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. 2017 Nov 8;146:70–89. doi: 10.1016/j.matcom.2017.10.006

Generalized reproduction numbers, sensitivity analysis and critical immunity levels of an SEQIJR disease model with immunization and varying total population size

Supatcha Siriprapaiwan a, Elvin J Moore a,b,*, Sanoe Koonprasert a
PMCID: PMC7127447  PMID: 32288111

Abstract

An SEQIJR model of epidemic disease transmission which includes immunization and a varying population size is studied. The model includes immunization of susceptible people (S), quarantine (Q) of exposed people (E), isolation (J) of infectious people (I), a recovered population (R), and variation in population size due to natural births and deaths and deaths of infected people. It is shown analytically that the model has a disease-free equilibrium state which always exists and an endemic equilibrium state which exists if and only if the disease-free state is unstable. A simple formula is obtained for a generalized reproduction number Rg where, for any given initial population, Rg<1 means that the initial population is locally asymptotically stable and Rg>1 means that the initial population is unstable. As special cases, simple formulas are given for the basic reproduction number R0, a disease-free reproduction number Rdf, and an endemic reproduction number Ren. Formulas are derived for the sensitivity indices for variations in model parameters of the disease-free reproduction number Rdf and for the infected populations in the endemic equilibrium state. A simple formula in terms of the basic reproduction number R0 is derived for the critical immunization level required to prevent the spread of disease in an initially disease-free population. Numerical simulations are carried out using the Matlab program for parameters corresponding to the outbreaks of severe acute respiratory syndrome (SARS) in Beijing, Hong Kong, Canada and Singapore in 2002 and 2003. From the sensitivity analyses for these four regions, the parameters are identified that are the most important for preventing the spread of disease in a disease-free population or for reducing infection in an infected population. The results support the importance of isolating infectious individuals in an epidemic and in maintaining a critical level of immunity in a population to prevent a disease from occurring.

Keywords: Epidemic model, Generalized reproduction number, Sensitivity analysis, Critical immunization level, SARS

1. Introduction

The SEQIJR model (see, e.g., [[6], [8], [9]]) is a generalization of the well-known SEIR model of an infectious disease in which a population at risk is separated into the four categories of susceptible (S), exposed (E), infectious (I) and recovered (R) (see, e.g., [[1], [12], [27], [29]]). A susceptible person is an uninfected person who can be infected through contact with an infectious or exposed person, an exposed person is someone who has come into contact with an infectious person but is asymptomatic, an infectious person is symptomatic, and a recovered person is someone who has recovered from the disease. It is assumed that a recovered individual cannot become infected again. In an SEQIJR model, two extra categories of quarantined (Q) and isolated (J) are added to the SEIR model. A quarantined person is an exposed person who is removed from contact with the general population and an isolated person is an infectious person who is removed from contact with the general population, usually by being admitted to a hospital. Quarantine and isolation are important in controlling epidemics because they are effective methods of reducing the contact between infected and susceptible populations.

In this paper, we consider a generalization of the basic SEQIJR model developed by Gumel et al. [6] for the severe acute respiratory syndrome (SARS) outbreaks of November 2002–July 2003. The SEQIJR model of Gumel et al. for the SARS outbreaks is useful for study for several reasons. Firstly, Gumel et al. suggested parameter values which can be used to obtain physically reasonable models for the four SARS outbreaks in Beijing, Hong Kong, Singapore and Toronto. Secondly, considerable data are available on the SARS outbreaks and mathematical models and the effectiveness of control measures have been studied by many authors (see, e.g., [[3], [7], [11], [13], [17], [18], [19], [20], [21], [26], [30]]).

In building their model, Gumel et al. assumed that there was no immunization (no vaccine was available for SARS), that susceptible people were born into the community at a constant rate and that the population was constant. They analyzed the effects of quarantine and isolation on the transmission of SARS and showed that an effective isolation policy was more important than quarantine or a combination of quarantine and a less effective isolation policy in reducing the transmission of the disease. A similar conclusion was obtained by Koonprasert et al. [9] in a numerical study of the effectiveness of quarantine and isolation. However, as noted by a number of authors (see, e.g., [[4], [5], [14], [15], [24], [28]]), the effectiveness of isolation requires strict hospital hygiene to reduce the hospital-based (nosocomial) spread of a disease.

The modifications introduced in our analysis include: (1) the existence of an immunization program, (2) changes in the total population due to births and deaths, (3) the derivation of generalized reproduction numbers for disease-free states and endemic equilibrium states, (4) a sensitivity analysis of the effect of variations in parameter values on the reproduction numbers for disease-free states and on the infected populations of the endemic equilibrium state, and (5) a derivation of a simple formula for the critical immunization level required to prevent spread of the disease in an initially disease-free population.

The organization of the paper is as follows. In Section 2, we define the mathematical model. In Section 3, we obtain analytical formulas for disease-free and endemic equilibrium points. Section 4 contains an analysis of the local stability of the equilibrium points and gives derivations of formulas for the generalized reproduction numbers. Formulas are derived for sensitivity indices in Section 5 and for critical immunization levels in Section 6. Section 7 and 8 contain results of numerical simulations using Matlab for the parameter values proposed by Gumel et al. [6] for four of the cities affected by the SARS outbreaks of 2002 and 2003, namely Beijing, Hong Kong, Singapore and Toronto. In Section 9, we give a discussion of results and conclusions.

2. The SEQIJR model

A flow chart of the dynamics of our SEQIJR model is shown in Fig. 1. The system of equations for the SEQIJR model that we use is given in Eqs. (1), (2), (3), (4), (5), (6). Definitions of the parameters in the model are given in Table 1. For the immunization program, we assume that a vaccine is available and that a fraction ν of the susceptible population acquires immunity from vaccination per unit time. The effect of the immunization can then be modeled as a direct transfer of a fraction νS of the susceptible people from the S to the R class per unit time. We also assume that people in the R class cannot be infected for a second time.

dSdt=ΠLSNDSS, (1)
dEdt=LSNDEE, (2)
dQdt=γ1EDQQ, (3)
dIdt=k1EDII, (4)
dJdt=γ2I+k2QDJJ, (5)
dRdt=νS+σ1I+σ2JμR, (6)

where

L=β(I+εEE+εQQ+εJJ) (7)

is an effective infectiousness rate of the S population due to contact with asymptomatic exposed and quarantined populations (E,Q) and symptomatic infectious and isolated populations (I, J), and where

DS=μ+ν,DE=γ1+k1+μ,DQ=k2+μ,DI=γ2+σ1+d1+μ,DJ=σ2+d2+μ (8)

are decay rate coefficients for S, E, Q, I, and J populations respectively.

