Dynamical resonance can account for seasonality ofinfluenza epidemics
Jonathan Dushoff*†‡, Joshua B. Plotkin§, Simon A. Levin*, and David J. D. Earn¶*Department of Ecology and Evolutionary Biology, Princeton University, Princeton, NJ 08540; †Fogarty International Center, National Institutes of Health,Bethesda, MD 20892; §Harvard Society of Fellows, 78 Mount Auburn Street, Cambridge, MA 02138; and ¶Department of Mathematics and Statistics,
McMaster University, Hamilton, ON, Canada L8S 4K1
Contributed by Simon A. Levin, October 6, 2004

Influenza incidence exhibits strong seasonal fluctuations in temperate regions throughout the world, concentrating the mortality
and morbidity burden of the disease into a few months each year.
The cause of influenza’s seasonality has remained elusive. Here weshow that the large oscillations in incidence may be caused byundetectably small seasonal changes in the influenza transmission
rate that are amplified by dynamical resonance.
demographic stochasticity 兩 disease dynamics 兩 susceptible, infectious,
recovered (SIR) 兩 immunity

he underlying cause of seasonal oscillations in influenza
incidence (1) remains unclear despite at least 80 years ofinvestigation (2). These oscillations are presumably due to somemechanism that causes seasonal changes in the effective transmission rate of the virus from person to person (often calledseasonal forcing). But laboratory experiments and epidemiological studies have failed to establish whether these transmission
changes are due to direct effects of temperature and humidity ontransmission (3), to changes in mixing patterns [e.g., school terms(4) or simply more time spent indoors], or to other factors, suchas increased viral production under winter conditions (evidenced by higher febrile reaction to identical influenza exposures; ref. 5). In fact, as we show here, it may be impossible toestablish the underlying cause of seasonality in influenza epidemics, because the large observed oscillations in incidence canbe generated by seasonal changes in the transmission rate thatare too small to measure.
For an infectious disease that induces permanent immunity,
simple deterministic models indicate that disease incidence has
an intrinsic tendency to oscillate, even in the absence of exogenous driving factors. But these intrinsic oscillations are always
damped (6). Such models are often called susceptible, infectious,
recovered (SIR) models, referring to the transition of an individual between susceptible, infectious, and recovered states.
For influenza, immunity is not permanent. After recoveringfrom one antigenic variant of the virus, a person is usually at least
partially susceptible to new variants within a few years, due togradual evolution of the virus (7). We model this process ofantigenic drift by simply allowing people to lose their resistanceto the circulating virus and hence to move back to the susceptibleclass after a few years [i.e., a SIR-susceptible (SIRS) model]. For
simplicity, we do not explicitly model the exposed population butinstead include people infected but not yet infectious in the ‘‘I’’box. Including an exposed class yields similar results.
In a SIRS model, the intrinsic period of oscillation is approximately T ⫽ 2 ␲公DL兾(R0-1), where D is the mean infectious
period, and L is the average duration of immunity (4, 6). Thebasic reproductive number R0 is defined as the expected numberof secondary infections that a single infectious individual willcause in a wholly susceptible population.
If the period of endogenous SIRS oscillations (T) is near theperiod of seasonal forcing (1 year), then these two factors may
resonate to produce greatly amplified oscillations in incidence(8). Appropriate parameter values for influenza (9–11), 6–10
www.pnas.org兾cgi兾doi兾10.1073兾pnas.0407293101

Fig. 1. The effect of seasonal forcing on oscillations in influenza incidence.
We simulate disease dynamics in a population of 500,000 people both with(blue) and without (red) demographic stochasticity. R0 varies sinusoidally
between 9.6 and 10.4. The black curve shows the (nearly invisible) oscillationsin incidence that these variations in R0 would cause if disease dynamics
responded instantaneously to changes in transmission, without resonance ortransient fluctuations. (a) Weak resonance. The intrinsic oscillatory period isT ⬇ 0.59 years (duration of infectiousness 0.02 yr, duration of immunity 4 yr).
(b) Strong resonance. T ⬇ 0.94 yr (duration of infectiousness 0.025 yr, duration of immunity 8 yr). When parameters are drawn at random from theranges given for R0, D, and L in the text, we find strong oscillations due toresonance (winter peak兾summer trough ⬎5) for 21% of parameter sets
(Fig. 2).

days for D, 4–8 years for L, and 4–16 for R0, yield an endogenous
oscillatory period, T, between 0.4 and 1.5 years. Therefore, theintrinsic period of oscillation for influenza is potentially close tothe (annual) period of seasonal forcing, in contrast to humanviral diseases that engender permanent effective immunity andtypically have intrinsic interepidemic periods of 2–5 years (4).
Whereas deterministic models treat the number of people ineach state as a continuous variable, in reality, individuals are
discrete, and the process is more appropriately modeled as aAbbreviations: SIR, susceptible, infectious, recovered; SIRS, SIR-susceptible.
‡To whom correspondence should be addressed. E-mail: dushoff@eno.princeton.edu.

