Response of a deterministic epidemiological system to a stochastically varying environment J. E. Truscott* and C. A. Gilligan Epidemiology and Modelling Group, Department of Plant Sciences, University of Cambridge, Downing Street, Cambridge CB2 3EA, United Kingdom Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved May 20, 2003 (received for review October 16, 2002) Fluctuations in the natural environment introduce variability into the Kingdom (7–10). Rhizomania is a highly persistent soil-borne biological systems that exist within them. In this paper, we develop disease of sugar beet with a serious economic impact for growers. a model for the influence of random fluctuations in the environment The disease is caused by the beet necrotic yellow vein virus, which on a simple epidemiological system. The model describes the infection is transmitted via the soil-borne endoparasitic slime-mold vector, of a dynamic host population by an environmentally sensitive patho- Polymyxa betae. A critical feature of the amplification of the gen and is based on the infection of sugar beet plants by the inoculum in the host plant is the ability of its fungal vector to attack endoparasitic slime-mold vector Polymyxa betae. The infection pro- the host plant’s roots. This process is characterized by a cutoff cess is switched on only when the temperature is above a critical temperature of Ϸ12°C, below which the fungus is unable to enter value. We discuss some of the problems inherent in modeling such a the roots (11, 12). As a result, the extent of disease in the host is very system and analyze the resulting model by using asymptotic tech- sensitive to climate and sowing date. A stochastically varying niques to generate closed-form solutions for the mean and variance temperature can cause the root system to move back and forth of the net amount of new inoculum produced within a season. In this through the critical temperature, thus switching the infectivity of way, the variance of temperature profile can be linked with that of the vector on and off. In this way, variability in temperature is the inoculum produced in a season and hence the risk of disease. We transmitted directly into the host-vector system and hence to the also examine the connection between the model developed in this quantity of new inoculum produced by the host. This mechanism is paper and discrete Markov-chain models for weather. discussed in more detail in Section 2.2. In Section 2.1, we introduce the deterministic system in detail and 1. Introduction discuss the effect of varying temperature on its evolution. We then ost analytical work on stochasticity in epidemics has focused derive a model of the full system as a stochastic differential equation Mon demographic variability (1, 2). However, in many in- (SDE) and discuss some of its assumptions and limitations in stances, environmental variation can be a critical influence on the Section 2.3. We analyze the resulting SDE in Section 3 by using development of an epidemic. A wide range of economically and asymptotic techniques to get closed-form solutions for the mean environmentally important fungi and invertebrates have a strong and variance of the solutions in terms of the properties of the sensitivity to environmental factors. Many species of fungi, includ- driving environmental variable. In Section 4, we show how these ing the damping-off fungi (Rhizoctonia solani), mildews (Blumeria results can be linked with a specific Markov-chain weather- graminis), and rusts (Puccinia spp.), have threshold temperatures generation model. In Section 5, we compare the results of the and humidity levels for germination to occur. Some pathogenic analysis with numerical simulations of the system and examine how nematodes that cause severe disease in staple crops exhibit a critical the analysis can lead to testable hypotheses about the behavior of sensitivity to soil moisture content, becoming inactive at low levels the system. (3). A variety of insect pests and parasites are strongly influenced 2. The Model by environmental switching, most notably by diapause, a suspension of development often triggered by changes in temperature, light 2.1. The Deterministic System. The biological system addressed is a levels, or humidity (4). Organisms with the sensitivities described susceptible-infectious epidemic driven by primary infection from a above may be acting directly as pathogens on the hosts or as reservoir of inoculum (X) in which the force of infection is switched biocontrol agents, predating pests on a host (5, 6). Environmental on and off above and below a critical temperature. The model variables such as temperature and moisture levels have both a presented below is motivated by work done on a sugar beet–P. betae predictable mean trend over time and a short time-scale random model described and analyzed in refs. 7–10. The