Fig. 1.

Fig. 1

The flow chart of the SEQIJR model (generalized from: Gumel et al. [6]).

Table 1.

Parameters for the SEQIJR model (rates are per day).

Param Definition
Π Rate of inflow of susceptible individuals into a region
or community through birth or migration.
μ The natural death rate for disease-free individuals
ν Rate of immunization of susceptible individuals
β Infectiousness and contact rate between a susceptible
and an infectious individual
εE Modification parameter associated with infection from an
exposed asymptomatic individual
εQ Modification parameter associated with infection from a
quarantined individual
εJ Modification parameter associated with infection from an
isolated individual
γ1 Rate of quarantine of exposed asymptomatic individuals
γ2 Rate of isolation of infectious symptomatic individuals
σ1 Rate of recovery of symptomatic individuals
σ2 Rate of recovery of isolated individuals
k1 Rate of development of symptoms in asymptomatic individuals
k2 Rate of development of symptoms in quarantined individuals
d1 Rate of disease-induced death for symptomatic individuals
d2 Rate of disease-induced death for isolated individuals

Adapted from Gumel et al. [6]

Although the basic SEQIJR model only contains 6 equations, one each for S,E,Q,I,J,R, we have found it convenient (see, e.g., [9]) to add 2 extra equations for easy computation of the total population size N(t)=S(t)+E(t)+Q(t)+I(t)+J(t)+R(t) at time t and the death rate D(t) due to disease at time t.

dNdt=Πd1Id2JμN, (9)
dDdt=d1I+d2J. (10)

The equations of the Gumel et al. model can be obtained from Eqs. (1), (2), (3), (4), (5), (6) by setting ν=0 and assuming that the total population is constant, i.e., that Π=d1I+d2J+μN.

3. Equilibrium analysis

Theorem 1

The system (1), (2), (3), (4), (5), (6) and (9) has two equilibrium states:

  • 1.

    Disease-free: (S1,E1,Q1,J1,R1,N1)=(ΠDS,0,0,0,0,νΠμDS,Πμ).

  • 2.
    Endemic:
    S2=1DS(ΠαSI2),E2=αEI2,Q2=αQI2,I2=Π(μαLDSαS)αS(μαLDSαN),J2=αJI2,R2=νΠμDSαRI2,N2=1μ(ΠαNI2), (11)
    where
    αE=DIk1,αS=DEαE,αQ=γ1DQαE,αJ=γ2+k2αQDJ,αR=1μ(νDSαSσ1σ2αJ),αN=d1+d2αJ,αL=β(1+εEαE+εQαQ+εJαJ),L2=αLI2. (12)

Proof

The equilibrium points are solutions of the system of nonlinear algebraic equations obtained by setting dSdt=dEdt=dQdt=dIdt=dJdt=dRdt=dNdt=0 in the system (1), (2), (3), (4), (5), (6) and (9) .

We begin by solving the system of nonlinear algebraic equations for S,E,Q,J,R and N as functions of I to obtain the equations in (11) and (12). Then, on substituting these solutions into (2), we obtain

αSI2(μαLDSαN)ΠI(μαLDSαS)=0. (13)

Therefore, two possible values for I are I1=0 and the solution for I2 in (11). The proof is complete.  □

For an endemic equilibrium point, all populations (S2,E2,Q2,I2,J2,R2,N2) must be non-negative and the infectious population I2 must be positive.

Theorem 2

The endemic equilibrium exists with all populations non-negative and positive infected populations if and only if αS>αN and μαLDSαS>0 .

Proof

We first prove that if I20 then αS>αN. From (11), we have

μN2=μ(S2+E2+Q2+I2+J2+R2),=ΠαSI2+μ(αE+αQ+1+αJ)+σ1+σ2αJI2. (14)

Then since μN2=ΠαNI2, we obtain

(αSαN)I2=μ(αE+αQ+1+αJ)+σ1+σ2αJI2,and therefore
αSαN=μ(αE+αQ+1+αJ)+σ1+σ2αJ>0,since I20. (15)

Therefore, if αSαN, then I2=0 and the endemic equilibrium does not exist.

We now prove that if αS>αN and μαLDSαS>0, then all populations are positive. If αS>αN and μαLDSαS>0, then μαLDSαN>0. So we have

I2=Π(μαLαSDS)αS(μαLDSαN)>0,E2=αEI2>0,N2=1μ(ΠαNI2)=ΠαL(αSαN)αS(μαLDSαN)>0, (16)
S2=1DS(ΠαSI2)=ΠαSαNμαLDSαN>0,R2=1μ(νS2+(σ1+σ2αJ)I2)>0.

Therefore, the endemic equilibrium exists and all populations are positive.

We next note that if μαLDSαS=0, then from (11) I2=0 and the endemic equilibrium does not exist.

Finally, if μαLDSαS<0 and N2=ΠαL(αSαN)αS(μαLDSαN)>0,

thenμαLDSαN>0andI2=Π(μαLαSDS)αS(μαLDSαN)<0.

The proof is complete.  □

4. Generalized reproduction numbers

We consider three cases for generalized reproduction numbers Rg with the property that a given population is locally asymptotically stable if Rg<1 and unstable if Rg>1. These generalized reproduction numbers are useful because they give a simple test for the local asymptotic stability of an initial population. Three useful cases are as follows:

  • 1.

    Basic reproduction number R0. This number is defined for a population that is 100% susceptible so that if R0<1 and infected people enter the population then the number of secondary infections is less than the number of primary infections, whereas if R0>1 then the number of secondary infections is greater.

  • 2.

    Reproduction number Rdf for disease-free equilibrium.

  • 3.

    Reproduction number Ren for endemic equilibrium.

The basic reproduction number R0 and disease-free reproduction number Rdf of the SEQIJR model can be obtained from the next-generation method of van den Driessche and Watmough [23] or by using Lyapunov’s first method, i.e., by checking the eigenvalues of the Jacobian of the linearized system about populations with all infected populations equal to zero (see, e.g.,[16]). However, as far as the authors are aware, the endemic equilibrium reproduction number Ren can only be obtained by checking the eigenvalues of the Jacobian of the linearized system about the endemic equilibrium point.

