Oscillations in SIRS model with distributed delays -...

Eur. Phys. J. B 81, 363–371 (2011) DOI: 10.1140/epjb/e2011-20054-9

Oscillations in SIRS model with distributed delays

S. Goncalves, G. Abramson and M.F.C. Gomes

Eur. Phys. J. B 81, 363–371 (2011)DOI: 10.1140/epjb/e2011-20054-9

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Oscillations in SIRS model with distributed delays

S. Goncalves1, G. Abramson2, and M.F.C. Gomes1,a

1 Instituto de Fısica, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 90501-970 Porto Alegre RS, Brazil2 Centro Atomico Bariloche, CONICET and Instituto Balseiro, 8400 S. C. de Bariloche, Argentina

Received 19 January 2011 / Received in final form 29 March 2011Published online 4 May 2011 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2011

Abstract. The ubiquity of oscillations in epidemics presents a long standing challenge for the formulationof epidemic models. Whether they are external and seasonally driven, or arise from the intrinsic dynamicsis an open problem. It is known that fixed time delays destabilize the steady state solution of the standardSIRS model, giving rise to stable oscillations for certain parameters values. In this contribution, startingfrom the classical SIRS model, we make a general treatment of the recovery and loss of immunity terms.We present oscillation diagrams (amplitude and period) in terms of the parameters of the model, showinghow oscillations can be destabilized by the shape of the distributions of the two characteristic (infectiousand immune) times. The formulation is made in terms of delay equations which are both numericallyintegrated and linearized. Results from simulations are included showing where they support the linearanalysis and explaining why not where they do not. Considerations and comparison with real diseases arepresented along.

1 Introduction

Many diseases that have affected and still affect humanscome and go with time in a well established way. Ex-amples are plenty and fill the bulletins of world and na-tional health organizations. Measles, typhus and choleraepidemic waves, just to cite a few, are even part of math-ematical biology books [1,2]. The common denominatorof such diseases is the cyclic natural history of them, inwhich a susceptible subject can go to infected, then toremoved, and finally back to the susceptible state. How-ever, the mere cyclic nature of the disease does not grantan oscillatory behavior of its epidemic, as can be exempli-fied by gonorrhea [3,4]. In 2009 a new variant of influenzaA H1N1, dubbed swine flu, appeared in the scene takingthe media to discuss on the wave behavior of influenza.In which way do these oscillations arise in a population,apparently synchronizing the infective state of many indi-viduals? Are they related to external driving causes, suchas the seasons? Or do they arise dynamically from thevery natural history of the disease? It is known that severalcauses can produce oscillations in model epidemic systems:seasonal driving [5], stochastic dynamics [6,7], a complexnetwork of contacts [8], etc.

In every infectious disease several characteristic timesclearly appear in the dynamics. In a SIRS type disease(see Fig. 1) one has an infectious time (during which theagent stays in category I, infected and infectious), and animmune time (during which the agent stays in category R,recovered from infection, and immune to re-infection until

a e-mail: [email protected]

S I R Scontagion

τI τ

Rtime

Fig. 1. (Color online) Timeline of an individual, showing thecourse of the disease after contagion.

it returns to the susceptible class S). A standard SIRSmodel uses the inverse of these characteristic times asrates in mass-action equations, showing damped oscilla-tions toward the endemic state in most typical situations.However, the standard SIRS model does not exhibit sus-tained oscillations for any value of the parameters. Wouldit be possible for a deterministic SIRS model to sustainoscillations, and if so how do the characteristic times re-late to the period of the epidemic? These are some of thequestions we address in this contribution.

As observed by Anderson and May [1] the mathemat-ically convenient treatment of the duration of the infec-tion as a constant rate is rarely realistic. It is more com-mon that recovery from infection takes place after somerather well defined time. It would seem a valid simplifica-tion to assume that recovery happens exactly at a giventime, instead of continuously at some rate. Both extremesare called type B and type A recovery respectively byAnderson and May. Most infections belong to an inter-mediate type between these two extremes (but closer totype B). These intermediate types have an infectious timedistributed with a shape between that of an exponential(type A) and that of a delta distribution (type B). An ac-curate model should implement the distributions based

364 The European Physical Journal B

on empirical data. And as suggested by Hoppenstaedtet al. [9], that could be done replacing the simple constantrate term by an integral, leading to integro-differentialequations. Similar considerations can be made about thetransition from recovered to susceptible term. It was al-most thirty years ago that Hethcote et al. [10] showedprecisely that the SIRS model with fixed delay in the re-covered to susceptible transition presents stable periodicsolution for certain parameter values. On the other sidethey demonstrated that fixed delay in the recovery frominfection term does not have the same effect when theother one is considered in the usual way. This implies thatthe immunity time plays the most important role in theemerging oscillations as we will show in detail. Recently,continuing with the work of Hethcote et al., Taylor andCarr [11] studied in detail the dynamics of the SIRS modelwith temporary immunity, but considering that a fractionof the population acquire permanent immunity. In otherwords, that is equivalent to a distribution of immunitytimes made of two delta peaks (at a finite time and atinfinite). That makes the analysis of the oscillations muchmore involved because of the extra parameter: the fractionof individuals who became permanently removed.

