SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES. Key words: SIRS epidemic models, sample path large deviations.

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1 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES R. G. DOLGOARSHINNYKH Abstract. We study infection spread in continuous time SIRS epidemic models. When infection is supercritical, the proportions of susceptible and infected individuals in the population tend to stabilize near an endemic level with positive proportion of infected individuals for a long time. The event when all of the infected individuals recover without introducing new infections into the population is a large deviations event for the scaled SIRS processes. We prove sample path large deviations principle for this problem. Key words: SIRS epidemic models, sample path large deviations. 1. Introduction. First introduced by Kermack and McKendrick 1927, SIR epidemic models have been extensively studied. In these models, all individuals in the population are in one of three possible states - susceptible, infected or recovered removed. When an individual recovers he is removed and plays no further role in the spread of infection. In the SIR models, the population is assumed closed and therefore the infection eventually dies out. Nagaev and Startsev 197 proved limiting results for the final outcome of a particular SIR epidemic process, the Reed-Frost process, when the number of initially infected individuals is large. A general framework that includes many discrete and continuous time SIR epidemic models was developed by Bahr and Martin-Löf 198. For these generalized SIR processes they obtain the limit theorems of Nagaev and Startsev 197 as well as for the case when the number of initially infected remains bounded as the population size grows. Sellke 1983 proved a distributional result about the final outcome of an epidemic using an elegant construction based on a coupling method. The SIRS epidemic models are extensions of SIR models that describe the infection spread when recovery from infection does not lead to permanent immunity introduced by Kermack and McKendrick 1932 and The dynamics of the SIRS epidemic processes differ from the SIR processes in that recovered individuals may loose immunity and re-enter the susceptible state. The SIRS models may also serve in the situation when upon recovery the individual is removed from the population, and new susceptibles enter the population in such a way that the population size remains approximately 1

2 2 R. G. DOLGOARSHINNYKH constant. For the SIRS processes, it is again the case that the infection will eventually become extinct. The time when there are no infected individuals left in the population for the first time is of central interest in epidemic modeling. In both SIR and SIRS epidemic models when the average number of secondary cases that are produced by one infected individual in a fully susceptible population sometimes called the reproduction number is less then one the infection dies out fast, and the epidemic is subcritical. When the reproduction number is greater than one the epidemics is supercritical, and in this case in the SIR processes the infection dies out in about ON time, where N is the population size see e.g. Bahr and Martin-Löf 198. However, in the supercritical SIRS processes, the number of infected individuals is likely to stay at an appreciable size for exponentially long time, about expcn, for some positive constant c. Under the SIRS epidemic models, the probability that the infection dies out in a fixed period of time is exponentially small in N. The strategy to study time until a rare event occurs was developed by Wentzell and Freudlin see e.g. Freidlin and Wentzell The first step is to describe the probability that the infection dies out in a fixed period of time. In this paper we prove the large deviations principle for the sample paths of SIRS epidemic processes. In the rest of this paper we derive the sample path large deviations principle LDP for the SIRS processes and describe the functional I. The problem is similar to the problem studied by Freidlin and Wentzell 1998, and in particular to the problem discussed in Ventsel 1976b. However, the hypothesis used by Ventsel assumption C do not hold for the SIRS model see Appendix A for details. Transition rates converge to zero as the number of infected in the population diminishes. To avoid problems near the boundary somewhat different techniques are needed here to establish the LDP. For the proof of the lower bound in Section 5 we mostly follow the strategy of Freidlin and Wentzell 1998 with minor technical modifications. Upper bound is proved in Section 6. In Section 3 we state the main theorem and outline the proof. Properties of the rate function are explored in Section 4, and we give an explicit characterization of the rate function, which could be used to obtain numerical estimates. We begin by describing infection spread in the SIRS epidemic models in Section Infection spread in the SIRS epidemic models. The SIRS models we consider are continuous time Markov processes. The population is closed and an individual passes through consecutive states of susceptibility, infection, recovery temporary immunity and back to susceptibility. For every population size N, let S t, I t, and R t be the number of susceptible, infectious, and recovered individuals respectively at time t so that R t =

