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 30, 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. The model contains only four parameters: m, the rate at will not change. The system samples only the random variable which susceptibles become resistant to infection; X, the initial between these periods. In Section 4, this approach is compared inoculum loading; ⌳m, the force of infection, and Q, the ampli- directly to discrete first-order Markov-chain models for generating fication factor for new inoculum. Temperature dependence stochastic data, as used in weather modeling (13, 14). The strong enters through the parameter, ⌳m, as follows: correspondence between the SDE formulation and such models allows us to directly include such environmental influences into ,ifT Ն T , SDE descriptions. ⌳ ͑T͒ ϭ ͭ m c [4] m 0, otherwise. To derive a continuous-time stochastic differential equation representation of the system described above, we first calculate the That is, for temperature less than the critical value, Tc, infection is mean and variance of the process. Consider the system described by effectively turned off, whereas for temperature above this value, the Eqs. 1–3 at time, t,instate(n, s, i). Let pϩ(t) be the probability that ⌬ force of infection is constant. the temperature is above Tc at time t, and let T, the effective correlation time, define the discrete time interval at which the 2.2. The Fluctuating Environment. In the present context, we focus on system checks the temperature. We can interpret ⌬T as a property temperature as the critical aspect of the environment. However, of either the deterministic system or the driving stochastic variable. more generally, a range of possible factors could introduce sto- Within the deterministic system, it can represent the time for the chastic effects into the system, such as rain, sunlight, and animal force of infection to respond to a change in temperature. In the case vector movement. These are discussed in more detail in Sections 4 of the vector for rhizomania, this could be the characteristic time and 6. During the growing season, the mean temperature rises scale for soil heating. From the viewpoint of temperature, it can considerably and passes through the critical temperature of the represent a correlation time over which the temperature remains parasite. The obvious deterministic approach to this changing unchanged. This possibility is discussed in detail in Section 4 with environment is to consider the parasite to be ‘‘switched off’’ until regard to discrete Markov-chain models. the mean temperature reaches Tc and ‘‘switched on’’ afterwards. The effective correlation time is assumed to be small compared However, if we consider temperature to be a stochastically varying to the time scale of evolution of the deterministic system. Consid- quantity, as experienced by the parasite, then the system will by ering just the changes over ⌬T of the stochastically varying ele- switched on and off randomly while the variation in temperature ments, s and i, spans Tc. Given that temperature can be taken to vary about the ds n ͑t͒ Ϫ ms mean between definable extremes, the rising temperature profile ͩ ͪ ϭ ͩ t ͪ⌬T, can be divided into three temporal phases (Fig. 1) with respect to di 0 the parasite. with probability, 1 Ϫ pϩ(t), and Y Phase 1, in which temperature is always below Tc and the infection process is always quiescent (t to t in Fig. 1). ͑ ͒ Ϫ Ϫ 0 1 ds ϭ nt t mXs ms ⌬ Y Phase 2, in which temperature varies across the critical temper- ͩ ͪ ͩ ͪ T, di Q mXs ature and the infection process switches between quiescent and active (t1 to t2 in Fig. 1). with probability, pϩ(t). This gives a mean of