4.1. The next-generation method

For the next-generation method, we write X=(E,S,Q,I,J,R)T and regard E as the next-generation “infected” population [23]. Then, (S,Q,I,J,R)T can be regarded as the remaining populations. Note that, in the SEQIJR model in this paper, an infected person always enters the E population first and then later enters the Q, I, J or R populations. Then the model can be written in the usual next-generation form as dXdt=F(X)V(X), where

F(X)=SLN00000,V(X)=DEEΠ+SLN+DSSγ1E+DQQk1E+DIIγ2Ik2Q+DJJνSσ1Iσ2J+μR. (17)

Then, a disease-free reproduction number is defined as the largest eigenvalue of the matrix FV1, where F and V are the Jacobian matrices of F(X) and V(X) at the disease-free equilibrium point ΠDS,0,0,0,0,νΠμDS. We have used the Maple software package to find an analytic formula for the largest eigenvalue of FV1, i.e., for Rdf, and obtained the result:

Rdf=μβDSDEεE+k1DI+εQγ1DQ+εJk2γ1DQDJ+εJk1γ2DIDJ. (18)

From the next-generation method, if the disease-free reproduction number Rdf<1, then the disease-free equilibrium point is locally asymptotically stable and if Rdf>1 then it is unstable.

We will now prove that Rdf can be rewritten in a much simpler form and that Rdf>1 also corresponds to the condition that the endemic equilibrium exists.

Theorem 3

An alternative formula for the disease-free reproduction number is Rdf=μαLDSαS and the endemic equilibrium exists if and only if Rdf>1 , i.e., if and only if the disease-free equilibrium point is unstable. The formula for the basic reproduction number is R0=αLαS .

Proof

From (18) and using the definitions of αS, αE, αQ and αJ in (12), we have

Rdf=μβDSDEεE+k1DI+εQγ1DQ+εJk2γ1DQDJ+εJk1γ2DIDJ=μβDSDEαEεEαE+1+εQγ1DIDQk1+εJk2γ1DIDQDJk1+εJγ2DJ=μβ1+εEαE+εQαQ+εJαJDSαS=μαLDSαS. (19)

Then, the formula for the endemic infectious population I2 in (11) can be written as:

I2=ΠDSμαLDSαN(Rdf1). (20)

Therefore, I2>0 if and only if Rdf>1, i.e., if and only if the disease-free equilibrium point is unstable.

To obtain the basic reproduction number R0, we consider the special case that the initial population is the 100% susceptible population. For this case, we have S=N, which corresponds to ν=0 or DS=μ. Thus the basic reproduction number R0 of the SEQIJR model is

R0=αLαS. (21)

The proof is complete.  □

4.2. Generalized reproduction numbers and asymptotic stability of equilibrium points

In the previous section, we used the next-generation method of [23] to obtain formulas for the reproduction number Rdf for the disease-free equilibrium point and for the basic reproduction number R0. We also showed that the endemic equilibrium point exists if and only if Rdf>1.

In this section, we derive generalized reproduction numbers Rg by using Lyapunov’s first method (see, e.g., [16]) of linearizing the system about an equilibrium point and checking that the real parts of all eigenvalues of the Jacobian at the equilibrium point are negative for Rg<1 and that the real part of at least one eigenvalue is positive for Rg>1.

The linearized model of the system at an equilibrium point (S,E,Q,I,J,R) of (1), (2), (3), (4), (5), (6) can be written in the form:

dxdt=Jx, (22)

where x=(S,E,Q,I,J,R)T(S,E,Q,I,J,R)T and J is the Jacobian matrix at an equilibrium point given by:

J=J(S,E,Q,I,J,R)=J11J12J13J14J15J16J11DSJ12DEJ13J14J15J160γ1DQ0000k10DI0000k2γ2DJ0ν00σ1σ2μ, (23)

where (we omit the on the populations in the matrix to save space)

J11=LN+LSN2DS,J12=βεESN+LSN2,J13=βεQSN+LSN2,J14=βSN+LSN2,J15=βεJSN+LSN2,J16=LSN2. (24)

As usual for linear systems, we can assume that Eq. (22) has a solution of the form x(t)=eλtv, where λ is an eigenvalue of J and v is the corresponding eigenvector.

For the 6 × 6 Jacobian matrix in (23) it is not possible to obtain useful analytical expressions for all six eigenvalues either by a direct analysis or by solving the characteristic equation

P(λ)=det(JλI)=0, (25)

where I is a 6 × 6 identity matrix. It is also not possible to obtain useful analytical tests for the real parts of eigenvalues to be negative from the six Routh–Hurwitz conditions (see, e.g., [16]). However, it is possible to check two of the six Routh–Hurwitz conditions for the real parts of the six eigenvalues to be negative. These conditions are usually stated in terms of determinants derived from the coefficients of the characteristic polynomial. However, we will use the two equivalent necessary conditions for the real parts of the six eigenvalues to be negative that the sum of the six eigenvalues (trace (J)) must be negative and the product of the six eigenvalues (det (J)) must be positive.

Now, for the disease-free populations, we have L1=0, Rdf=μαLDSαS<1, and S1N1=μDS. Then,

trace(J1)=μβεEDSDSDEDQDIDJμ<0 (26)

since βεEαE<αL and μαL<DSαS, and therefore

μβεEDS<μαLDSαE<αSαE=DE.

For the endemic population, L2>0 and then

trace(J2)=L2N2+βεES2N2DSDEDQDIDJμ. (27)

Now, from Eq. (16), we have

S2N2=αSαL,and thereforeβεES2N2=βεEαSαL<αSαE=DE, (28)

and trace(J2)<0.

To evaluate the determinant of the Jacobian J in (23), we first used the symbolic algebra toolbox in Matlab to find the following analytic expression:

det(J)={μDEDIDJDQLN+μDIDJDQDSLS+μDIDJDSLSγ1.+μDJDQDSLSk1+DJDQDSLSk1σ1+μDIDSLSγ1k2+μDQDSLSγ2k1+DIDSLSγ1k2σ2+DQDSLSγ2k1σ2+μDEDIDJDQDSN2μDEDIDJDQLS+νDEDIDJDQLS+μDJDQDSNSβk1+μDIDJDQDSNSβεE+μDIDJDSNSβεQγ1.+μDIDSNSβεJγ1k2+μDQDSNSβεJγ2k1}N2. (29)

Then, after considerable straightforward algebra, we obtained the following equivalent expression for the determinant.

det(J)=DEDIDJDQDSNαS{μαSN+μLαSDS.+LSN.σ1+σ2αJ+μ(1+αE+αQ+αJ)}(1Rg), (30)

where a generalized reproduction number can be defined as:

Rg=SNμαLN+αSLμαSN+μLαSDS+LSNσ1+σ2αJ+μ(1+αE+αQ+αJ). (31)

Formulas for the disease-free reproduction number (Rdf) and the endemic equilibrium reproduction number (Ren) can be easily obtained from Eq. (31). The results are summarized in the following theorem.