3 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 3 Figure 1. Trajectories of 1 for θ = 3, ρ = 1 started at various initial points such that i >. N S t I t for all t. Let γt N = s N t, i N t = S t /N, I t /N be re-scaled processes of population proportions and let rt N = R t /N = 1 s N t i N t. If s = S/N, i = I/N, and r = R/N then the transition probabilities of the N th SIRS Markov process are given by P N{ S t+h, I t+h = S 1, I + 1 S t, I t = S, I } = Nθsih + oh, P N{ S t+h, I t+h = S, I 1 St, I t = S, I } = Nρih + oh, P N{ S t+h, I t+h = S + 1, I S t, I t = S, I } = Nrh + oh, for positive constants θ and ρ. The law of large numbers for density dependent Markov processes see for example Ethier and Kurtz 1986, 11.2 gives that as N the sample paths, γ N = s N t, i N t t [,T ], converge almost surely in supnorm topology on K := {s, i : s, i, s + i 1} to trajectories of the following dynamical system: 1 ds t dt = r t θi t s t di t dt = θi ts t ρi t, provided convergence of the initial conditions. The dynamical system 1 has two fixed points, s, i = ρ/θ, 1 ρ/θ/1 + ρ and s, i = 1,. When ρ θ, the only fixed point in the triangle K is s, i = 1,. In this case, the epidemic is subcritical and the infection dies out fast. When ρ < θ, the epidemic is supercritical and the first of the two fixed points lies in the interior of the triangle K. We show in Appendix B that in the supercritical case all trajectories of 1 started at i > converge to the interior fixed point, s, i = ρ/θ, 1 ρ/θ/1 + ρ, as t. That is, starting with a positive proportion of infecteds, as population size grows the scaled SIRS processes follow trajectories of 1 to a neighborhood of s, i, the endemic level of infection. Figure 1 gives an example of the mean field. Upon reaching a neighborhood of the endemic level the population proportions fluctuate around it. By showing convergence of respective infinitesimal operators one can prove that the processes Ns t s, Ni t i converge weakly to a Gauss-Markov process. Therefore, the typical fluctuations around the endemic level are on the order of 1/ N. However, over any fixed time period there is a small but positive chance that all of the currently infected individuals recover before introducing new infections into the population so that eventually the infection becomes extinct. We are

4 4 R. G. DOLGOARSHINNYKH interested in studying τ, the time when there are no infected individuals left in the population for the first time. To study time to extinction, τ, the first step is to show that there is a functional Iγ such that for any fixed path γ and δ >, N large P sup N γ t γ t < δ t T e NI γ. The above together with elementary properties of the functional I suggest that the probability of exit over the next fixed time interval is exponentially small in N. The situation is similar to that arising in Laplace s method to calculate the asymptotics as N of integrals of the form b a e Nfx gxdx. If x is the only minimum of the function fx on the interval [a, b] and gx is continuous and positive then the major contribution to the integral comes from the neighborhood of x. If the minimum of Iγ over the paths is attained at some path γ, then by analogy to Laplace s method we may expect that for N large the major contribution to the probability that the infection dies out during the next fixed time period comes from a neighborhood of γ. Similarly to flipping a coin until the first success, it is plausible that the time to extinction, τ, is approximately proportional to the inverse of the probability of extinction in a fixed time interval. In particular, we expect that 1 lim N N log EN τ = V, { where V = inf T, s Iγ : γ = s, i, γ T = s, }. This approach has been successfully carried out by Wentzell and Freidlin in studying time of exit from a domain for a large class of processes. 3. Statement of the theorem and an outline of the proof. Let X be a metric space and let B X be the Borel σ-field on X. A function I : X [, ] is a rate function if the level sets Γs := {x : Ix s} are closed, and I is a good rate function if the level sets are compact. Definition. A family of probability measures {µ n, n N}, satisfies the large deviations principle LDP with a good rate function I if for all F B X 1 inf Ix lim inf x F n n log 1 µn F lim sup n n log µn F inf Ix, x F where F denotes the interior of F and F the closure of F. We call the leftmost inequality in theorem 1 the lower bound and the rightmost inequality the upper bound. An introduction to the theory of large deviations can be found in Dembo and Zeitouni 1992 and in Freidlin and Wentzell Let D = D[, T ], K be the set of functions f : [, T ] K, which are right continuous and have left limits and let AC[, T ] D, be the subspace

