing the direct contact between individuals, proportional conditions of$s,e,i$ and $r$ continues approximately in 170 days from starting point, unidentified infected $e$ and identified infected $i$ are in maximum level on that day, and after that $e$and $i$ start to decline. Susceptible $s,$ and recovered$r,$ do not cross over each other even in their minimum (for susceptible s) and maximum (for recovered r) levels in day 215, such that 0.49 of population are susceptible yet, and 0.44 of population have been recovered, and they are temporarily immune from disease. These proportions are approximately 0.026 for unidentified infected e, and 0.035 for identified infected.

4.1.2. Transmission rates of $\beta$ and $\alpha$

One strategy to decrease the transmission rate$\beta$ 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 $\beta =$0.1785, while other parameters are constant. Table 7a shows numerical results of $\beta =0.1785,\alpha =0.011\rho =0.0$0 effect, on the ordinary differential equation of $seirs$ model, disease-free equilibrium${\overline{X}}_{DFE}$, endemic equilibrium${\overline{X}}_{EE}$ and eigenvalues ${\mathit{\lambda }}_{\mathit{i}}$of a Jacobian $J\left({\overline{X}}_{DFE}\right)$and $J\left({\overline{X}}_{EE}\right)$ along stability.

Table 7a.

Parameters, eigenvalues and stability of COVID-19 ($\mathit{\beta }=0.1785,\mathit{\alpha }=0.011,\mathit{\rho }=0.0$0).

Point	$s$	$e$	$i$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	stability
$DFE$	1	0	0	-0.0111	-0.3486	0.0128	Unstable
$EE$	0.5103	0.0193	0.0135	-0.0061 - 0.0105i	-0.0061 + 0.0105i	-0.3361 +0.0000i	Stable

From the eigenvalue${\lambda }_3,$with non-negative real part in $\beta =0.1785,\alpha =0.011,\rho =0.0$0 the diseases-free equilibrium${\overline{X}}_{DFE}$ of $seirs$ is locally unstable (at least one $\lambda$ has non-negative real part) and endemic equilibrium point${\overline{X}}_{EE}$ is globally asymptotically stable.

There are possibilities that could be presumed, such as transmission rate of $\alpha$decreases [13]. Suppose that, 90 days of transmission from $r$to $s$compartment, increases to 120 days ($\alpha =$0.0083), while transmission rate $\beta$ is as previously mentioned in 4.1. Table 7b shows increase in temporary immunity duration to 120 days $(\beta =0.468741,\alpha =0.0083,\rho =0.0$0) effect on the ordinary differential equation of $seirs$ model disease-free equilibrium${\overline{X}}_{DFE}$, endemic equilibrium${\overline{X}}_{EE}$ and eigenvalues ${\mathit{\lambda }}_{\mathit{i}}$of a Jacobian $J\left({\overline{X}}_{DFE}\right)$and $J\left({\overline{X}}_{EE}\right)$ along stability.

Table 7b.

Parameters, eigenvalues and stability of COVID-19 ($\mathit{\beta }=0.468741,\mathbf{\alpha }=0.0083,\mathbf{\rho }=0.0$0).

Point	$s$	$e$	$i$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	stability
$DFE$	1	0	0	-0.0083	-0.4601	0.1244	Unstable
$EE$	0.3284	0.0280	0.0331	-0.0099 - 0.0362i	-0.0099 + 0.0362i	-0.3398 + 0.0000i	Stable

From the non-negative real part of eigenvalues${\lambda }_3$, disease-free equilibrium ${\overline{X}}_{DFE}$ is locally unstable, and endemic equilibrium${\overline{X}}_{EE}$of $seirs$ model is globally asymptotically stable (all 3 eigenvalues${\lambda }_i$ have negative real parts). Fig. 4 a, illustrates the simulation result of effect of simultaneously decrease in direct contact per each person per day to 5, and risk of infectious to 0.0714 because of wearing masks, with $\beta =0.1785,\alpha =0.011,\rho =0.0$0 for 360 days. Fig. 4b, shows increase of the temporary immunity duration to 120 days while the other effective parameters are constant as they were in 4.1., with $\beta =0.468741,\alpha =0.0083,\rho =0.0$0, for 360 days.

Fig. 4.

