Modal series solution for an epidemic modelL. Acedo a, Gilberto González-Parra a,b,∗, Abraham J. Arenas a,ca Instituto de Matemática Multidisciplinar, Universidad Politécnica de Valencia Edif. 8G, 2o , 46022 Valencia, Spainb Departamento Cálculo, Universidad de los Andes, Mérida, Venezuelac Departamento de Matemáticas y Estadística, Universidad de Córdoba, Montería, Colombia
a r t i c l e i n f o
Article history:Received 6 July 2009Received in revised form 1 November 2009Available online 13 November 2009
Keywords:SIRS epidemic modelExact solutionModal expansion infinite series
a b s t r a c t
In this article, we generalize a recently proposed method to obtain an exact generalsolution for the classical Susceptible, Infected, Recovered and Susceptible (SIRS) epidemicmathematical model. This generalization is based upon the nonlinear coupling of twofrequencies in an infinite modal series solution. It is shown that these series provide anonstandard approach in order to obtain an accurate analytical solution for the classicalSIRS epidemic model. Numerical results of the SIRS epidemic model for real and complexfrequencies are included in order to test the validity and reliability of the method. Thismethod could be applied to a wide class of models in physics, chemistry or engineering.
At the beginning of the second quarter of the past century, Kermack and McKendrick [1] proposed a system of ordinarydifferential equations to determine the time evolution of the susceptible, infected and recovered populations of individualsexposed to an infectious disease. This model is generally known by the acronym SIR and similar names have been given toother compartmental models such as SIS, SIRS, SEIR, etc., by disposing the different populations considered in the modelsin the same order as the flow from one compartment to another takes place. The importance acquired throughout the pastand present centuries by this simple model cannot be underestimated. This is probably the most widely used model inMathematical Epidemiology with applications to many epidemics such as: the bubonic plague [2], measles [3], cholera [4],rubella [5], respiratory syncytial virus [6,7], hepatitis [8], influenza [9] andmany other. Random perturbations has also beenadded to these models [10].From another point of view, this model has also been used to illustrate Fokker-Planck relaxation to an equilibrium point
in Statistical Physics [11]. A connection with the kinetic of reactions can also be easily developed, if we consider S and I astwo different chemical species which react as follows: S + I → I + I . Thus, relaxation towards chemical equilibrium canalso be described by very similar models which incorporate the binary reaction term SI into the dynamics. In order to makereliable predictions, numerical integration methods such as Runge–Kutta can be efficiently implemented virtually in anycomputer language. Nevertheless, exact solutions play a very important role in any nonlinear theory: Onsager’s solution of2D Ising model [12], the Schwarzschild and Kerr solutions of Einstein’s Field Equations [13] exemplify the creation of newconcepts such as criticality or Black Holes which were disclosed more easily; thanks to these solutions.Closed-form expressions for infected and susceptible individuals are known time ago for the SI epidemic model [14].
Recently, some authors have also found special solutions of the SIR model by means of modal series [15,16]. We usedmodal series in terms of a single frequency to obtain a solution for a particular set of initial conditions [16]. The homotopyperturbation method provides a similar approach, but oscillations in the transient regime are never observed because asingle real time scale is taken into account [15]. On the other hand, many numerical methods have been presented forsolving epidemics models [17].
∗ Corresponding author at: Departamento Cálculo, Universidad de los Andes, Mérida, Venezuela.E-mail addresses: [email protected] (L. Acedo), [email protected] (G. González-Parra), [email protected] (A.J. Arenas).
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The wide interest of the SIR model demands a more complete mathematical treatment encompassing the differenttransient regimes for the two degrees of freedom corresponding to the initial conditions for the infected and susceptibleindividuals.Our procedure to solve the problem is as follows: (i) We rewrite the system of equations for the functions I(t), S(t)
and R(t) corresponding, respectively, to the infected, susceptible and recovered individuals as a single integro-differentialequation for I(t). (iii) Derive the integro-differential equation in order to obtain a relationship to compute the twofrequencies. (ii) Propose a modal expansion series with two frequencies of the following form:
I(t) =∞∑n=0
∞∑m=0
An,me−(nω+mΩ)t , Ω > ω > 0, (1)
where the frequencies and the coefficientsAn,m, n,m = 1, 2, . . . are determined by a recurrence relation in terms ofA0,1,A1,0 and the parameters of the model. In this way, we obtain a series solution where the frequencies Ω and ω may becomplex numbers as in Ref. [16].This article is organized as follows. In Section 2, the SIRS epidemiological mathematical model and the reduction of the
system to a single nonlinear equation in terms of I(t) is presented. In Section 3, the general analytical global solution in seriesform for the classical SIRS epidemic model is obtained. Numerical results of the SIRS epidemic model for real and complexfrequencies are shown in Section 4. Section 5 is devoted to the conclusions and guidelines for future work.