9068 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.1436273100 Truscott and Gilligan Downloaded by guest on September 30, 2021 ͑ ͒ Ϫ ͑ ͒ Ϫ for the deterministic system, where nt t pϩ t mXs ms ⌬ ͩ ͑ ͒ ͪ T, [5] pϩ t Q mXs mX r t ϭ , ϭ , ϭ , and a variance of m m m
2 2 2 2 ͑ pϩ Ϫ pϩ͒ X s ⌬TЈ ͩ m ͪ⌬ 2 ͱ 2 ˜ ͑ Ϫ 2 ͒ 22 2 2 T . [6] ͑͒ ϭ ⌬TЈ͑pЈϩ͑͒ Ϫ pЈϩ͑͒͒, ⌬T ϭ , i ϭ Qi, pϩ pϩ Q mX s m
We want these variances to be matched by the SDE, and using the identity, dW(␣) ϭ ͌␣dW(). The parameter, Јϭ Јϭ ds A B 0 in phase 1 and in Phase 3. ͩ ͪ ϭ ͩ sͪdt ϩ ͩ sͪdW͑t͒, Because the stochastic effects of Phase 2 enter through the force di Ai Bi of the infection process and the parameter, m, the small-noise limit where dW(t) represents the infinitesimal Wiener deviate at time, t. is analyzed by taking the limit in which the parameter, 3 0. We Only one infinitesimal Wiener process is required, because there is look for solutions of Eqs. 10–12 as a series in , i.e., only a single stochastic process operating. Considering again the ͑͒ ϭ ͑͒ ϩ ͑͒ ϩ2 ͑͒ period, ⌬T, and assuming a slow evolution for the deterministic s sa sb sc .... processes, we arrive at a mean of Substituting these series into the SDE gives for the first two terms A ͩ sͪ⌬T, A Ϫ Ϫ Ј i ͑͒ ϭ ϩ ͵ ͑Ј͒ Ј [16] sa s1e e e n d , and 0 Var͑ds͒ ϭ B2⌬T, Var͑di͒ ϭ B2⌬T, s i Ϫ Ј Ј ͑͒ ϭ Ϫ͵ ϩ͑Ј͒ ͑Ј͒ Ј Ϫ ͵ ͑Ј͒ ͑Ј͒ ͑Ј͒ sb e ͭ e p sa d e sa dW ͮ , ͗ 2͘ϭ using the identity, dWi dt (15). By comparison with Eqs. 5 and 0 0 6, we have [17] ϭ ͑ ͒ Ϫ Ϫ ϭ As nt t pϩ mXs ms, Ai pϩQ mXs, where n1, s1 represent values at the start of Phase 2, and is ϭ Ϯ ͱ͑ Ϫ 2 ͒⌬ ϭ Ϯ ͱ͑ Ϫ 2 ͒⌬ Bs mXs pϩ pϩ T, Bi Q mXs pϩ pϩ T. measured from the start of Phase 2. We are interested only in the first term of ˜i, which is ˜i Therefore, the SDE describing the dynamics of Phase 2 is b ϭ ͑ ͒ dn nt t dt, [7] ˜ ͑͒ ϭ ͵ ϩ͑Ј͒ ͑Ј͒ Ј ϩ ͵ ͑Ј͒ ͑Ј͒ ͑Ј͒ [18] ib p sa d sa dW . ϭ ͑ ͑ ͒ Ϫ Ϫ ͒ Ϫ ⌽͑ ͒ ͑ ͒ ds nt t pϩ mXs ms dt mXs t dW t , [8] 0 0 ϭ ϩ ⌽͑ ͒ ͑ ͒ di pϩQ mXsdt Q mXs t dW t , [9] It is easy to calculate the mean and variance of these quantities by using the identity, where ⌽(t) ϭ ͌(pϩ(t) Ϫ pϩ(t)2)⌬T. We have taken advantage of t t t the degree of freedom in the signs of Bs and Bi to allow for material ͳ͵ G͑tЈ͒dW͑tЈ͒͵ H͑tЈ͒dW͑tЈ͒ʹ ϭ ͳ͵ G͑tЈ͒H͑tЈ͒dtЈʹ removed from s to be added to i. ,
t0 t0 t0 3. Analysis By using Eqs. 1–3 for Phases 1 and 3 and Eqs. 7–9 for Phase 2, it (15), giving
is possible to describe the evolution of a single realization of the system through all three phases in the form of a stochastic integral Ϫ Ј Var͑s͑͒͒ ϭ2e 2 ͵ e2 2͑Ј͒s2͑Ј͒dЈ, [19] (15). This solution is, in general, too complex to yield any practically a useful information about the statistics of the system, such as mean 0
and variance. To simplify the solution, we consider an asymptotic limit in which the stochasticity is small, thereby allowing a solution Var͑˜i͑͒͒ ϭ2͵ 2͑Ј͒s2͑Ј͒dЈ, [20] through asymptotic techniques. Nondimensionalizing with respect a to time and grouping parameters gives 0
dn ϭ nd, [10] ͑ ͑͒ ˜͑͒͒ ϭ2 Ϫ͵ Ј2͑Ј͒ 2͑Ј͒ Ј [21] Cov s , i e e sa d . ϭ ͑ Ϫ Ϫ ͒ Ϫ ͑͒ ͑͒ ds n pϩ s s d sdW , [11] 0 ˜ ϭ ϩ ͑͒ ͑͒ di pϩ sd s dW , [12] The O(2) for these terms reflects the fact that the infection process for the SDE and and hence its products are O( ). As yet, we have made no assumptions about the form of pϩ() and hence (Ј). The simplest dn assumption is that pϩ() is constant through Phase 2, in which case ϭ fЈ͑n, ͒, [13] d can be removed from the integrals in Eqs. 19–21. The simplest approximation to the time-dependent behavior illustrated in Fig. 1 ds is a linear rate of change for pϩ, which captures the smooth ϭ fЈ͑n, ͒ Ϫ s Ϫ Јs, [14] d transition in pϩ from 0 at the start of Phase 2 to Phase 1 at the end.