Our contribution can be regarded as an extension ofthe work of Hethcote et al. with time delays, in whichwe use arbitrary distributions of both infectious and im-mune times. While doing this we want to keep the prob-lem as simple as possible in terms of parameters, thus weavoid the use of vital dynamics. We start by consideringthe most extreme case: fixed delay in both terms (deltadistributions). This is the simplest mathematical case interms of delays, and it is closely related with the firstcase studied by Hethcote et al. [10]. Then, we consider amono-parametric family of models in which the times aredescribed by continuous distributions – being possible togo continuously from a type B to a type A model for ex-ample. In other words we go from the SIRS model withdeterministic delays to the classical constant rates SIRS,taking all the intermediate situations in between. In thisway, while we cover previous results, we can go furtherconsidering the most general situation for a SIRS model.

For delayed models, we analyze the onset of sustainedoscillations and characterize them with the parameters ofthe system. Linear analysis, together with numerical so-lutions of the nonlinear model, provide a clear character-ization of the phenomenon. Stochastic numerical simula-tions provide further support to our analysis. Moreover,we show the effect of the shape of the distributed delayson the stabilization of the oscillations. Besides, the oscil-lations period, which may play an important role in thedesign of intervention policies, are shown to satisfy generalrules in terms of the SIRS parameters.

2 SIRS model with arbitrary recovery and lossof immunity dynamics

As mentioned above, the usual formulation of a SIRSmodel implies that recovery from infection, I → R, pro-ceeds at a rate which is independent of the moment of

infection. Also, the loss of immunity R → S is also justproportional to the current sub-population and proceedsat its own rate. The mathematical formulation of such amodel is usually presented in terms of differential equa-tions as follows:

ds(t)dt

= −β s(t) i(t) +r(t)τr

, (1)

di(t)dt

= β s(t) i(t) − i(t)τi

, (2)

dr(t)dt

=i(t)τi

− r(t)τr

, (3)

where s(t), i(t) and r(t) stand for the corresponding frac-tions of susceptible, infectious and recovered individualsin the population (s(t) + i(t) + r(t) = 1). The parame-ters of the model are β, the contagion rate per individual,and τi and τr, the characteristic infectious and immunityperiods respectively1.

The analysis of more general systems – in which the re-covery from infection and loss of immunity processes obeymore general and more realistic dynamics – is more in-volved, leading to non-local integro-differential equations.Before proceeding to the most general situation, we ana-lyze the simplest case of fixed times.

2.1 SIRS with fixed infectious and immunity times

Let us assume that the disease is characterized by an in-fectious time τi as well as an immunity time τr. That is, anindividual that becomes infectious at time t will determin-istically recover at time t+τi, becoming immune, and willsubsequently lose its immunity at time t+ τi + τr ≡ t+ τ0,becoming susceptible again. The process is schematicallydepicted in Figure 1. This system can be represented bythe following set of equations for the fraction of suscep-tible and infectious sub-populations (bear in mind thatr(t) = 1 − s(t) − i(t), so that just two equations describethe dynamics):

ds(t)dt

= −β s(t) i(t) + β s(t − τ0) i(t − τ0), (4)

di(t)dt

= β s(t) i(t) − β s(t − τi) i(t − τi). (5)

Before proceeding with the detailed analysis of the aboveformulation of the SIRS model we show in Figure 2 aresult in advance, comparing the time evolution of thefraction of infectives in the two scenarios: the SIRS withtwo fixed time delays and the standard SIRS. The lastone produce the well know behavior of damped oscilla-tions toward the endemic state. Using the same parame-ters (β = 0.4, τi = 5, τr = 50) the SIRS with two delays

1 Note on nomenclature: the infectious time is frequentlycalled as recovery time as well, because it marks the passagefrom I to R which is the recovery from infection. While we pre-fer the first name, in order to avoid ambiguities with labels weassociate them to the classes, i.e.: τi is the time in the infectiveclass, τr is the time in the recovered class.

S. Goncalves et al.: Oscillations in SIRS model with distributed delays 365

0 100 200 300 400time (arbitrary units)

0

0.1

0.2

0.3

frac

tion

of I

nfec

tives

SIRS standardSIRS with delay

Fig. 2. (Color online) Time evolution of the fraction of infectedindividuals for the SIRS model. Comparison between the nu-merical solutions for the standard and the two fixed time delaysformulation. Parameters are β = 0.4, τi = 5, τr = 50.

shows clearly the sustained peaked oscillation in the infec-tive fraction. The time delay in the removed to susceptibletransition, instead of the usual continuous rate transition,is the responsible for the oscillation as we will see in thenext sections. In general, delay equations applied to SIRlike models have noticeable effects on the dynamics, thatis different forms of the infection time distribution yielddifferent dynamics for the infectives and susceptibles [12].The example of Figure 2 is a remarkable case.