2. SIRS epidemic mathematical model
In this section, the SIRS epidemicmathematicalmodel is presented. Thismodel is based on a systemof first order ordinarydifferential equations and it has been used in the modeling of several infectious diseases, where the parameters need to beestimated by epidemiological data [18]. In this model, the variables represent subpopulations of Susceptibles (S), Infected(I) and Recovered (R). Thus, the model describes the dynamics of the different classes [19,20]. For the sake of simplicity, theSIRS model is written as:
S(t) = µ− µS(t)− βS(t)I(t)+ γ R(t), S(0) > 0,
I(t) = βS(t)I(t)− αI(t), I(0) ≥ 0,
R(t) = νI(t)− δR(t), R(0) ≥ 0,
(2)
where α = ν+µ and δ = γ +µ and the hypothesis assumed in this model can be seen in Ref. [16] for the interested reader.On the other hand, assumingwithout loss of generality that S(t)+I(t)+R(t) = 1,we can reducemodel (2) to a systemof onlytwo equations. It iswell known that system (2) has two equilibriumpoints: The disease free F∗0 = (S
∗
0 , I∗
0 , R∗
0) = (1, 0, 0), and
the unique endemic point given by E∗e = (S∗e , I∗e , R∗e ) =
(µ+ν
β,(µ+ν)(β−µ−ν)
β(γ+µ+ν), νµ+γI∗e). Furthermore, we defineR0 =
β
µ+ν, as
the basic reproductive number associated with the model (2), and the uniqueness of E∗e is guaranteed wheneverR0 > 1.Next, taking the last two equations corresponding to infected and recovered populations one gets the following system:
where a = α − β = ν + µ− β , b = βR(t = 0), c = β and d = βν. In a previous article, we impose a restrictive conditionto eliminate the terms of the form e−δt in order to simplify this equation [16]. Without loss of generality, we assume thatI(t) > 0, and dividing Eq. (6) by the factor e−δt I(t) and deriving we have,
δ I(t)I(t)+ I(t)I(t)− I2(t)+ cI(t)I2(t)+ (d+ cδ)I3(t)+ aδI2(t) = 0. (7)In this way, we propose a series solution with two frequencies of the following form:
I(t) =∞∑n=0
∞∑m=0
An,me−(nω+mΩ)t , Ω > ω > 0, (8)
L. Acedo et al. / Physica A 389 (2010) 1151–1157 1153
where the frequenciesΩ ,ω and the real coefficientsAn,m, n,m = 1, 2, . . . are determined by a recurrence relation in termsof coefficientsA0,1,A1,0 and the parameters of the model. Additionally, the frequenciesΩ and ωmay be complex numbersas in Ref. [16].
3. General solution in modal series form of the SIRS model
In this section, we obtain a global solution in series form for the SIRS epidemic model (2). As we have mentioned inthe previous section, our approach is to seek series solutions of the form (8), where An,m and ω, Ω are real coefficients todeterminate.In order to compute the series solution from (7), we rely on the following equations containing the Cauchy product of
where we have taken into account that e−(nω+mΩ)t (n,m = 0, 1, . . .) are a linearly independent base of exponentialfunctions. For n = m = 0, one gets
A20,0 aδ +A30,0(d+ cδ) = 0, (16)and assuming thatA0,0 6= 0 (becauseA0,0 corresponds to the endemic point), it follows that
A0,0 = −aδ
(d+ cδ)= limt→∞
I(t). (17)
For n = 1,m = 0 and n = 0,m = 1, one gets the following equation,
A0,0A1,0 aδ +A1,0A0,0(aδ + ω[ω − δ])+A20,0A1,0(3d+ 3cδ − ωc) = 0. (18)Again, we assume thatA1,0 6= 0. Thus, we obtain the equation for ω and the parametersA1,0 andA0,1 remain arbitrary. Itis important to remark that these parameters are free of choice in order to satisfy the two initial conditions for I(t) and R(t).Using Eq. (15), we can obtain the coefficients of the series solution (8) by means of the following equation:
1154 L. Acedo et al. / Physica A 389 (2010) 1151–1157
where (′)means that we do not take into account any term of the sum whereAn,m appears, and
∆n,m = aδ + (nω +mΩ)(δ + cA0,0 − (nω +mΩ)
). (20)
Now, from Eq. (8) and if we choose arbitrary parametersA0,1 andA1,0, we have that I(0)must be
I(0) =∞∑n=0
∞∑m=0
An,m < 1, (21)
since that model is scaled and this is the initial condition for the infected individuals. Using Eq. (4) and assuming thatnω +mΩ − δ 6= 0 for all n,m, one gets that the analytical solution for R(t) given by
R(t) = e−δt(R(0)+ ν
∞∑n=0
∞∑m=0
An,m
nω +mΩ − δ
)− ν
∞∑n=0
∞∑m=0
An,me−(nω+mΩ)t
nω +mΩ − δ, (22)
where An,m are obtained from (19), for any choice of A0,1 and A1,0. The consistency of the method requires that thedenominator in both sums of Eq. (22) must be nonzero for any pair of integers n,m and for any combination of parametersof the epidemic model. This is shown in the Appendix. Now, from the integro-differential equation (6) for t = 0, we findthat the initial conditions satisfy
I(0) = −(a+ b)I(0)− cI(0)2, (23)