͑͒ ϭ ͞⌬ BIOLOGY di˜ pϩ 2 [22] POPULATION ϭ Јs, [15] d and
Truscott and Gilligan PNAS ͉ July 22, 2003 ͉ vol. 100 ͉ no. 15 ͉ 9069 Downloaded by guest on September 30, 2021 ͗i͑t͒͘ ϭ Xs pϩt, [27] ͑͒ ϭ ͱ ͩ Ϫ ͪ⌬ Ј m a ⌬ 1 ⌬ T , [23] 2 2 ͑ ͑ ͒͒ ϭ 2 2 2 ͑ Ϫ ͒⌬ Var i t mX sa pϩ 1 pϩ Tt, [28] ⌬ where 2 is the duration of Phase 2. We can make a further simplification to the above expressions for in dimensional form. The parameter grouping mXsa isarate variance when Phase 2 is short in comparison to the rate of change appearing in both expressions, converting time into the units of i. Hence we can make a direct comparison between the time spent in of sa. We can approximate an infectious state in Phase 2 and the time spent raining in the dsa Markov-chain model, and we can associate the parameters of the s ͑͒ Ϸ s ͑0͒ ϩ ͑0͒, [24] a a d two models: N where is measured from the start of Phase 2. The derivative of sa SDE model Markov-chain model at the start of the phase can be expressed in terms of n and sa N through Eq. 14. With this approximation, the integrals in Eqs. 19–21 pϩ , [29] become analytically tractable. For example, the expression for the 1 ϩ p Ϫ p ⌬ N 1 0 ϭ ⌬ variance of ı˜ (Eq. 20) becomes T Ϫ ϩ t0 tc. [30] 1 p1 p0 2 dsa ͑˜͑͒͒ ϭ2͵ ͩ Ϫ ͪ⌬ Јͩ ͑ ͒ ϩ ͑ ͒ͪ Ј Both pϩ and are absolute probabilities, whereas the quantities in Var i ⌬ 1 ⌬ T sa 0 0 d , 2 2 d 30 0 Eq. are both functions of correlation parameters. We can use this basis, therefore, to construct SDE models for which is simply the integral of a polynomial in . systems governed by discrete Markov-chain stochastic processes The evolution of the mean and variance of s and i in the over relatively long periods of time. Simulations show that the deterministic third phase is addressed in the Supporting Appendix, relative error in the variance as predicted by the SDE model which is published as supporting information on the PNAS web site, decreases with increasing time. This decrease would of course be www.pnas.org. Essentially, the development of these quantities in accompanied by a corresponding increase in error from the as- Phase 3 can be expressed as functions of the mean, variance, and ymptotic approximation. covariance of s and i at the end of Phase 2. 5. Numerical Results 4. Comparison of Results with Discrete First-Order The theory developed in Section 3 was tested against numerical Markov-Chain Models simulations by using a fourth-order Runge–Kutta algorithm to Discrete first-order Markov-chain models are widely used in model integrate Eqs. 1–3 over the three phases of the system’s evolution. weather-generating systems as a simple and accurate method of During the stochastic phase, the temperature of the system for any generating daily rainfall patterns (13). The seminal example is a correlation period was chosen randomly according to the proba- model of rainfall in Tel Aviv (14). We will use this model to bility, pϩ. For all the following simulations, the temperature phases illustrate the connection between first-order Markov-chain models were defined as and that derived in Section 3. Using the relationships between SDE and Markov-chain models derived in this section, we can correctly Phase 1 : 0 Յ Ͻ 1, parameterize an SDE to simulate the stochastic influence of a Phase 2 : 1 Յ Ͻ 1.5, discrete Markov-chain environmental variable. Phase 3 : 1.5 Յ Յ 3. The rainfall model assumes the