Theorem 4

Necessary conditions for stability of the two equilibrium points of the SEQIJR model are:

  • 1.
    Disease-free equilibrium.
    Rdf=μαLDSαS<1,R0=αLαS<1(ν=0). (32)
  • 2.
    Endemic equilibrium.
    Ren=μN2+αSI2μN2+I2αS+μαLDSαNDS<1. (33)
    Further, the endemic equilibrium exists if and only if Ren<1 .

Proof

For a disease-free population, we have L1=0 and S1N1=μDS, and therefore from (31) the reproduction numbers for the disease-free cases are as stated in part 1 of the theorem. These results are in agreement with the results from the next-generation method.

A simple formula for the endemic equilibrium number can be obtained by substituting the expressions in Eqs. (11)(15) and (16) for the endemic equilibrium populations into Eq. (31). We then obtain the result given in part 2 of the theorem.

Since the endemic equilibrium exists only if μαLDSαN>0, it can be seen from Eq. (33) that Ren<1 when the endemic equilibrium exists and that Ren1 when I20, i.e., at the transition between the endemic and disease-free equilibrium points.  □

Note: In this example, it is possible to obtain the reproduction numbers from the trace and determinant of the Jacobian at an equilibrium point because the change in stability occurs due to a single real eigenvalue changing value from negative to positive. A change in value of the determinant cannot be used to detect a change in stability if the change is due to a complex conjugate pair of eigenvalues crossing the imaginary axis, e.g., if an Andronov–Hopf bifurcation occurs (see, e.g., [10]). Therefore, a negative trace and a positive determinant for the Jacobian of a system of 6 equations are necessary but not sufficient conditions for the real parts of all eigenvalues to be negative and for an equilibrium point to be locally stable. However, for any given set of parameter values it is very easy to check if Rg<1 corresponds to asymptotic stability of an equilibrium point by numerically computing all eigenvalues of the Jacobian matrix with any of the standard mathematical software packages such as Matlab, Mathematica, Maple etc.

5. Sensitivity analysis

Sensitivity analysis is important as it can be used for determining the parameters which are of most importance in reducing the level of a disease (see, e.g., [[2], [22]]). The normalized sensitivity index for a quantity Q with respect to a parameter h is defined by:

Φ(Q|h)=hQQh=hhlog(Q). (34)

The sensitivity indices can be calculated in at least three ways, namely, (1) by direct differentiation of formulas for Q, (2) by the method in Chitnis et al. [2] of linearizing the Eqs. (1), (2), (3), (4), (5), (6) about the equilibrium point to obtain a system of linear algebraic equations for the indices and then solving the set of equations or (3) by a Latin hypercube sampling technique (see, e.g., [25]). In this section, we will use the method of direct differentiation as it gives analytical expressions for the indices. However, as a check on our formulas we have compared numerical values from our formulas with numerical results we obtained using the method of Chitnis et al.

5.1. Sensitivity analysis of the disease-free reproduction number

The normalized sensitivity indices can be computed from the formula in (19) as

Φ(Rdf|h)=hhlog(μ)+log(αL)log(αS)log(DS). (35)

We used the symbolic algebra package in Matlab to derive normalized sensitivity indices for the fourteen parameters of the model from Eq. (35) and then simplified the expressions to obtain the list shown in Table 2.

Table 2.

Normalized sensitivity indices Φ(Rdf|h) for disease-free reproduction number.

Param Normalized sensitivity index
β 1
μ νDSμDEμDI+μβk1DQDJαL[εEDQDJ+εQDJ(γ1k1αQ)
+εJ(k2(γ1k1αQ)k1DQαJ)].
ν νDS
εE βεEαEαL
εQ βεQαQαL
εJ βεJαJαL
γ1 βαQDJαL[DJεQ+k2εJ]γ1DE
γ2 βγ2k1DQDJαL[DQDJεE+γ1DJεQ+(k1DQ+k2γ1)εJ]γ2DI
σ1 βσ1k1DQDJαL[DQDJεE+γ1DJεQ+k2γ1εJ]σ1DI
σ2 βσ2αJεJDJαL
k1 βDJαL[DJαEεE+DJαQεQ+k2αQεJ]+γ1+μDE
k2 βk2αQDQDJαL(DJεQμεJ)
d1 βd1k1DQDJαL[DQDJεE+γ1DJεQ+k2γ1εJ]d1DI
d2 βd2DJαLαJεJ

The sensitivity indices for the basic reproduction number R0 can be obtained from the results in Table 2 by setting ν=0, i.e., DS=μ.

5.2. Sensitivity analysis of the endemic equilibrium point

In this section, we derive formulas for the sensitivity indices for the endemic equilibrium populations. In particular, we derive indices for the endemic infectious population I2. The sensitivity indices for the other endemic populations can then be easily obtained from the sensitivity indices for I2 using the formulas in Eqs. (11) and (12).

The formulas for the sensitivity indices for I2 can be computed from the formula in (11) as:

Φ(I2|h)=hhlog(Π)+log(μαLDSαS)log(αS)log(μαLDSαN).=hhΠΠ+ξhμαLDSαSψhμαLDSαNωhαS, (36)

where

ξh=h(μαLDSαS),ψh=h(μαLDSαN),ωh=hαS. (37)

The formulas for the coefficients ξh, ψh, ωh of the sensitivity indices in (37) are shown in Table 3. The formulas for the natural birth rate Π and natural death rate μ are not included in Table 3 because these parameters are very difficult to change for human populations. As stated previously, the indices can also be computed using the method of Chitnis et al. [2]. As a check on our results, we have compared numerical results obtained from the formulas in Table 3 with results obtained from the method of Chitnis et al.

Table 3.

Formulas for endemic sensitivity coefficients in Eq. (37).