where b = βR(0). Thus, one gets that,
R(0) =−a− cI(0)− I(0)
I(0)
β, (24)
and the initial condition for the recovered population R(t) is now defined. Notice that we have two degrees of freedom inthe series:A0,1 andA1,0 that are related to I(0) and R(0).In the complex case,Ω = ω∗. It can be showneasily that the solution of interest in epidemiologymust verify the condition
of hermicity of the coefficients:
Ai,j = A∗j,i, ∀i, j = 0, 1, . . . (25)
because I(t), R(t) and S(t)must be real functions of t . Nevertheless, we still have the freedom to choose two independentinitial conditions manifested in the real and imaginary parts ofA0,1 = A∗1,0.It is important to remark that the epidemic disease free case can be seen as a special case (A0,0 = 0), where only a single
frequency appears. Thus, we propose a solution of the following form:
I(t) =∞∑n=0
Ane−nωt , ω > 0,A0 = 0, (26)
instead of Eq. (1). The recurrence formula is now given by:
n∑l=0
AlAn−l(aδ − ωδl+ 2ω2l2 − ω2nl)+n∑l=0
n−l∑m=0
(d+ cδ − cmω)AmAlAn−m−l = 0, (27)
withA0 = 0. The first nontrivial identity obtained from Eq. (27) corresponds to n = 2:
δA21(ω − a) = 0, (28)
and consequently, ω = a assuming A1 6= 0. Notice that in this case a > 0 [18,19] and the series in Eq. (26) decreases forlarge times as expected. By extracting from the first sum in Eq. (27) the terms l = 1, n − 1, we rewrite it as the followingrecurrence relation:
An−1 =1
A1Γn
n−2∑l=2
AlAn−l[aδ(1− l)+ a2l(2l− n)] +1
A1Γn
n∑l=0
n−l∑m=0
(d+ c(δ −ma))AmAlAn−m−l, (29)
whereΓn = a(n−2)[δ−a(n−2)], and n ≥ 3. In this case,A1 is freely chosen and the rest of the coefficients are determinedby Eq. (29). In Section 4, we will confirm numerically that the exact solutions developed here for the real and complex casesare indeed correct. In addition, the solution of the epidemic disease free case is also shown.
L. Acedo et al. / Physica A 389 (2010) 1151–1157 1155
0.114
0.112
0.11
0.108
0.106
0.104
Infe
cted
pop
ulat
ion
0.102
0.1
0.098
0.0960 20 40
Time t
60 80
Infected (Analytical)
Infected (Runge–Kutta)
Fig. 1. Comparison of the approximated solutions of the Runge–Kutta fourth order variable step size method and the approximated exact solution withn,m ≤ 10 terms to the SIRS epidemic model (2). The parameter values are β = 0.5, µ = 0.041, ν = 0.4, γ = 1.8, α = 0.441 and δ = 1.841. The seedcoefficients for the recurrence relation in Eq. (18) areA1,0 = 0.005 andA0,1 = 0.01.
Fig. 2. Comparison of the approximated solutions of the Runge–Kutta fourth order variable step size method and the modal series with n,m ≤ 4 andn,m ≤ 20 for the SIRS epidemic model (2). The parameter values are β = 60, µ = 0.041, ν = 36, γ = 1.8, α = 36.041 and δ = 1.841. Here, the seedcoefficients for the recurrence relation in Eq. (18) areA1,0 = A∗0,1 = 0.02+ 0.01i.
4. Numerical simulations
In this section, numerical results for the solution of the SIRS epidemic mathematical model using Runge–Kutta fourthorder variable step size method and our analytical series method are compared in order to support our theoretical resultsand show his accuracy and reliability. The comparisons are made using parameters values such that the real and complexcases appear. The initial conditions can be computed with the seed coefficients and the recurrence relation in Eqs. (18), (20)and (23). In addition, the graphics of the solution of the epidemic disease free case is also shown.In Fig. 1, it can be seen the solutions representing the infected I(t) and recovered R(t) populations for of the SIRS epidemic
mathematical model (2). It is clear from Fig. 1 that an excellent agreement exists between the two sets of results, the fourthorder variable step size Runge–Kutta type method solution and the approximated exact solution in series form developedin this article.In Fig. 2, it can be observed the solutions representing the infected I(t) and recovered R(t) populations for the SIRS
epidemic mathematical model (2) for the complex case. In this case, we also find an excellent agreement between fourthorder Runge–Kutta method and our modal series method. Notice that the agreement of the modal series with n,m ≤ 4 isgood for large times. Convergence towards the exact solution is fast and with n,m ≤ 20 the solution is very accurate.The last numerical example includes the disease free case. Fig. 3 shows the solutions representing the infected I(t) and
recovered R(t) populations. These populations become extinct as the time is increased. In this case also the series solutiongives a very accurate global solution as it can be seen in Fig. 3.