weather to be in one of two states on any day: raining or dry. Hence the day is the fundamental unit The parameters used were those presented in Table 1 unless stated of time, t0, over which the state of the system remains unchanged. otherwise. A monomolecular growth function was used for simu- The state on any day depends only on that of the previous day lations for algebraic tractability. through the conditional probabilities, Table 2 below compares values for variance of inoculum pro- duction at the end of Phase 3 between the numerical solution and ϭ ͑ ͉ ͒ p1 p wet today wet the previous day , the theoretical predictions from Section 3 for different correlation periods, ⌬T. Predictions for the mean were found to be very p ϭ p͑wet today ͉ dry the previous day͒. 0 accurate for all values of the period. For variance, the accuracy It can be shown that over a period of n days the mean and variance increases as the correlation period gets smaller. These results also of the cumulative length of time of rainy days, r, are given by illustrate the problem discussed in Section 2.3: as the correlation period is decreased toward zero, the variance in the system also ϭ r n t0, [25] vanishes. 1 ϩ p Ϫ p Fig. 2 shows realizations of the stochastic process produced by the ͑ ͒ ϭ ͑ Ϫ ͒ 1 0 2 simulator. The thicker lines represent the mean and two standard Var r 1 n Ϫ ϩ t0 , [26] 1 p1 p0 deviations on either side as calculated from theory (Eq. 20). The as n gets large, where is the absolute probability of rain on region between these lines represents approximately a 98% confi- any day, dence interval for the realizations, assuming a normal distribution. This assumption can be seen to be approximate from the fact that p ϭ 0 the lower curve goes negative at the start of the second phase. ϩ Ϫ . 1 p0 p1 These expressions can be directly compared to those for the mean Table 1. Dimensional parameter values and variance of i in Section 3. The quantity, i, represents the Name Symbol Value accumulation of inoculum generated by the stochastically switching infection process and is entirely analogous to the accumulation of Growth rate, dϪ1 r 0.1 Ϫ rainy days in the Markov-chain weather model. If we consider the Force of infection, d 1 ⌳X 0.007 ͞ dynamics of the underlying model to be static through Phase 2, we Susceptible interval, d 1 m14 Correlation period, d ⌬T 1 can express the mean and variance of i as
9070 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.1436273100 Truscott and Gilligan Downloaded by guest on September 30, 2021 Table 2. Table of comparisons between simulation and theory for a range of values for ⌬T Inoculum variance Correlation period, ⌬T, days Simulation Theory %
1.0 2.20 ϫ 10Ϫ5 2.30 ϫ 10Ϫ5 4.3 0.5 1.12 ϫ 10Ϫ5 1.15 ϫ 10Ϫ5 2.7 0.2 4.51 ϫ 10Ϫ6 4.59 ϫ 10Ϫ6 1.7
Nevertheless, they function well as bounds on the behavior of the realizations. Fig. 3 shows more clearly the variance of generated inoculum over the three phases of evolution. In particular, it shows the continuing change in variance in the third deterministic phase. Simulation and theory are in close agreement through the entire range of pϩ. The shape of the curve reflects the dependence of the variance of inoculum on 2 (Eq. 20), where ϭ ͌⌬TЈpϩЈ (1 ϪpϩЈ ), giving a characteristic parabolic shape. A linear rate of change in Fig. 3. Variance in infected root over all three phases as predicted from theory. probability (Eq. 22) is a more realistic form for pϩ to take, as discussed in Section 3. Fig. 4 was generated by using this linear form there is a quite high probability (P Ͼ 0.4) of breaching the