(a). Proportional population in $\mathbf{s},\mathbf{e},\mathbf{i}$ and $\mathbf{r}$ with $\mathit{\beta }=0.1785,\mathit{\alpha }=0.011,\mathit{\rho }=0.0$, (b). Proportional population in $\mathbf{s},\mathbf{e},\mathbf{i}$ and $\mathbf{r}$ with $\mathit{\beta }=0.468741,\mathit{\alpha }=0.0083,\mathit{\rho }=0.0$.

Decreasing transmission rete $\beta$ (Fig. 4a) causes the infectious disease spreading slowly amongst susceptible individuals s with$R_0=1.1596$, such that unidentified infected e, and identified infected $i,$ do not reach their maximum points in 360 simulated days. At the end of simulated time about 0.94 of population are still susceptible s, and these ratios for recovered r, unidentified e, and identified infected $i,$is 0.05, 0.006 and 0.004 respectively.

Increase in temporary immunity duration from 90 to 120 days, do not alter the $R_0$ value, as the same as, decreasing transmission rate $\beta ,$ and proportional conditions of $s,e,i$and $r$ remain approximately as mentioned in 4.1., without any control strategies, with$R_0=3.0125$. In day 75, unidentified $e,$and identified infected $i$are in their maximum points, and after that they start to decline. Susceptible $s,$and recovered $r,$reach there steady-states values in the same weeks number, where unidentified $e,$and identified infected $i$are maximum. After approximately 109 days from starting point, the majority of proportional population, are recovered individuals $r$(0.765), and proportions of susceptible $s,$ unidentified $e$and identified infected $i,$is 0.2, 0.009 and 0.035, respectively.

4.2. Transmission rates of $\beta ,\alpha ,$and $\rho$

Combining control strategies such as restrictive measures to decrease direct contact, wearing masks to reduce risk of infectious (in order to decrease transmission rate $\beta ),$ 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 $R_0=1.1596)$, 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${\overline{X}}_{DFE}$, endemic equilibrium${\overline{X}}_{EE}$, eigenvalues ${\lambda }_i$ of Jacobian matrix $J\left({\overline{X}}_{DFE}\right)$ and $J\left({\overline{X}}_{EE}\right)$ 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$seirs$ model.

Table 8a.

Parameters, eigenvalues and stability of COVID-19, ($\mathit{\beta }=0.1785,\mathit{\alpha }=0.0083,\mathit{\rho }=0.0$).

Point	$s$	$e$	$i$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	stability
$DFE$	1	0	0	-0.0083	-0.3486	0.0128	Unstable
$EE$	0.8624	0.0057	0.0068	-0.0046 - 0.0095i	-0.0046 + 0.0095i	-0.3360 +0.0000i	Stable

Table 8b.

Parameters, eigenvalues and stability of COVID-19 ($\mathit{\beta }=0.1785,\mathbf{\alpha }=0.0083.\mathbf{\rho }=0.0$14).

Point	$s$	$e$	$i$	${\mathit{\lambda }}_1$	${\mathit{\lambda }}_2$	${\mathit{\lambda }}_3$	stability
$DFE$	0.8557	0	0	-0.0970	-0.3351	-0.0006	Stable
$EE$	0.8624	-0.0018	-0.0021	0.0006	-0.0974	-0.3356	Unstable

Description of these conditions $\beta =0.1785,\alpha =0.0083,\rho =0.0$0, with$R_0=1.1596,$is such as mentioned for Table 7a, and it shows that at least there is one non-negative eigenvalue ${\lambda }_i,$ in disease-free equilibrium ${\overline{X}}_{DFE}$ of rescaled $seirs$ model. Thus disease-free equilibrium ${\overline{X}}_{DFE}$ is locally unstable and endemic equilibrium ${\overline{X}}_{EE}$ is globally asymptotically stable.

Beside strategies to decrease transmission rate $\beta ,$ and increase the duration of temporary immunity (transmission rate $\alpha )$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 $\left(\beta =0.1785,\alpha =0.0083,\rho =0.014\right)$ as has been shown in Table 8b, the disease-free equilibrium ${\overline{X}}_{DFE},$ has no non-negative real part eigenvalues ${\lambda }_i$ along its stability and is globally asymptotically stability, with $R_0=9923.$

From the negative real part eigenvalues${\lambda }_i,$ the disease-free equilibrium${\overline{X}}_{DFE}$ is globally asymptotically stable, because there is not any non-negative real part eigenvalues for${\lambda }_i,$ and endemic equilibrium${\overline{X}}_{EE}$of $seirs$ is locally unstable (${\lambda }_1$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 $\left({14}^{-1}*0.9*x=0.014\right)$ of the susceptible individuals to stabilize the disease-free equilibrium${\overline{X}}_{DFE}$ globally asymptotically.