5. Discussion and conclusions
In this article, we generalize the modal series solution to obtain an exact global solution for the classical SIRS epidemicmathematicalmodel. This generalization is based on the introduction of two frequencies in themodal infinite series solution.It has been shown that these solutions provide a reliable order to obtain an accurate analytical solution for this type ofsystem.
1156 L. Acedo et al. / Physica A 389 (2010) 1151–1157
Fig. 3. Comparison of the approximated solutions of the Runge–Kutta fourth order variable step size method and the series with 80 terms to the SIRSepidemic model (2). The parameter values are β = 0.6, µ = 0.041, ν = 0.6, γ = 1.8, α = 0.641 and δ = 1.841. The seed coefficient in this case isA1 = 0.03.
The coupling of the characteristic frequencies of a nonlinear system, disclosed by the modal series approach to thesolution of the SIRS model, is an important insight whose applications to the other nonlinear models must be pursuedfurther.The implementation of thismethod for the case of systemsmodeled by equationswith quadratic terms similar to the SIRS
epidemicmodels, such as the kinetic of chemical reactions, nonlinear oscillations, more epidemicmodels (SEIRS,MSEIR, etc.)is presently under study and will be published elsewhere. More general nonlinear differential models including nonlineardamping or anharmonicity with terms of cubic or quarter order in the unknowns can also be solved with this method. Inthese cases, a different Borel basis is probably needed and more complicated recurrence relations are found. Work alongthese lines is also in progress and will be published elsewhere.
Appendix
In this appendix, we will show that the series in Eq. (22) is well defined in the sense that the denominator cannot bezero in any meaningful applicable of the SIRS model. Indeed, the second order Eq. (18) for the frequencies ω andΩ can berewritten as follows
ω2 − ω(δ + ζ )+ δζ + dA0,0 = 0, (30)
where, we have taken into account that A1,0 6= 0 in order to simplify Eq. (18) and ζ = −acδd+cδ . Notice that for the roots of
Eq. (30) it follows that,
ω +Ω = δ
(1−
acd+ cδ
),
ωΩ = −aδ. (31)The following condition
nω +mΩ = δ, (32)must be avoided in order to keep bounded the proportion of recovered individuals, R(t). We will show that this conditionis not fulfilled for the realistic parameters of the SIRS model and n,m integers.From Eq. (31), we haveΩ = − aδ
ω, and Eq. (32) yields to a system of equations for δ and ω as follows:
ω −aδω= δ
(1−
acd+ cδ
)= ξ, (33)
ω −aδω=δ
n−aδω
(1−
mn
)= η. (34)
The endemic point is given by Eq. (17), I(∞) = I(t −→ ∞) = − aδd+cδ > 0, and c = β > 0 (infection rate), d = βν > 0
(infection rate times recovery rate), δ = γ +µ > 0 (loss immunity rate plus death rate). Consequently, wemust have a < 0in order to obtain a positive value for the infected individuals in the limit t −→∞. From these inequalities, we deduce that
ξ = δ
(1+
I(∞)cδ
)> δ, (35)
and
η =δ
n−δ
ωa(1−
mn
)≤δ
n< δ, (36)
L. Acedo et al. / Physica A 389 (2010) 1151–1157 1157
for m ≥ n, because a(1 − mn ) ≥ 0, and ω > 0 in order to avoid divergence in the series (8). From Eqs. (35) and (36), we
deduce that ξ 6= η, but this contradicts (33) and (34). To analyze the case m < n, we consider the analogous system forΩobtained from (33) and (34). Thus,
Ω −aδΩ= δ
(1+
I(∞)cδ
)= ξ ′, (37)
Ω −aδΩ=δ
m−aδΩ
(1−
nm
)= η′. (38)
But ξ ′ > δ, and η′ ≤ δm < δ for n > m. Thus, Eqs. (33) and (34) are incompatible. This proves that for the parameters that
appear in realistic applications of Eq. (22), the denominator must be different from zero. Ifω andΩ are complex conjugates,it is clear that (32) could only be verified ifm = n and this is also precluded by Eqs. (33) and (34).
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