threshold for pϩ and the linear approximation for s (Eq. 24). It shows the a for certain values of r. error in the variance of the infected roots against the length of Phase 2. Clearly, the variance error increases with the length of Phase 2, 6. Conclusion showing that the improvement in the accuracy of the SDE described The development of a model for this system falls into two distinct in Section 3 is outweighed by the accumulating errors from the parts: first, the development of a conceptual model for the real asymptotic approximation and the assumption of linearity in sa. system, and second, the development of a mathematical description The inclusion of environmental variability allows us to make of the conceptual model. useful statistical statements about how disease will develop in the Fokker–Planck equations and stochastic differential equations host. Such predictions can be tested against experiments in the field. are two different ways of representing the same underlying con- Ͼ As an example, we consider the probability, P(i ic), that the tinuous-time stochastic process (15, 16). Stochastic differential inoculum generated in a season exceeds some critical level. ic, such equations describe the evolution of the random variables repre- as that resulting in symptoms in the host or detection in the senting the state of the system, whereas Fokker–Planck equations environment. In Fig. 5, these are calculated as a function of growth describe the evolution of the probability distribution for the state of rate and force of infection, respectively, from the analysis in Section the system. From an analytic standpoint, they have a well-behaved 3. The probability of exceeding the threshold as a function of force deterministic limit as the effect of stochasticity is reduced. For of infection shows an error function shape, reaching P ϭ 0.5 as the systems such as the present one, in which the deterministic solutions mean of i ϭ ic. For a deterministic representation, the probability are available, there is the possibility of generating asymptotic series willswitchfrom0to1atthispoint.Fig.5B shows P(i Ͼ ic)asa around the small noise limit. In the analogous limit for Fokker– function of the growth rate, r. For low values of r, the slow Planck equations, no such convenient form is obtained. generation of new roots limits the quantity of inoculum generated. Results from Section 3 show that it is possible to construct an For high values, the majority of roots have already entered the SDE model for the stochastic-switching model developed in Section 2.3. Moreover, a closed-form solution to this system of equations resistant class before Tc is reached and the disease process can can be found in the asymptotic limit and hence an analytical develop strongly. Hence P(i Ͼ ic) has a local maximum as a function of r. For these particular parameter values, the deterministic value approximation for the stochastic process. This approach has broad for i never exceeds ic. The stochastic model, however, indicates that
Fig. 2. A set of realizations of the evolution of inoculum generated by the Fig. 4. Dependence of variance in infected root at end of Phase 3 on the
simulator. Superimposed on these in broader lines are the mean (central line) and duration of the stochastic Phase 2. The duration of Phase 3 was held constant at BIOLOGY POPULATION the mean Ϯ 2 standard deviations (outer lines) as generated from theoretical 1.5 units. The probability, pϩ, is a linear function of time (Eq. 22). Crosses and solid results. line indicate results from simulation and theory, respectively.