Par ξ ψ ω
β μαL μαL 0
ν ναS ναN 0
εE μβεEαE μβεEαE 0
εQ μβεQαQ μβεQαQ 0
εJ μβεJαJ μβεJαJ 0
γ1 γ1αE(DS γ1αE(DSd2k2DJDQ. γ1αE
+μβDQ(εQ+εJk2DJ)) +μβDQ(εQ+εJk2DJ))
γ2 γ2k1(DEDS+μβεJk1DJ γ2k1(DSd2DJ(k1+γ1k2DQ)+μβεJk1DJ DEγ2k1
+μβ(εE+εQγ1DQ+εJγ1k2DJDQ)) +μβ(εE+εQγ1DQ+εJγ1k2DJDQ))
σ1 σ1k1(DEDS σ1k1(DSd2γ1k2DJDQ σ1DEk1
+μβ(εE+εQγ1DQ+εJγ1k2DJDQ)) +μβ(εE+εQγ1DQ+εJγ1k2DJDQ))
σ2 σ2μβεJαJDJ σ2αJDJ(DSd2μβεJ) 0
k1 αE(DS(k1DE) αE(DSd2γ1k2DJDQ (k1DE)αE
+μβ(εE+εQγ1DQ+εJγ1k2DJDQ)) +μβ(εE+εQγ1DQ+εJγ1k2DJDQ))
k2 μβk2αQDJDQ(εJ(DQk2) k2αQDJDQ(μβεQDJ 0
εQDJ) +(μβεJDSd2)(DQk2))
d1 d1k1(DEDS d1k1(DS(k1+d2γ1k2DJDQ) d1DEk1
+μβ(εE+εQγ1DQ+εJγ1k2DJDQ)) +μβ(εE+εQγ1DQ+εJγ1k2DJDQ))
d2 d2μβεJαJDJ d2αJDJ(μβεJ+DS(DJd2)) 0

6. Critical immunization levels

From the disease-free reproduction number, we can easily find a formula for a critical immunization level required to prevent the spread of the SEQIJR disease. Since, Rdf=μαLDSαS<1 is required to stop the disease spreading and DS=μ+ν, we have the critical immunization level

νc=μαLαSμ=μ(R01), (38)

where R0 is the basic reproduction number. Further, the critical ratio of immune (recovered) population to total population is

R1N1=νcμ+νc=R01R0. (39)

7. Numerical results for Beijing

For the models given in Eqs. (1), (2), (3), (4), (5), (6) and (9), We have computed the equilibrium points, generalized reproduction numbers, sensitivity indices, dynamical behavior and the critical immunization levels by using the parameter values estimated by Gumel et al. [6] for the SARS epidemics in Beijing, Canada, Hong Kong and Singapore. In this section, we will show the results for Beijing. We will then briefly discuss any differences between the Beijing results and the results for the other three regions in Section 8.

The meanings of the parameters in the SEQIJR model are defined in Table 1 and the parameter values that we use are shown in Table 4. To examine the effect of immunization, we will use values of ν=0.01 corresponding to a disease-free equilibrium, ν=0.000002 corresponding to an endemic equilibrium, and a critical value of ν corresponding to a value for the disease-free reproduction number of Rdf1. Substituting the parameters for Beijing in Table 4 into the system of differential equations for disease transmission in Eqs. (1), (2), (3), (4), (5), (6) and (9), we obtain Eqs. (40).

dSdt=408S0.23I+0.1886JN0.000034+νS
dEdt=S0.23I+0.1886JN0.200034E,
dQdt=0.1E0.125034Q,
dIdt=0.1E0.546834I, (40)
dJdt=0.5I+0.125Q0.047234J,
dRdt=νS+0.0413I+0.0431J0.000034R,
dNdt=4080.0055I0.0041J0.000034N.

Table 4.

Parameter values for SARS outbreaks in Beijing, Canada, Hong Kong and Singapore [6].

Parameter Value Unit
Beijing Canada HongKong Singapore
Π 408 136 221 136 day−1
μ 0.000034 0.000034 0.000034 0.000034 day−1
β 0.23 0.2 0.15 0.21 day−1
εE 0 0 0 0 None
εQ 0 0 0 0 None
εJ 0.82 0.36 0.84 0.2 None
γ1 0.1 0.1 0.1 0.1 day−1
γ2 0.5 0.5 0.5 0.5 day−1
σ1 0.0413 0.0337 0.0337 0.0337 day−1
σ2 0.0431 0.0386 0.0386 0.0386 day−1
k1 0.1 0.1 0.1 0.1 day−1
k2 0.125 0.125 0.125 0.125 day−1
d1 0.0055 0.0079 0.0079 0.0079 day−1
d2 0.0041 0.068 0.0068 0.0068 day−1

7.1. Disease-free equilibrium point and asymptotic stability

For the parameter values in Table 4 and an immunization rate ν=0.01, the disease-free equilibrium point is

(S1,E1,Q1,I1,J1,R1)=(40661,0,0,0,0,1.196×107), (41)

and the endemic equilibrium point does not exist as some populations are negative.

The disease-free and basic reproduction numbers are:

Rdf=0.01366<1R0=4.031>1. (42)

That is, the disease-free equilibrium is locally asymptotically stable and the population with 100% susceptible population will be unstable.

These conclusions can also be checked by using Matlab to compute the eigenvalues of the Jacobian (23) for the disease-free equilibrium and the 100% susceptible population. These eigenvalues are:

Disease_free equilibriumλ1=0.000034,λ2=0.0100034,λ3=0.046160,λ4=0.126390,λ5=0.199712,λ6=0.546874.100% susceptibleλ1=0.000034,λ2=0.000034,λ3=0.08867,λ4=0.20190.06012i,λ5=0.2019+0.06012i,λ6=0.5580.

Since the real parts of all eigenvalues for the disease-free equilibrium are negative and the real part of one eigenvalue of the 100% susceptible population is positive, the disease-free equilibrium state will be locally asymptotically stable and the 100% susceptible population will be unstable.

7.2. Dynamical solutions for disease-free equilibrium

Using the Matlab program we have numerically integrated the system of Eqs. (40) for the four sets of initial conditions given in Table 5 to obtain the dynamical solutions shown in Fig. 2. The graphs show that, for the four different sets of initial conditions, the four infected populations first decrease and then increase before finally converging to zero, i.e., the initial points are not locally asymptotically stable even though there is long-term convergence to the disease-free equilibrium point. However, it can also be seen that at the longer times, where the population values are close to the disease-free equilibrium point, the solutions converge asymptotically to this equilibrium state. This behavior for the longer times in Fig. 2 is in agreement with Eq. (41) and the values of the reproduction numbers and eigenvalues given above.