model comprises component. Both aspects of this variation can be transmitted to the the following system of ordinary differential equations: disease process through the sensitivity of the organisms involved. dn Hence the variability of the environment is fed through to the state ϭ f͑n, t͒, [1] of the epidemic. dt In this paper, we derive solutions for the evolution of the mean ds ϭ ͑ ͒ Ϫ Ϫ ⌳ and variance for newly generated infectious material for a broad f n, t ms mXs, [2] class of epidemics under the influence of stochastic switching. Our dt model is based on a dynamically changing host population as di ϭ ⌳ susceptibles are produced or become infected or resistant; inocu- Q mXs, [3] lum germinates in response to an environmental threshold and dt amplifies after infection of the host plant. This analysis depends, in with initial conditions, n(0) ϭ s(0) ϭ n , i(0) ϭ 0. The variables part, on the effective correlation time of the stochastically switching 0 n, s, and i represent the total population, the susceptible system. This correlation time is the mean time over which the population at risk from the disease, and the inoculum generated switching process remains in a single state. We explicitly link our results with Markov-chain weather-generating models, enabling us to connect the statistics of environmental variation with those of This paper was submitted directly (Track II) to the PNAS office. resulting epidemic. The results presented here have broad appli- Abbreviation: SDE, stochastic differential equation. cability to a range of pests and diseases. BIOLOGY *To whom correspondence should be sent at the present address: Department of Infectious POPULATION Our work on this problem arises from modeling the amplifica- Disease Epidemiology, Faculty of Medicine, Imperial College, London W2 1PG, United tion, transmission, and invasion of rhizomania disease in the United Kingdom. E-mail: [email protected]. www.pnas.org͞cgi͞doi͞10.1073͞pnas.1436273100 PNAS ͉ July 22, 2003 ͉ vol. 100 ͉ no. 15 ͉ 9067–9072 Downloaded by guest on September 29, 2021 Y Phase 3, in which the temperature is constantly above Tc and the infection process is always active (t2 to t3 in Fig. 1). Clearly the dynamics of Phase 1 are straightforward, assuming deterministic initial conditions. Phase 3 is once again deterministic but must describe the evolution of the mean and variance of the variables as they stand at the end of Phase 2. Phase 2, however, requires a new approach to integrate the stochasticity into the dynamics. 2.3. Stochastic Differential Equation (SDE) Model. An SDE is used to model the continuous-time stochastic process. The SDE comprises the deterministic mean behavior to which a noise term is added in the form of an infinitesimal Wiener process. The deterministic solution of Eqs. 1–3 is easily calculated in terms of the function, f(n, t). It is now necessary to formulate the infection switching in terms of the Wiener process. Problems arise from the fact that the Fig. 1. Representation of the stochastic variation of temperature with time in a switching process is binary; i.e., it is either ‘‘on’’ or ‘‘off.’’ In trying season, showing the relationship between the critical temperature, Tc, and the various phases of the evolution of the system. to formulate a continuous-time representation of switching based on a Wiener process, one must take a limit as dt 3 0. As this limit is approached, the high-frequency elements in the Wiener process by the infected population, respectively. The function, f(n, t), spectrum dominate, causing the variance and hence stochastic represents the rate of introduction of new hosts at any time, t, terms in the equation to vanish. Viewed from a physical or and encompasses very many commonly used growth functions. biological standpoint, the nature of the problem is clear. In the For numerical work, we use a monomolecular function, f(n, t) ϭ context of the sugar-beet rhizomania system, switching at high r(1 Ϫ n), which is both representative and convenient for frequencies would require soil to heat up and the behavior of the calculation. Clearly, Eq. 1 can be solved independently of the parasite to change almost instantaneously. A more accurate de- other equations, and hence we can replace the function, f, in Eq. scription of the process would require either that the sharp step 2 with nt(t), the derivative of the total population. We assume from ‘‘off’’ to ‘‘on’’ at Tc be made into a smooth function or that the that newly generated inoculum is released as fully active for the physical system be given some inertia in changing state. Of these next season. In the context of rhizomania, the variables n, s, and two, the second can be approximated by assuming that there exists i represent the fraction of the total root length in a particular an effective correlation period over which the state of the system state.