Truscott and Gilligan PNAS ͉ July 22, 2003 ͉ vol. 100 ͉ no. 15 ͉ 9071 Downloaded by guest on September 30, 2021 Fig. 5. Probability of inoculum generated in a sea- son exceeding critical value, ic, as a function of force of infection, (A), and growth rate, r (B). Probability, P(T Ͼ Tc), was kept constant through Phase 2 (pϩ ϭ 0.5). Parameter values: r ϭ 0.1, ϭ 0.02, ic ϭ 0.22. Other parameters as in Table 1.
applicability to a range of epidemics and growth processes subject in plants vulnerable to frost damage and infection. In this case, the to environmentally controlled stochastic switching. In the present cutoff temperature is 0°C. Periods of rainfall or high humidity can example, analytic tractability rests on the simplicity of the under- also be triggers for infection and, as has been illustrated in Section lying epidemic model. For more complex systems, numerical tech- 4, rainfall patterns are accurately simulated by Markov-chain niques would be required. models and hence SDEs. Other candidates include insolation and Numerical results in Section 5 show that the solutions for the the movement of animal vectors. statistics of the inoculum, i, calculated in Section 3 and the By explicitly including stochastic effects into a deterministic Supporting Appendix describe those of the simulation very well. model, we address properties of the system that are not recoverable Predictions for the mean value of the inoculum are very accurate, from a purely deterministic description. The probability of exceed- whereas that for the variance are typically Ͻ5% out for the ing a threshold shown in Fig. 5 A and B is an example of such a parameter values used. This accuracy is maintained across the property. Most biological experiments yield results of a statistical whole range of pϩ. There is an implicit assumption in the construc- result of this nature based on many replicates, which can be tested tion of the SDE that the Wiener process can be used to describe against the probability distributions predicted by the model. Hence what is the essentially binomial process of switching. One problem we can examine directly the nature and strength of the interaction can be seen in Eq. 12, where dW(t) can be negative and large enough between environmental phenomena and the biological systems they to allow di to be negative. Clearly, in the real system, di Ն 0. Hence, control. although the statistics predicted by the theory are reliable, individ- The work presented here illustrates some of the difficulties ual realizations may be unrepresentative. As the stochastic period inherent in representing the influence of stochastic processes on 3 lengthens, the accuracy of the estimate of variance declines (Fig. 4), time-continuous systems. Simply taking the limit as dt 0togeta because of the accumulation of errors from the approximation. continuous representation can fail to capture the variability of the In Section 4, we explicitly compare the behavior of the SDE system. It can ‘‘disappear’’ in the limit. This is the result of the fact model to that of a discrete first-order Markov-chain model. The that the variance of the Wiener process is proportional to time. ⌬ concept of Markov chains underlies many stochastic processes, and Table 2 illustrates this point well. As the correlation period, T,is discrete first-order chains are widely used to model weather phe- made smaller, and variance shrinks proportionally. From a practical nomena. Strong parallels can be drawn between such models and point of view, the problem lies with the infinitesimal properties of the SDE systems in terms of their statistics, in particular, the the Wiener distribution combined with the step-function nature of correlation period and the conditional probabilities of the Markov the switch. It could be said that the Weiner process is capable chain. That is, of finite changes in infinitesimally small periods of time. Because of inertia, no natural process behaves in this way. Hence for a 1 ϩ p Ϫ p ⌬ ϭ 1 0 switching process, it is necessary to include inertia artificially. In the T Ϫ ϩ t0, 1 p1 p0 present case, we included the effective correlation time to represent thermal inertia. An alternative approach is to replace the switching where t0 is the length of the discrete time unit. Using this relation- function with a smooth alternative. This would better represent the ship and the approach outlined in this paper, we can include response of the system to an environmental influence. What form the stochastic influence of such Markov-chain environmental these functions should take and what effect their form will have on phenomena on evolving deterministic systems subject to stochastic the transmission of variance into the system are the subjects of switching. further work. For the specific model treated in this paper, we have considered stochastic forcing through fluctuating temperature in detail. This We gratefully acknowledge funding from the Biotechnology and Bio- forcing is mediated through an extreme sensitivity to temperature logical Sciences Research Council (J.E.T.) and the Royal Society and in the parasite. A similar sensitivity to temperature would be found Leverhulme Trust (C.A.G.).
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