Table 5.

Initial population values used for numerical integration.

Set S(0) E(0) Q(0) I(0) J(0) R(0)
1 12×106 10 0 3 0 0
2 13×106 1×105 1×105 1×104 4×103 2×105
3 14×106 2×105 2×105 2×104 5×103 3×105
4 15×106 5×105 4×105 4×104 1×104 6×105

Fig. 2.

Example of convergence to a disease-free equilibrium for four different sets of initial conditions. The E, Q, I, J populations rapidly converge to zero and the S and R populations converge to equilibrium values S1=40661 and R1=1.196×107 (Beijing).

Fig. 2(a).

Fig. 2(a)

(a) Susceptible population.

Fig. 2(b).

Fig. 2(b)

(b) Exposed population.

Fig. 2(c).

Fig. 2(c)

(c) Quarantined population.

Fig. 2(d).

Fig. 2(d)

(d) Infectious population.

Fig. 2(e).

Fig. 2(e)

(e) Isolated population.

Fig. 2(f).

Fig. 2(f)

(f) Recovered population.

7.3. Sensitivity analysis of disease-free reproduction number

Using the analytical results given in Table 2, we have computed the normalized sensitivity indices shown in Table 6 of the disease-free reproduction number Rdf=0.01366<1 for the parameter values of Table 4 and ν=0.01. For these parameter values, the sensitivity indices Φ(Rdf|β), Φ(Rdf|εE), Φ(Rdf|εQ), Φ(Rdf|εJ), Φ(Rdf|μ), Φ(Rdf|k1) and Φ(Rdf|k2) are positive and the remaining indices are negative.

For these parameter values, Rdf=0.01366<1 and the solution converges to a disease-free equilibrium point. Then, a reduction in the value of Rdf corresponds to a faster approach to equilibrium, i.e., to a faster disappearance of the disease. The values in column 2 of Table 6 are the sensitivity indices and the values in column 3 are % changes in parameter values required to give a 1% decrease in Rdf. For example, in order to get a 1% decrease in the value of Rdf, it is necessary to decrease the values of β, μ, εJ, k1 and k2 by 1.000%, 1.004%, 1.055%, 198.8% and 7428%, respectively. Also, in order to get a 1% decrease in the value of Rdf, it is necessary to increase the values of ν, σ2, d2, σ1, γ2, d1 and γ1 by 1.003%, 1.156%, 12.2%, 26.2%, 112.2%, 196.9% and 205.8%, respectively. Since the natural death rate μ cannot be easily changed, the most effective methods of reducing Rdf are to reduce the value of the infectiousness and contact rate β, increase the value of the immunization rate ν, reduce the value of the isolation modification parameter εJ and increase the value of the isolation recovery rate σ2.

Table 6.

Normalized sensitivity indices Φ(Rdf|h) and required % changes in parameter for 1% reduction in Rdf for disease-free equilibrium state (Beijing).

Param Φ(Rdf|h) % change Param Φ(Rdf|h) % change
μ +0.996 1.004% ν 0.997 +1.003%
β +1.000 1.000% εE +0.000 +0.000%
εQ +0.000 +0.000% εJ +0.948 1.055%
γ1 0.0049 +205.8% γ2 0.0089 +112.2%
σ1 0.038 +26.2% σ2 0.865 +1.156%
k1 +0.0050 198.9% k2 +0.0001 7428%
d1 0.0051 +196.9% d2 0.082 +12.2%

7.4. Endemic equilibrium point and asymptotic stability

For the Beijing parameter values in Table 4 and ν=0.000002, we obtain the endemic equilibrium point

(S2,E2,Q2,I2,J2,R2,N2)=(2.78×106,1539.51,1231.277,281.532,6238.638,8.414×106,1.12×107).

The value of the disease-free reproduction number is Rdf=3.807>1 and the value of the endemic reproduction number is Ren=0.3756<1. Further, the eigenvalues of the Jacobian for the disease-free equilibrium and for the endemic equilibrium are:

Disease-free

λ1=0.000034λ2=0.000036λ3=0.06618λ4=0.5585λ5=0.021340.059345iλ6=0.02134+0.05935i.

Endemic

λ1=0.000034,λ2=0.000062+0.00190i,λ3=0.0000620.00190iλ4=0.5498,λ5=0.1847+0.03302i,λ6=0.18470.03302i.

Therefore, the disease-free equilibrium point is unstable and the endemic equilibrium point is locally asymptotically stable.

As shown in Gumel et al. [6] (see also [9]), an endemic equilibrium point is also obtained for ν=0.

7.5. Dynamical solutions for endemic equilibrium

Using Matlab, we integrated the system of Eqs. (1), (2), (3), (4), (5), (6) for ν=0.000002 for the four sets of initial conditions given in Table 5. Since the dynamical behavior for the four infected populations are similar, we will only show the solutions for the infectious population (I) and for the susceptible (S) and recovered (R) populations. The solutions for the S, I and R populations are shown in Fig. 3 first for a short integration time to show the initial behavior and then for a long integration time to show the convergence to an endemic equilibrium. It can be seen that the convergence to the endemic equilibrium is very slow and that many disease outbreaks occur during this period. This type of behavior was very common in “childhood” diseases such as measles, mumps, poliomyelitis, rubella, whooping cough etc. before vaccines became available for these diseases. Although it is difficult to see in Fig. 3, we found that epidemics can recur when the fraction of the susceptible population exceeds a certain critical level due to births or if the population of the immune recovered population falls below a certain critical level due to deaths. Similar results for the dynamical behavior of populations in the endemic region have also been reported by Koonprasert et al. [9] in a study on the effectiveness of quarantine and isolation in an SEQIJR model.

Fig. 3.

Example of convergence to an endemic equilibrium for four different sets of initial conditions (Beijing). The graphs in the left column show the initial behavior for 0t500 days. The graphs in the right column show the long-time behavior for 20000t120000 days.

Fig. 3(a).

Fig. 3(a)

(a) Susceptible population.

Fig. 3(b).

Fig. 3(b)

(b) Infectious population.

Fig. 3(c).

Fig. 3(c)

(c) Recovered population.

