Mass vaccination to control chickenpox: The influence of zoster. (population dynamics/infectious diseases/mathematical modeling/varicella/immunization).
Proc. Natl. Acad. Sci. USA Vol. 93, pp. 7231-7235, July 1996 Medical Sciences
Mass vaccination to control chickenpox: The influence of zoster (population dynamics/infectious diseases/mathematical modeling/varicella/immunization)
NEIL M. FERGUSON, RoY M. ANDERSON, AND GEOFF P. GARNETT* Wellcome Centre for the Epidemiology of Infectious Disease, Department of Zoology, Oxford University, South Parks Road, Oxford OX1 3PS, United Kingdom
Communicated by Robert May, University of Oxford, Oxford, United Kingdom, February 20, 1996 (received for review November 17, 1995)
ABSTRACT The impact of transmission events from patients with shingles (zoster) on the epidemiology of varicella is examined before and after the introduction of mass immunization by using a stochastic mathematical model of transmission dynamics. Reactivation of the virus is shown to damp stochastic fluctuations and move the dynamics toward simple annual oscillations. The force of infection due to zoster cases is estimated by comparison of simulated and observed incidence time series. The presence of infectious zoster cases reduces the tendency for mass immunization to increase varicella incidence at older ages when disease severity is typically greater.
allow detailed investigation of the effects of different vaccination policies, where age structure and mixing patterns play a critical role (13). We therefore adapted a stochastic realistic age-structured (RAS) model design that has been applied successfully to investigate the pre- and postvaccination epidemiology of the measles virus (14-16) and used it to explore the potential impact of varicella vaccination (3, 4). The RAS model incorporates discrete age structure in the form of 20 yearly cohorts and one group representing those over 20 years of age. The age-specific force of infection Aj is the rate at which susceptible individuals in age cohort i acquire infection and is defined as
Approval for the use of a varicella (chickenpox) vaccine was given by the U.S. Food and Drug Administration in March 1995 (1, 2). The predictions of mathematical models of varicella transmission dynamics played a part in the decision to license the vaccine, and they will have a role in the design of any mass vaccination programs (3, 4). To date, however, all such models of varicella transmission have been based upon a somewhat simplified view of the natural history of varicella infection. Following the primary disease stage of varicellazoster virus (VZV) infection, the infectious agent becomes dormant in the dorsal root ganglia (5, 6), from where it can reactivate many years later, causing herpes-zoster (shingles). The first indication that the two diseases might be connected was the 1888 observation, by von Bokay, of varicella cases acquired through contact with zoster. However, beyond acknowledging its role in the continued presence of VZV in small isolated populations (7, 8), the importance of infections acquired from zoster has not been considered in studies of the transmission dynamics of the virus. This is largely due to ignorance of the relative infectiousness of zoster compared with varicella. By using a stochastic version of a simple, seasonally forced, compartmental [SEIR-susceptible (X), exposed (H), infectious (Y) recovered (Z)] transmission model, Rand and Wilson (9) demonstrated that chaotic repellors can be stabilized by stochastic fluctuations. Stochastic dynamics with a positive Lyapunov characteristic exponent created large deviations from the deterministic orbit. Fluctuations of such magnitude had not been observed in the similar stochastic models of Olsen and colleagues (10-12). Their models produced incidence time-series dominated by an annual cycle in good agreement with data on varicella incidence from Bornholm and Copenhagen in Denmark, as a result of using an immigration rate of infectives (21 per year for N = 106, and 2 per year for N = 50,000), much larger than that used in the Rand and Wilson model. The effect of this immigration term is to destroy the chaotic repellor, and consequently reduce the fluctuations that would otherwise be seen. While SEIR models may be able to replicate the observed incidence patterns of varicella quite accurately, they do not
A =
lfijyj+4'
[1]
Here 3ij is the transmission coefficient from cohortj to cohort i, and Yj is the number of infectives in cohort j (see Appendix for model details). 4' represents a constant "background" force of infection, generated by the additional varicella infections caused by herpes-zoster cases. The limited data that exist on zoster incidence indicates that it has a white noise spectrum, with no discernible dynamical structure (17, 18), justifying our assumption that the force of infection due to zoster can be presumed to be constant (i.e., nonseasonal). However, since zoster incidence data tell us little about the magnitude and age structure of its contribution to VZV transmission, it is necessary to explore the effect of a wide range of 4' values on model dynamics. In general, the introduction of a constant rate of infective immigration into the SEIR or RAS models has a profound (and often unremarked upon) effect on their dynamics (1921), effectively imposing a minimum bound on Y, and thereby destroying all attractors with orbits dropping below that bound. In the case of measles models, where immigration is used as a crude representation of spatial effects (coupling a population to an external reservoir of infection), the result is the destruction of those attractors that produce the large amplitude chaotic behavior seen in the SEIR model with large seasonal forcing. For chickenpox, non-negligible immigration destroys the chaotic repellor seen in the SEIR model (9), thereby stabilizing the dynamics and moving them toward the deterministic annual limit cycle. Seasonality is introduced by reducing the value of the transmission coefficient 132 (see Appendix, Eq. 4), describing mixing within primary school children, outside school term time. The magnitude of seasonal forcing is calculated as the difference, AP3, between the in-school and out-of-school mean ,B values (see Appendix). The deterministic version of this model generates a annual limit cycle for the range of values of AP3 and 4' relevant to varicella in developed countries. The magnitude of the oscillations increased with AP3, but varying Abbreviations: VZV, varicella-zoster virus; SEIR model, susceptible, exposed, infectious, recovery model; RAS model, realistic agestructured model; FOI, force of infection; WAIFW, who acquires infection from whom matrix. *To whom reprint requests should be addressed.
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peak heights. The power ratio curves obtained by varying A4 for different values of N (Fig. lb) encompass three different dynamical regimes. For a low force of infection from zoster (A4 < 10-5), when the population is small (N < 2 x 106) all the curves converge to r 0.575, corresponding to long periods of disease extinction, interspersed by large epidemics (see Fig. 2f for typical spectrum). Around the critical population size of N 2 x 106, the dynamics becomes much less stable (reflected by the larger error bars on the curve), intermittently switching between cycles of different periodicities (1-4 years) and amplitudes (Fig. lc). For large populations N . 5,000,000, extinctions become very rare and the dynamics show a strong annual oscillatory component for the entire range of zoster forces of infection. An intermediate zoster force of infection (4' = 10-3) is sufficient to prevent frequent extinctions, and the dynamics are then dominated by large-scale triennial and biennial stochastic fluctuations (see Fig. 2e for typical spectrum). As A4 increases further, stochastic fluctuations are increasingly damped and annual oscillations dominate the dynamics. Increasing A4 is directly equivalent to increasing N and the dynamics move rapidly toward the deterministic limit. For large populations (N < 107), the power ratio curve flattens off and r