In equations (4, 5), the first terms represents the conta-gion of susceptible by infectious ones, which occurs locallyin time at a rate β. The second terms account for loss ofinfectivity (5) and loss of immunity (4). Both terms corre-spond to the individuals infected at some earlier time: t−τi

and t − τ0 respectively, and who have proceeded throughthe corresponding stage of the disease. These equationsmust be supplemented with initial conditions, appropri-ate for interesting epidemiological situations. A reason-able choice, which we use in the remaining of the paper,is an introduction of infectious subjects into a completelysusceptible population:

s(0) = 1 − i0, i(0) = i0, r(0) = 0. (6)

The system (4, 5) has the drawback that any pair of con-stants (s0, i0) satisfies them, apparently indicating thatany pair of values are equilibria. The origin of this prob-lem lies in the fact that (4, 5) together with (6) do notconstitute a well-posed differential problem. Due to thenon-locality in time, extended initial conditions must beprovided. Mathematically, it is usual to provide arbitraryfunctions s(t) and i(t) in the interval [−τ0, 0). From anepidemiological point of view, however, it is more reason-able to provide just the initial conditions at t = 0, andcomplementary dynamics in the intervals [0, τi): no lossof infectivity or immunity, just local contagion (only thefirst term in 4, 5); and [τi, τ0): plus transitions from I to R(absence of the second term of 4).

Indeed, this is the most reasonable choice for the nu-merical solution of the system, and it is the one we havefollowed in the numerical results shown below. For theanalysis of equilibria, however, an integral representation

of the system results into a better-posed problem and thedifficulties for the calculation of the equilibria disappear.

An integral equation equivalent to (5) is:

i(t) = c1 + β

∫ t

t−τi

s(u) i(u) du, (7)

the interpretation of which is immediate: the integral sumsover all the individuals that got infected since time t−τi upto time t. These are all the infectious at time t, since thoseinfected before have already recovered. The integrationconstant c1 is, in principle, arbitrary, but it is easy to seethat it must be zero since no other sources of infectiousexist beyond those taken into account by the integral term.

Complementing (7) it is convenient to write the equa-tion for 1 − r = s + i, which cancels out the first termsin (4, 5):

s(t) + i(t) = c2 − β

∫ t−τi

t−τ0

s(u) i(u) du, (8)

where, again, c2 is an integration constant. In this case wehave c2 = 1 since no other sources of R exist.

The system (7, 8) can be solved for the equilibria ofthe dynamics, s∗ and i∗. One obtains:

s∗ =1

βτi, i∗ =

βτi − 1βτ0

, (9)

which coincides with the equilibria found numerically byintegrating (4, 5), and also corresponds to the same equi-libria that can be found in a constant-rate SIRS model (1),where the rates of recovery and loss of immunity are 1/τi

and 1/τr respectively.

2.2 SIRS with general distribution of infectiousand immunity times

The idea behind the fixed-time delays can be generalizedto describe more complex dynamics. Let us start with theinfected individuals, which is simpler. Consider a prob-ability distribution function G(t), representing the prob-ability (per unit time) of losing infectivity at time t af-ter having become infected at time 0. Observe that thefixed-time dynamics is included in this description, whenG(t) = δ(t− τi). G(t) can be used as an integration kernelin a delayed equation for the infectious. Indeed, the indi-viduals that got infected at any time u < t and cease tobe infectious at time t are:

β

∫ t

0

s(u)i(u)G(t − u)du,

so that the differential-delayed equation for i(t) is:

di(t)dt

= βs(t)i(t) − β

∫ t

0

s(u)i(u)G(t − u)du, (10)

analogous to (5).


A second kernel H(t) must be considered for the loss ofimmunity process. The differential equation for susceptiblecan then be written as:

ds(t)dt

= −βs(t)i(t) + β

∫ t

0

[∫ v

0

s(u)i(u)G(v − u)du

]

×H(t − v)dv, (11)

where the second term corresponds to individuals thatget infected at earlier times, then lose their infectivity atintermediate times with probability G, and finally recoverat time t with probability H .

The difficulty with initial conditions that we faced inthe fixed-times system is also found here, and can besolved in the same way. Integral equations for i(t) ands(t) + i(t) result:

i(t) = c1 + β

∫ t

0

s(u)i(u)du

− β

∫ t

0

[∫ v

0

s(u)i(u)G(v − u)du

]dv, (12)

s(t) + i(t) = c2 + β

∫ t

0

∫ x

0

[∫ v

0

s(u)i(u)G(v − u)

× du H(x − v) − s(v)i(v)G(x − v)]dv dx.