7.6. Sensitivity analysis of the endemic equilibrium point

In the sensitivity analysis we do not consider variations in Π, μ, εE and εQ. The parameter Π represents the natural birth rate and the parameter μ represents the natural death rate for the population. These parameters cannot be changed easily for human populations. Also, we omit the modification parameters for infection from the exposed population εE and the quarantined population εQ since these parameters have been assumed to be zero. We therefore only consider sensitivity indices for the remaining eleven parameters.

Using the analytical results given in Table 3 and Eq. (36), and the parameter values in Table 4 with ν=0.000002, we obtain the values shown in Table 7 for the normalized sensitivity indices of the endemic equilibrium populations. As noted above, the values of Rdf=3.807>1 and Ren=0.3756<1 show that the endemic equilibrium is locally asymptotically stable. Since the sensitivity indices are functions of parameter values, the values in Table 7 will change if parameter values are changed. The interpretation of these indices is that a positive index means that I2 increases if the parameter value is increased and a negative index means that I2 decreases if the parameter value is increased with the magnitude giving the rate of change.

Table 7.

Sensitivity indices for endemic equilibrium state (Beijing).

Parameter Sensitivity indices
S2 E2 Q2 I2 J2 R2 N2
β −1.0237 +0.3326 +0.3326 +0.3326 +0.3326 +0.3062 −0.0237
εJ −0.9703 +0.3152 +0.3152 +0.3152 +0.3152 +0.2902 −0.0225
ν +0.0013 −0.0185 −0.0185 −0.0185 −0.0185 +0.0013 +0.0013
γ1 +0.0055 −0.5017 +0.4983 −0.5017 +0.0206 −0.0009 +0.0006
γ2 +0.0101 −0.0033 −0.0033 −0.9176 +0.0376 −0.0017 +0.0012
k1 −0.0057 −0.4981 −0.4981 +0.5019 −0.0204 +0.0012 −0.0007
k2 −0.0002 +0.0001 −0.9997 +0.0001 +0.0002 +0.0002 −0.00001
σ1 +0.0418 −0.0136 −0.0136 −0.0891 −0.0497 −0.0089 +0.0037
σ2 +0.9481 −0.3080 −0.3080 −0.3080 −1.2205 −0.2013 +0.0832
d1 +0.0014 −0.0005 −0.0005 −0.0105 −0.0053 −0.0054 −0.0037
d2 +0.0214 −0.0070 −0.0070 −0.0070 −0.0938 −0.0880 −0.0608

In order to reduce the spread of the disease, it is necessary to reduce the infectious population (I2) and isolated population (J2) since these populations are capable of direct transmission of the disease to uninfected people. It is also desirable to reduce the number of deaths from the disease, i.e., to increase the total surviving population N2.

From the indices in Table 7, the most effective methods of reducing I2 are, in order, to increase the isolation rate (γ2), reduce the transfer rate from exposed to infectious population (k1), increase the quarantine rate (γ1) and reduce the contact rate between infectious and susceptible people (β). The most effective methods of reducing J2 are, in order, to increase the recovery rate of isolated individuals (σ2), reduce the contact rate β, and reduce the modification parameter εJ. It can be seen from Table 7 that increasing the death rates d1 and d2 of the infectious and isolated individuals would also reduce the infectious and isolated populations. This method is clearly not an acceptable control method for a human disease. However, this method was adopted as an effective control method in the case of the spread of avian flu among chickens in China and “mad-cow disease” in England, and it has also been used as an effective method of controlling some other diseases among animals and plants.

The most effective methods for increasing the total surviving population N2, i.e., to reduce the death rate, are, in order, increase the recovery rate of infectious individuals (σ2), reduce the contact rate β and reduce the modification parameter εJ, i.e., to reduce the level of nosocomial disease as discussed in the introduction.

8. Comparison of results for Beijing, Canada, Hong Kong and Singapore

Using the parametervalues given in Table 4, we have computed and compared the disease-free and endemic results for the four regions. The overall behavior of the results for all four regions is similar, but with some differences in detail. We will compare the results for the reproduction numbers, the sensitivity indices and the critical immunity levels.

8.1. Reproduction numbers and sensitivity indices

The results for the basic reproduction numbers and the disease-free and endemic reproduction numbers are shown in Table 8. For each case, we have numerically integrated the differential equations and checked the asymptotic stability of equilibrium points by computing the value of the reproduction numbers and by computing the eigenvalues of the Jacobian matrices. In all cases we have shown that the numerical results agree with the theoretical analysis.

Table 8.

Reproduction numbers.

Beijing Canada Hong Kong Singapore

Disease-free ν=0.01

R0 4.031 1.708 2.804 1.082
Rdf 0.01366 0.005787 0.009502 0.003668

Endemic ν=0.000002

Rdf 3.807 1.613 2.649 1.022
Ren 0.3756 0.6887 0.4863 0.9786

For each of the four regions, we have also computed the sensitivity indices of the disease-free reproduction number Rdf and the endemic equilibrium populations for all of the parameters in the model.

From these calculations, we can rank the indices in order of effectiveness in reducing a disease. The results for the disease-free equilibrium Rdf are shown in Table 9. The table shows the top three indices for each region for a disease-free case (Rdf<1), and a critical immunization case Rdf1.

Table 9.

Ranking for best parameters to reduce Rdf.

Region Rdf<1 Rdf1
Beijing β,ν,(μ),εJ β,εJ,σ2
Canada β,ν,(μ),εJ β,εJ,σ2
Hong Kong β,ν,(μ),εJ β,εJ,σ2
Singapore β,ν,(μ),εJ β,εJ,σ2

For Rdf<1, the rankings for all regions are the same. The most effective methods of reducing Rdf are, in order, to reduce the infectiousness and contact rate β between infectious and susceptible people, increase the value of the immunization rate ν and reduce the modification parameter εJ for the infectiousness and contact rate of isolated people. It can be seen from Table 9 that reducing (index is positive) the natural death rate μ is the third most effective method of reducing the value of Rdf. However, since the natural death rate cannot be varied easily for human populations, this is not an acceptable control method for a human disease. For Rdf1, the most effective methods of reducing Rdf are, in order, to reduce the infectiousness and contact rate β, reduce the isolation modification parameter εJ and increase the recovery rate of isolated individuals σ2.