© 2004 by The National Academy of Sciences of the USAPNAS 兩 November 30, 2004 兩 vol. 101 兩 no. 48 兩 16915–16916

MEDICAL SCIENCES

TTable 1. Markov chain transition rates
EventInfection
RecoveryImmunity loss

ChangeRate(S, I) 3 (S ⫺ 1, I ⫹ 1)
(S, I) 3 (S, I ⫺ 1)
(S, I) 3 (S ⫹ 1, I)

␤(t)IS兾N
I兾D(N ⫺ S ⫺ I)兾L

stochastic Markov chain. The resulting demographic stochasticity causes fluctuations in disease incidence to persist (12).
Demographic stochastisticity is more important if the average
duration of immunity is long compared with the duration ofinfection (13). In the case of influenza, the ratio between thesetwo time scales is on the order of a few years to 1–2 weeks,or ⬇100.
Methods
The red curves in Fig. 1 represent solutions to the deterministic
SIRS epidemic model defined by the equations
dS兾dt ⫽ 共N ⫺ S ⫺ I兲兾L ⫺ ␤共t兲IS兾N[1]

dI兾dt ⫽ ␤共t兲IS兾N ⫺ I兾D[2]

where t is time in years, N is the total population size, andN⫺S⫺I gives the number of resistant (immune) individuals. Thecontact rate ␤(t) varies sinusoidally according to the formula␤共t兲 ⫽ ␤0共1 ⫹ ␤1 cos共2 ␲ t兲兲.

[3]

In this formulation, R0 equals D ␤0. The parameter values
corresponding to Fig. 1a are n ⫽ 500,000, L ⫽ 4 yr, D ⫽ 0.02yr, ␤0 ⫽ 500 per year, and ␤1 ⫽ 0.02. The parameter values inFig. 1b are n ⫽ 500,000, L ⫽ 8 yr, D ⫽ 0.025 yr, ␤0 ⫽ 400 peryear, and ␤ 1 ⫽ 0.02. The blue curves in Fig. 1 represent solutions
to the discrete-state continuous-time Markov chain in which
events occur at the exponential rates given in Table 1.
The Markov chain was simulated by using the standard Gillespie
algorithm (14).
ResultsFig. 1 shows the effect of resonance, using simple deterministic
and stochastic models of inf luenza dynamics in a population of
500,000 people, with very small sinusoidal forcing of thetransmission rate. If resonance is weak (i.e., the endogenous
period T is far from 1 year), then the resulting oscillations inincidence are much smaller than are actually observed forinf luenza. If resonance is strong, however, the resulting dynamics exhibit regular annual epidemics in which incidence
1. Earn, D. J. D., Dushoff, J. & Levin, S. A. (2002) Trends Ecol. Evol. 17,334–340.
2. Huntington, E. (1920) Ecology 1, 6–23.
3. Schulman, J. L. & Kilbourne, E. D. (1962) Nature 195, 1129–1130.
4. Anderson, R. M. & May, R. M. (1991) Infectious Diseases of Humans: Dynamicsand Control (Oxford Science, Oxford).
5. Nelson, R. J., Demas, G. E., Klein, S. L. & Kriegsfeld, L. J. (2002) SeasonalPatterns of Stress, Immune Function and Disease (Cambridge Univ. Press,Cambridge).
6. Wilson, E. B. & Worcester, J. (1945) Proc. Natl. Acad. Sci. USA 31, 294–298.
7. Cox, N. J. & Bender, C. A. (1995) Semin. Virol. 6, 359–370.

16916 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0407293101

Fig. 2. Magnitude of observed oscillations (ratio of peak-to-trough incidence) in the stochastic forced SIRS epidemic model plotted against theapproximated period of endogenous oscillations in the SIRS model[2␲兹DL兾(R0 ⫺ 1)], for 2,000 sets of parameters randomly chosen from theranges given in the text (shown are the 1,560 trials where the disease persisted
for at least 20 years after being started from the deterministic equilibrium).
Underlying variation in transmission rate is ⫾ 4% (i.e., ␤1 ⫽ 0.04 in Eq. 3).
Strong resonance occurs when the approximate endogenous period is near
1 year.

increases by a factor of three or more from winter to summer.
Fig. 2 shows the endogenous period and the resulting size ofoscillations for parameters chosen at random from the plausible range for inf luenza. Due to resonance, large oscillations(average peak-to-trough ratio ⬎5) occur for about one-fifth ofparameter sets, even when the transmission rate varies by only
a few percent.
Researchers have worked to identify and measure a seasonal
component of influenza transmission with the goal of explaining
large annual fluctuations in incidence. But, as we have seen here
using simple models, these large fluctuations may be caused byexogenous seasonal changes in transmission that are too small todetect, amplified by the endogenous population dynamics of thehost–pathogen system.
We are grateful for valuable discussions with Viggo Andreasen,
Tom Reichert, and Lone Simonsen and helpful comments from
Herbert Hethcote and an anonymous reviewer. This work was supportedby the National Institutes of Health (Grant 1-R01-GM60729-01, toS.A.L.) and the Canadian Institutes of Health Research (to D.J.D.E.).
8. Dietz, K. (1976) in Lecture Notes in Biomathematics: Mathematical Models inMedicine, eds. Berger, J., Bühler, W., Repges, R. & Tautu, P. (Springer, Berlin),
pp. 1–15.
9. Douglas, R. G., Jr. (1975) in The Influenza Viruses and Influenza, ed. Kilbourne,
E. D. (Academic, New York), pp. 395–447.
10. Fine, P. (1982) in Influenza Models: Prospects for Development and Use, ed.
Selby, P. (MTP Press, Lancaster), pp. 15–85.
11. Pease, C. M. (1987) Theor. Popul. Biol. 31, 422–452.
12. Bartlett, M. S. (1957) J. R. Stat. Soc. A120, 48–70.
13. Nasell, I. (1999) J. R. Stat. Soc. Ser. B 61, 309–330.
14. Gillespie, D. T. (1976) J. Comput. Phys. 22, 403–434.

Dushoff et al.