Fig. 5a, illustrates the simulation result for $\beta =0.1785,\alpha =0.0083,\rho =0.0$0, and Fig. 5b, shows the simulation result for $\beta =0.1785,\alpha =0.0083,\rho =0.0$14, in 360 days.

Fig. 5.

(a). Proportional population in $\mathbf{s},\mathbf{e},\mathbf{i}$ and $\mathbf{r}$ with $\mathit{\beta }=0.1785,\mathit{\alpha }=0.0083,\mathit{\rho }=0.0$, (b). Proportional population in $\mathbf{s},\mathbf{e},\mathbf{i}$ and $\mathbf{r}$ with $\mathit{\beta }=0.1785,\mathit{\alpha }=0.0083,\mathit{\rho }=0.014$.

Decrease in transmission rate $\beta ,$and increase in temporary immunity duration shown in Fig. 5a, reduces the speed of infectious spread, but does not stable the disease-free equilibrium${\overline{X}}_{DFE}$ globally asymptotically.

Considering the vaccination $\rho$effect on the ordinary differential equation $seirs$ model stabilizes the disease-free equilibrium${\overline{X}}_{DFE}$ globally asymptotically and makes the endemic equilibrium ${\overline{X}}_{EE}$locally unstable. In other words, the basic reproduction number $R_0$ 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.

5. Conclusions

In this paper transformed subsystem of ordinary differential equation $seirs$has been used, to focus on the simulation of vital dynamics (birth $b$ and death $d$ rates) with different rates and transmission rates $\beta ,\alpha$ and $\rho$. From the experimental results, various scenarios has been simulated to examine stability (locally and globally) of disease-free equilibrium$DFE{\overline{X}}_{DFE}$ and endemic equilibrium points $EE{\overline{X}}_{EE}$ when $R_0$ 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 $\beta ),$ and also increase in temporary immunity duration (without vaccination $\rho =0.00)$, at least one eigenvalue${\lambda }_i$ of Jacobian matrix of disease-free equilibrium$DFE{\overline{X}}_{DFE},$ has non-negative real part which implies that $DFE{\overline{X}}_{DFE}$ is locally unstable, and endemic equilibrium $EE{\overline{X}}_{EE}$ 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 $R_0$ less than or equal unity $\left(R_0\le 1\right).$ 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 $\beta ,$ increasing temporary immunity duration and vaccination) in poorer countries such as Iran, makes $R_0$ less than unity, and all Jacobian matrix $J\left({\overline{X}}_{DFE}\right)$ eigenvalues ${\lambda }_i$ of disease-free equilibrium with negative real part, which implies that disease free equilibrium$DFE{\overline{X}}_{DFE}$ is globally asymptotically stable, and endemic equilibrium $EE{\overline{X}}_{EE}$ is locally unstable.

Briefly findings of this research could be stated as the initial conditions for decrease in transmission rates of $\beta$ and $\alpha ,$ reached the corresponding disease-free equilibrium ${\overline{X}}_{DFE}$ locally unstable, and globally asymptotically stable for endemic equilibrium${\overline{X}}_{EE}$. The initial conditions for decrease in transmission rate$s\beta$and$\alpha ,$ and increase in $\rho ,$ reached the corresponding disease-free equilibrium ${\overline{X}}_{DFE}$ globally asymptotically stable, and locally unstable in endemic equilibrium${\overline{X}}_{EE}$.

For future research, two kinds of researches can be done. Firstly, investigating a mathematical model to find the critical point (bifurcation) where$R_0$ is equal unity, what would be the proportions of $s,e,iandr$ 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.

Funding

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

CRediT authorship contribution statement

Mohammad Qaleh Shakhany: Writing – original draft, Conceptualization, Methodology, Software. Khodakaram Salimifard: Conceptualization, Methodology, Validation, Supervision, Writing – review & editing.

Declaration of Competing Interest

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.