(13)

These equations can be used to find closed expressions forthe equilibria which, depending on the functional forms ofG and H , can be solved analytically. In general one findstwo sets of solutions, the disease free one: s∗ = 1, i∗ = 0,and the endemic one bifurcating from it:

s∗ =1

βΣ1, i∗ =

βΣ1 − 1β(Σ1 − Σ2)

, (14)

with:

Σ1 =∫ ∞

0

[1 −

∫ v

0

G(v − u)du

]dv, (15)

Σ2 =∫ ∞

0

∫ x

0

[H(x− v)

∫ v

0

G(v−u)du−G(x− v)]dv dx.

(16)We observe that using either Dirac deltas or exponentialfunctions for the kernels (corresponding to fixed-times andconstant rates, respectively), these integrals can be solvedanalytically to find the equilibria.

Between the two extremes of constant rates and fixeddelays, as mentioned, realistic systems are expected todisplay some extended probabilities distributions for in-fectious and immunity times. A convenient interpolationbetween the exponential and delta distributions that char-acterize those regimes can be achieved by gamma distri-butions:

Gpi(t) =ppi

i tpi−1e−pit/τi

τpi

i (pi − 1)!, (17)

Hpr(t) =ppr

r tpr−1e−prt/τr

τprr (pr − 1)!

. (18)

These distributions have mean τi and τr respectively,for any value of the parameter pi,r. Besides, they inter-polate between exponential (when pi,r = 1) and Diracdelta distributions (when pi,r → ∞), with smooth bell-shaped functions for intermediate values of pi,r. It can beshown that the equilibria (14) are identical to the classicalones (9) for any pi,r.

3 Sustained oscillations in SIRS with delays

Standard SIRS systems, without delays (or equivalently,with G1(t) and H1(t) as delay kernels) have either nodesor stable spirals as equilibria. That is, oscillations appearin them as transient regimes damped towards the fixedpoints. SIRS systems with delays, on the other hand, canexhibit sustained oscillations. These appear as a Hopf bi-furcation of the spiral points, controlled by the parametersτi, τr and β. A linear stability analysis of the fixed-timescase can exemplify how this happens.

Assuming that the system (4, 5) is close to equilibrium,one sets s(t) = s∗ + x(t), i(t) = i∗ + y(t), obtaining inlinear approximation a linear delay-differential system forthe departures from equilibrium:

x(t)/β = −i∗x(t) − s∗y(t)+i∗x(t − τ0) + s∗y(t − τ0), (19)

y(t)/β = i∗x(t) + s∗y(t)−i∗x(t − τi) − s∗y(t − τi). (20)

From this system, proposing exponential solutions x(t) =c1e

λt and y(t) = c2eλt, a transcendental characteristic

equation is obtained:

λ2 + λβ[s∗(e−λτi − 1) − i∗(e−λτ 0 − 1)

]= 0. (21)

Equation (21) can be solved numerically for complex λ,obtaining from its real part the bifurcation line from thestable spirals. This line is shown in Figure 3 along withthe amplitude of oscillations represented by a colour (gray)map. The amplitude is the result of the numerical integra-tion of the full nonlinear system (4, 5). Figure 3 condensesthe bifurcation phenomenon as a function of the key pa-rameters τr/τi, the ratio of immunity to infective time,and R0 = βτi, known as the basic reproductive rate. Theblack region represents the non oscillating endemic solu-tion. It can be seen that the linear analysis, representedby the line, defines almost exactly the transition.

3.1 Distributed delays: the general case

The fact that we have qualitatively different results re-garding the nature of the endemic state for a delta oran exponential distribution gives rise to a fundamentalquestion. Is the existence of an oscillatory endemic stateparticular to the delta distribution? Or is there a criti-cal shape of the distributions G and H necessary for theemergence of the oscillations? Using the Gamma functions


0

0.1

0.2

0.3

0.4

2

4

6

8

10

0.5 1 1.5 2 2.5 3 3.5

R0

τr / τi

Fig. 3. (Color online) Bifurcation diagram of the SIRS modelwith fixed times in the space defined by τr/τi and R0. Theyellow (white) line shows the linear result. The squared rootamplitude of the infectives oscillations is shown as colour (gray)coded shades above the transition line. The black region meanszero amplitude, representing non-oscillatory endemic states.

defined in (17) we can check the existence of such so-lutions for different shapes by controlling the parameterpi,r. In this way, we can verify if there is a critical shapepi,r = pc(β, τi, τr) beyond which the system has sustainedoscillations at the endemic state. Linearizing the generalsystem (10, 11) in the same way presented in the previ-ous section, one has the following integral characteristicequation:

λ2 + λβi∗[1 −

∫ t

0

∫ t−v

0

H(v)G(u)e−λ(u+v)dudv

]