Table 10 shows the rankings for the most effective methods for reducing the endemic infectious populations I2 in the four regions for Rdf1 and for Rdf>1. For Rdf1, the rankings for Beijing, Canada, Hong Kong are the same. The most effective methods of reducing I2 are, in order, to reduce the contact and infectiousness rate β, increase the recovery rate of isolated individuals σ2 and increase the value of the immunization rate ν. For Singapore, the most effective methods of reducing I2 are, in order, to reduce the infectiousness and contact rate β, increase the recovery rate of isolated individuals σ2 and increase the isolation rate γ2.

Table 10.

Ranking for best parameters to reduce I2.

Region Rdf1 ν value for Rdf1 Rdf>1 ν value for Rdf>1
Beijing β,σ2,ν 0.0001 γ2,k1,γ1 0.000002
Canada β,σ2,ν 0.000024 β,σ2,γ2 0.000002
Hong Kong β,σ2,ν 0.000061 γ2,β,k1 0.000002
Singapore β,σ2,γ2 0.0000028 β,σ2,γ2 0.000002

For Rdf>1, the most effective methods of reducing I2 for Beijing are, in order, to increase the isolation rate γ2, reduce the transfer rate from exposed to infectious k1 and increase the quarantine rate γ1. For Canada the most effective methods of reducing I2 are, in order, to reduce the infectiousness and contact rate β, increase the recovery rate of isolated individuals σ2 and increase the isolation rate γ2. For Hong Kong the most effective methods of reducing I2 are, in order, to increase the isolation rate γ2, reduce the infectiousness and contact rate β and reduce the transfer rate from exposed to infectious k1. For Singapore the most effective methods of reducing I2 are the same as for the Rdf1 case.

8.2. Critical immunity levels

The critical values of the immunization rate ν and the critical ratio of recovered to total population can be calculated from Eqs. (38) and (39) and parameter values in Table 4 for the SARS epidemics in Beijing, Canada, Hong Kong and Singapore in 2002 and 2003. The results for the four regions are shown in Table 11. The results show that there is a big difference between the critical immunization levels and critical immunized ratio in the four regions. The difference is due to the different values of the basic reproduction number R0=αLαS in the four regions. The difference in R0 values in turn is mainly due to differences in the values of the infection factor β and the modification factors for infection from the quarantined population (εQ) and the isolated population (εJ).

Table 11.

Critical immunization levels for prevention of disease.

Beijing Canada Hong Kong Singapore
Critical ν 1.03×104 2.407×105 6.135×105 2.803×106
Critical R/N ratio 0.7519 0.4415 0.6434 0.07616

9. Discussion and conclusions

We have studied an SEQIJR model for infectious disease transmission which generalizes the model developed by Gumel et al. [6] for the SARS epidemics of 2002 and 2003 by including immunization and a varying total populations size.

We have shown analytically that this model has a disease-free equilibrium state which always exists and an endemic equilibrium state which exists if and only if the disease-free equilibrium is unstable. We have defined generalized reproduction numbers Rg for any given equilibrium state such that the state is locally asymptotically stable if Rg<1 and unstable if Rg>1. As special cases, we have obtained simple formulas for the reproduction numbers for a 100% susceptible population (the basic reproduction number R0=αLαS) and for the disease-free equilibrium Rdf=μαL(μ+ν)αS by both the next-generation method [23] and from the eigenvalues of the Jacobian of the linearized model about the disease-free equilibrium. In these formulas, αL can be interpreted as the rate of new infections from all infected individuals and αS as the rate of reduction of the susceptible population due to transfers to other population groups including to the recovered immune group. The effect of the immunization rate (ν) in reducing the reproduction number and increasing the stability of disease-free equilibrium states can be clearly seen by comparing the R0 and Rdf formulas. We have also obtained simple formulas for a reproduction number Ren for the endemic equilibrium state and shown that the endemic equilibrium is locally asymptotically stable if Ren<1 and does not exist if Ren>1.

We have also derived formulas for sensitivity indices of the disease-free and basic reproduction numbers, and the endemic equilibrium populations for changes in the values of model parameters.

For the numerical study of the model, we have used values of model parameters estimated by Gumel et al. [6] for the SARS epidemics in 2002–2003 in Beijing, Canada, Hong Kong and Singapore. For each of the four regions we have considered parameter values corresponding to a disease-free equilibrium and an endemic equilibrium. From the computed sensitivity indices for each of the regions, we have shown that the most important parameters for reducing infection are, in order, (1) for high immunization rates (ν0.01) reduce the overall infection rate (β), increase the immunization rate, and reduce the modification factor for infection by isolated individuals (εJ), (2) for critical immunization rates with Rdf1, reduce β, reduce εJ and increase recovery rate of isolated individuals (σ2), (3) for low immunization rates (endemic equilibrium) the most important for Beijing and Hong Kong is to increase the isolation rate of infectious individuals (γ2) and the most important for Canada and Singapore is to reduce β.

The numerical results for the reproduction numbers and the sensitivity indices clearly support the importance of isolation and to a lesser extent quarantine in controlling the spread of a disease.

We have obtained very simple formulas in terms of the basic reproduction number R0 for critical values of the immunization rate and the ratio of the recovered (immune) and total populations that are required to stop the spread of a disease. Our numerical results for the four regions considered in this paper clearly support the importance that is now widely accepted, except by anti-vax campaigners, of maintaining a critical immunization level of “herd immunity” to stop the spread of a disease. The differences in the critical values of the immunization rate in the four regions clearly show the importance of factors such as reducing infection rates, and increasing isolation and quarantine, in stopping an epidemic that can occur when the immunization ratio falls below a critical level.

Unfortunately, in 2002 and 2003 there was no known immunization available for SARS. Therefore, the immunization results in this paper are not relevant for the SARS outbreaks of 2002–2003. However, the inclusion of immunization in the study of an SEQIJR model is important as the SEQIJR model is a very general model which can be applied to a number of other diseases that are also mainly transmitted by person to person contact and for which immunization is available, e.g., influenza and “childhood diseases” such as measles, mumps, whooping cough etc. A similar model with the addition of a bird reservoir of the disease could also be used to analyze avian influenza. SEIR models have, of course, been studied by many authors and there is now an extensive literature on them. However, we would note that the formulas for generalized reproduction numbers, sensitivity indices and critical immunization levels for the SEQIJR model could easily be applied to SEIR models by removing the quarantined and isolated compartments and suitably redefining some of the parameters.

Acknowledgments

This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. This research was also supported by the Science and Technology Research Institute of King Mongkut’s University of Technology North Bangkok . We would also like to thank colleagues in the Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok for encouragement and critical discussions.

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