− λβs∗[∫ t

0

G(u)e−λudu

]= 0. (22)

Using the distributions (17) in the equation above andtaking the limit t → ∞ we get:

λ2 + λβi∗[1 −

(1 +

λτi

pi

)−pi(

1 +λτr

pr

)−pr]

− λβs∗[1 −

(1 +

λτi

pi

)−pi]

= 0. (23)

As expected, for pi,r = 1 and pi,r → ∞ we recover thecharacteristic equations of the constant rates and fixeddelays models, respectively. Solving (23) numerically forcomplex λ, one finds that for every set of parameters (pro-vided that R0 > 1) where pr > 1 there is always a criticalshape pi = pc above which the endemic state consists ofsustained oscillations. Conversely, any value of pr > 1 canpresent sustained oscillations in some region of parameterspace. Instead, the constant rate model pr = 1 is a par-ticular case where there is no such solution for any set ofparameters, as was demonstrated by Hethcote et al. [10].

Figure 4 shows the real part of the solution of (23) asa function of the shape (pi,r = p) of the two time distri-butions, and for fixed values of the parameters β, τi andτr. In other words we can appreciate (for those β, τi, τr)the critical p value (pc) to have oscillations in the SIRSdynamics, i.e. the value at which Re[λ] = 0.

0 1 2 3t

0

0.5

1

1.5

2

Gp(t

)

p=1p=3p=15

0 5 10 15 20p

-0.2

-0.1

0

0.1

Re{

λ}

β = 2 τi = 0.8 τ

r = 16

β = 1 τi = 5 τ

r = 10

Fig. 4. (Color online) Real part of the eigenvalue λ as a func-tion of the shape parameter pr = pi = p. Inset: examples ofthe distribution Gp with τi = 1.

10 15 20p

-0,02

0

0,02

Re{

λ}

2000 2020 2040 2060 2080 2100time (arbitrary units)

0,2

0,4

frac

tion

of in

fect

ives

p=12p=13p=14

Fig. 5. (Color online) Numerical integration of the SIRS gen-eral model with β = 1, τi = 5, τr = 10 near the critical shapepi,r = pc predicted by the linear analysis of the system (inset,12 < pc < 13).

The critical shape pc obtained by numerical calcula-tion of (23) gives an accurate prediction of the emergenceof oscillations in the full nonlinear system (10, 11), as canbe seen in Figure 5. The meaning of the critical shape issimple: the time distributions have to be narrower thanthe one represented by pc in order to have sustained os-cillations.

The distributions of infectious and immunity timesused so far share a common shape given by the value ofp – that made the analysis, restricted to only one shapeparameter, simpler. In real situations, though, it is reason-able to expect that these uncorrelated kernels have differ-ent shapes, not necessarily of the same relative width. Weexplore this more general scenario, presenting an oscilla-tion diagram in terms of pi and pr in Figure 6, for twosets of the SIRS parameters. Some interesting conclusioncan be extracted from such diagram:

– If the immunity time distribution is not sharp (pr < 5for the parameters used in Fig. 6) there is no oscillationindependent of the type of infection time distribution.

– For a relatively narrow distribution of immunity timesthere is a critical infective time width above which (i.e.a critical pi below which) no oscillations are obtained.

– The broader one of the time distribution is, the nar-rower the other one has to be in order to haveoscillations.


1 5 10 15 20 25 30p

i

1

5

10

15

20

p r

β = 1 τi= 2 τ

r= 10

β = 1 τi= 2 τ

r= 20

non oscillatory region

Fig. 6. (Color online) Oscillation diagram in the pi, pr planefor two sets of parameters: β = 1, τi = 2, τr = 10 (circles) andβ = 1, τi = 2, τr = 20 (triangles).

– Longer immunity times (in units of infectious time)makes the oscillatory region wider in the pi, pr space.

In the case pi,r → ∞ there is a critical value of R0τr/τi

above which the endemic solution is always a stable cy-cle. The linear analysis also shows that for finite pi,r thebifurcation is more involved. Figure 7 shows that the bi-furcation line encloses a region of oscillating solutions.Then, for a given value of τr/τi, there is a second criticalR0, larger then the previous one, where the endemic solu-tion ceases to be cyclic. This phenomenon is verified in thenumerical solution of the nonlinear system. For any givenτr/τi, if one increases p the region of oscillation grows (asshown in Fig. 7), so that in the limit pi,r → ∞ the uppercritical R0 disappears. In such a situation, the only wayto break the oscillations is by decreasing R0. On the otherhand, by decreasing pi,r the oscillatory region shrinks, dis-appearing completely when pi,r = 1 (exponential distribu-tions, constant rates).

Yet, the interplay between the uncorrelated shapes pi

and pr and the SIRS parameters is very interesting. Again,in Figure 7 we see for example that when pi = pr = 5(equal very broad shapes) the oscillation region is verysmall. But increasing pi to 20 (narrow infective times dis-tribution), keeping the other fixed, expands the regionconsiderably. Similarly, but in other part of the same dia-gram, when pi = pr = 20 (equal very narrow shapes), theregion of sustained oscillations is relatively large. Now, byletting the distribution of infective times to be broader(pi = 5) again with the other distribution fixed, we seethe oscillatory region to shrink significatively. So, whileit is true that the immunity times distribution must benecessarily different from an exponential in order to haveoscillations, in general both distributions are relevant todefine the actual parameter region of oscillations. The nar-rower one or the two are, the wider the oscillatory regionis.

Three examples of dynamics of the SIRS with dif-ferent shapes sets are displayed in Figure 8. The curvescorrespond to the numerical solution of systems with thesame epidemic parameters but three different shape sets:(pi = pr = ∞), (pi = 1, pr = ∞), and (pi = ∞, pr = 1).

0 5 10 15 20τ

r / τ

i

0

2

4

6

8

10

R0

p → ∞p = 20p = 12p = 5p

i= 20 p

r= 5

pi= 5 p

r= 20

non oscillatory region

Fig. 7. (Color online) Oscillation diagram in the τr/τi, R0

plane for the SIRS model with different distribution for infec-tious and immunity times controlled by the pi,r-shape factorof the Gamma distribution function. For each pair, pi, pr theoscillatory region is enclosed by the corresponding line. Thecurves with only one p are for pi = pr = p.

0 100 200 300 400time (arbitrary units)

0

0.1

0.2

0.3

frac

tion

of I

nfec

tives

τi: delta - τ

r: delta

τi: delta - τ

r: exp.

τi: exp. - τ

r: delta

Fig. 8. (Color online) Fraction of infectious as a function oftime for three distributions of τr and τi with different shapesbut same parameters β = 0.4, τi = 5, τr = 50.

The first two show the sustained oscillations because theSIRS parameters are inside the oscillatory region, while inthe last case we know that there is no possible region ofoscillation so we have the damped behavior. Besides, wesee that enlarging the τi distribution alone, while it short-ens the period of the oscillation, it does not destabilizesthe oscillation. Remarkably it shows an enhancement ofthe number of infected during the low part of the cycles.This behavior diminishes the probability of extinction ina discrete population realization of the system, a problemthat is common in simulations as we will see in the nextsection.

From the imaginary part of (23) it is possible to obtainthe period of oscillations in the linear approximation. InFigure 9 we show the result of this calculation for differ-ent shapes of the distributions of infectious and immunitytimes. For finite pi,r, the period of oscillation has a depen-dency on the parameters that approaches the one foundfor pi,r → ∞ (delta distributions, fixed times) in the lowercritical value of R0, given by

T = 3/4τi + 2τr. (24)


0 5 10 15 20τ

r / τ

i

0

10

20

30

40

50

T /

τ i

p → ∞p = 20p = 12p = 5y = 3/4 + 2 xy = 3/4 + x

R0

Fig. 9. (Color online) Period of oscillations as a function ofτr/τi. Each curve corresponds to a different shape of the timedistributions, as shown in the legend where pi = pr = p. Thereare no oscillations above and below the dashed lines.

For a fixed value of τr/τi, this period decreases as R0 in-creases up to the upper limit of oscillation, being boundedfrom below by 3/4τi + τr, as shown in Figure 9. There areno oscillations above and below these two straight lines,for any value of pi,r. This remarkable lock of the period ofdisease oscillation is related to the one obtained by Taylorand Carr [11] in the case of exponential distributed τi

studied by them.

4 Simulation

A complete picture of the general SIRS dynamics needsto contemplate straightforward numerical simulations. Webelieve that simulations represent the most accurate im-plementation of the real system, which is discrete andstochastic. Therefore, as a test of our analytical andnumerical results with the generalized SIRS model, wepresent here results from a probabilistic discrete model.In this model a finite number of agents meet at randomand contagion proceeds in a probabilistic way. The phasediagram for such a system, with delta distributions for τi

and τr, is shown in Figure 10. The temporal evolution ofthe system is shown in Figure 11 along with the numericalsolution of the deterministic model.

The agreement between the two implementations ofthe same model is very satisfactory, as can be seen byboth the bifurcation diagram and the temporal solution.Regarding the diagram the agreement is not as perfect aswith the numerical solution of the deterministic equations(Fig. 3) because of fluctuations that arise in simulationdue to finite size. These cause large amplitude oscillationswhich become extinct but that contribute with a non-zerovalue for the amplitude. As for the temporal evolution,eventually, as time goes by, a small drift develops in thesimulation due to the stochastic nature of its dynamics.Far from being disappointing this is a desirable feature,since in real systems epidemic oscillations are not exactlyperiodic. Yet, the extinctions, that are normally observedin the simulations, are related with the discreteness ofthem. Long time ago discussed by Bartlett [13] and others

0

0.1

0.2

0.3

0.4

1

2

3

4

5

2 4 6 8 10

R0

τr / τi

Fig. 10. (Color online) Diagram of the oscillation ampli-tude, in the τr/τi, R0 plane, obtained by simulation withN = 1000 agents in the SIRS model with fixed times. Ver-tical axis is R0 = βτi, horizontal axis is the ratio τr/τi, andthe colour (gray) map represent the squared root amplitude ofthe i(t) at the steady state. The black region means zero am-plitude, representing non-oscillatory endemic or non-epidemicstates. Superimposed is the critical line obtained by the linearanalysis as in Figure 3.

0 2 4 6 8time (years)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08fr

actio

n of

Inf

ectiv

esEquationsSimulations

Fig. 11. (Color online) Evolution of an epidemic with stochas-tic and deterministic models. Time distributions are deltas atτi = 10 days and τr = 180 days, and β = .13 per day. With thetime scale in years, the annual oscillation is clearly observed(T = 2τr + 3/4τi = 367 days).

it manifests here as being very sensitive to the distanceto the lower threshold in the bifurcation diagram, whichrelates to the amplitude of the oscillation. Greater am-plitudes drive the system closer to the absorbing state ati = 0. To illustrate this effect we show in Figure 12 threedynamics obtained by simulation with N = 105 for threedifferent values of β, close to the onset of epidemics and os-cillations. However we saw in the previous section (Fig. 8)that a distribution of infectious times with finite widthcan increase the lower values of the infectives cycle, whicheventually may prevent the extinction in the simulationcounterpart.

5 Conclusions

We have analyzed a general SIRS epidemic model, inwhich the infective state as well as the immune state


0 2 4 6 8 10time (years)

0

0.01

0.02

0.03

0.04

0.05

0.06

frac

tion

of I

nfec

tives

β = 0.11β = 0.12β = 0.13

Fig. 12. (Color online) Evolution of the epidemic in thestochastic model, showing extinctions of the infected popula-tion. Time distributions are deltas at τi = 10 and τr = 180days, N = 105.

last prescribed times drawn from distributions. Thesedistributed-time transitions constitute a generalization ofthe standard SIRS model, in which the transitions out ofthe infective and the immune states happen at a constantrate. The generalization allows for situations in whichthese states last for certain fixed times – which is moresimilar to many real diseases than the constant rate as-sumption. Between the two extremes of constant rates andfixed times, we have also analyzed the intermediate situa-tions of broader or narrower distributions of the transitiontimes.

The generalization runs along the proposals made byprevious authors, by implementing the differential equa-tions of the model as non-local in time. For exampleHethcote et al. have shown that cyclic models have a tran-sition to oscillatory behavior when the immunity time isdistributed [10].

Our contribution shows how the oscillating state arisesas a function of all the parameters of the model, complet-ing a phase diagram that provides a thorough and gen-eral view of the possible behaviors. We show, moreover,that the linear analysis, the numerical integration of themodel, and its stochastic implementation, all converge tothe same general picture, within the inherent limitationsof each one.

For delta distributed delays (i.e., for fixed times spentin the infective and recovered classes), the phase diagramof Figure 3 shows that the region of oscillation is boundedfrom below by the basic reproductive number R0 as afunction of the scaled immunity time τr/τi. We see that,as τr decreases toward zero, the minimum value of R0

diverges. This result is in agreement with the fact thatin the SIS model there are no sustained oscillations. Onthe other hand, the amplitude of the oscillations growsboth with R0 and τr/τi. This kind of behavior may berelevant in the analysis of real epidemics where changesin the parameters are occurring due to interventions, ad-vances in treatment or natural causes. For example, let usimagine an epidemic in the endemic non-oscillating region(shaded black in the diagram), for which, as a desirableconsequence of the treatment of the disease, there is anincrease in the duration of the immune state, at constant

R0. As a result, the epidemic may start to oscillate. Such atransition may manifest itself as an initial increase of theinfectious fraction, due to the onset of oscillation, whichshould be properly understood in the right context.

We have also shown that the transition to oscilla-tions behaves as a critical phenomenon depending on thewidth of the time distributions. Figure 4 shows this be-havior. The remarkable stabilization of the oscillationsof pertussis after the introduction of massive vaccination([1] Sect. 6.4.2) may be related to a narrowing of the im-munity time distributions.

For distributed delays, corresponding in our model tovalues of the parameter 1 < pi,r < ∞, we have found aremarkable reentrance phenomenon in the phase diagram.As exemplified by Figure 7, this reentrance means that theregion of oscillations is also bounded from above by a curveof R0 vs. τr/τi. One sees, also, that the region of oscillationshrinks with lowering pi,r, until its disappearance at thecritical value pc. Only the case pi,r → ∞, corresponding todelta distributed times is unbounded. This is at variancewith the SIR case analyzed by [14] who claims that thereis no significant change between large values of pi,r = p(p = 10–20) and p → ∞. On the other hand our modelsupports their result in that the region of oscillations isalready large for p in this range. An important general re-sult that can be derived from this diagram is the fact that,whenever τr � τi, the minimum value of R0 is very closeto 1, implying that such systems – provided that pi,r is suf-ficiently large – are very prone to sustained oscillations.This implies that oscillations due to the implementationof real time distribution into the SIRS formalism are not amarginal or delicate effect as someone could think before.

Our analysis also shows the behavior of the period ofoscillation and its dependence on system parameters. It isinteresting to discuss this aspect of our model in connec-tion with real infectious diseases, like influenza, syphilis,and pertussis. While the spread of influenza – and prob-ably the others too – can be understood only by takinginto consideration several factors like human mobility, theinteraction between different age classes, and the contactnetwork, we want to show here some practical results re-stricted to the ideas presented so far.

In the case of influenza, where the period of oscilla-tion is more or less a year and the infectious time is ofthe order of 10 days, our model predicts oscillations forany R0 > R0min = 1.09. This minimum value is lowerthan the usual estimate for this disease, which is around1.4 [15]. Indeed, this value is so close to R0 = 1 that onecan conclude that there will always be oscillations for thisdisease as long as it remains endemic. Besides we see thatusing a value of τr = 180 days for the loss of immunity,the region of oscillation lays between 1.22 < R0 < 2.4for a rather wide distribution with pi,r = p = 10, and1.16 < R0 < 3.4 for a narrower distribution with p = 30.In the first case, the corresponding period of oscillationis 188 < T < 342 (in days), while on the second one itis 190 < T < 360. Therefore, one needs a narrow dis-tribution to satisfy the observed period of oscillation forinfluenza. In any case, the predicted region of oscillation


includes the observed values of R0. We want to stress that,although an infection by a specific strain of the influenzavirus confers permanent immunity after the infection pe-riod, the mutation rate of the virus can be thought of asan effective immunity period and therefore can also bemodeled with the SIRS dynamics.

For syphilis, in the pi,r → ∞ case, using the period of132 months measured in United States [3] and the usualvalue of τi = 6 months, we have an R0min = 1.17 andτr ≈ 6 years. Greater values of R0 (which are expected forsyphilis in [3]), with fixed τi, predict even longer immunetimes (which is also the case of syphilis). For pertussis,using the period of 4 years and τi = 1−2 months, we findR0min = 1.06−1.15, which is smaller than the estimationsobtained for mixing models [15].

In the open field of oscillatory epidemics, we believethat there is a multiplicity of causes that can concur toproduce the observed phenomena. Even when the consis-tency of the present formulation with the observed datain the three cases of oscillating diseases just consideredis very appealing, we do not intend to provide an exclu-sive and definitive answer to this matter. The constructionof valid theoretical models for real diseases should incor-porate all the relevant mechanisms, therefore a thoroughtheoretical investigation of any concurrent possible causedeserves due attention. We hope our work has revealedone good candidate.

This work was supported by a cooperative agreement betweenBrazilian Coordenacao de Aperfeicoamento de Pessoal de NıvelSuperior and Argentinian Ministerio de Ciencia y Tecnologiaagencies (grant CAPES-MINCyT 151/08 – 017/07), and byBrazilian agency Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico (grant CNPq PROSUL-490440/2007).

S.G. and M.F.C.G acknowledge financial support from CNPq,Brasil. G.A acknowledges support from Argentinian agen-cies Agencia Nacional de Promocion Cientıfica y Tec-nologica (PICT 04/943), Consejo Nacional de InvestigacionesCientıficas y Tecnicas (PIP 112-200801-00076), and Universi-dad Nacional de Cuyo (06/C304).

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1. R.M. Anderson, R.M. May, Infectious Diseases of Humans:Dynamics and Control (Oxford University Press, 1991)

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(2005)4. B. Grenfell, O. Bjørnstad, Nature 433, 366 (2005)5. G. Abramson, V.M. Kenkre, Phys. Rev. E 66, 011912

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(2007)7. J.P. Aparicio, H.G. Solari, Math. Biosci. 169, 15 (2001)8. M. Kuperman, G. Abramson, Phys. Rev. Lett. 86, 2909

(2001)9. F.C. Hoppensteadt, Mathematical Theories of

Populations: Demographics, Genetics and Epidemics,in Regional Conference Series on Applied Mathematics(SIAM, Philadelphia, 1975)

10. H.W. Hethcote, H.W. Stech, P. Van Den Driessche, SIAMJ. Appl. Math. 40, 1 (1981)

11. M.L. Taylor, T.W. Carr, J. Math. Biol. 59, 841 (2009)12. M.F.C. Gomes, S. Goncalves, Physica A 388, 3133 (2009)

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