5 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 5 of absolutely continuous functions. For any f, g D let f g T = sup f t g t, t T where denotes Euclidean distance in R 2. Let B be the Borel σ-field on D and let P N be the probability measure on D, B induced by the N th SIRS Markov Chain. To define the rate function for the re-scaled SIRS processes, for γ = s t, i t t T AC[, T ] let 2 and 3 Sγ λ, µ, ν = where for x, m fλ t, θs t i t + fµ t, ρi t + fν t, r t dt, Iγ = inf Sγ λ, µ, ν, λ, µ, ν 4 fx, m = x logx/m x + m, and fx, m = + for all other x, m. The infimum is taken over all λ, µ, ν measurable with respect to Lesbegue measure on [, T ] such that λ t, µ t, ν t and satisfy almost surely 5 and Iγ = + for γ / AC[, T ]. ds t dt = ν t λ t di t dt = λ t µ t ; Theorem 1. The probability measures P N, N 1, satisfy the large deviations principle with a good rate function Iγ. We say that nonnegative λ, µ, ν satisfying 5 are allowed for γ. We often omit the index N to simplify the notation. In the definition of fx, m we assume the convention loga/ = + for any positive a and log/ = log =. Before we proceed, we construct the SIRS processes from standard Poisson processes, which will be useful in many proofs that follow. We do that by applying random time changes to a collection of Poisson processes see for example Ethier and Kurtz 1986, Ch. version of γ satisfying t s t = s y 1 6 t θs u i u du + y 3 t t i t = i + y 1 θs u i u du y 2 VI, 4. In particular, there is a r u du ρi u du,

6 6 R. G. DOLGOARSHINNYKH where y i = N 1 Y i N for i = 1, 2, 3, and Y 1, Y 2, and Y 3 are independent rate one Poisson processes. Note that this construction can not be used to apply the contraction principle see for example Section of Dembo and Zeitouni 1992 because this transformation is not continuous Lower bound: sketch of the proof. The proof is based on the method of tilting. Let λ, µ, ν be some nonnegative real-valued functions on [, T ] and let Q N be the measure on D induced by a continuous time Markov chain with state space K N := {s, i : Ns, Ni Z 2 +, s + i 1} and the following instantaneous transition rates: 7 q{s t, i t s t 1/N, i t + 1/N} = Nλ t, q{s t, i t s t, i t 1/N} = Nµ t, q{s t, i t s t + 1/N, i t } = Nν t. For γ D let U δ γ = {γ : γ γ T < δ}. It is enough to show the local exponential lower bound see for example 1.2 of Dembo and Zeitouni That is, we need show that for any fixed path γ and any η >, small δ > and large N 8 P N U δ γ dp = E N Q dq 1I { U δ γ } e NI γ+η, where E N Q denotes the expectation under the measure Q = QN. We choose functions λ, µ, ν such that as N goes to infinity the random paths are likely to stay near the fixed path γ under Q. Then 8 suggests that the behavior of the likelihood ratio dp/dq in a neighborhood of γ under Q determines the probability that we are interested in. To control the likelihood ratio it is convenient to restrict our attention to certain nice subsets of U δ γ with high Q probability. Suppose we can show that, under Q, dp/dq exp { N I γ + η/2 } on U δ γ B for some collection of events B = {B N }. Moreover, technical estimates of Appendix C allow us to consider only certain nice γ. Then dp E N Q dq 1I { U δ γ } dp E N Q dq 1I { U δ γ } B Therefore if Q N U δ γ B > c > e NI γ+η/2 Q N U δ γ B. 1 N log PN U δ γ I γ η/2 + 1 N log QN U δ γ B I γ η for all large N and the lower bound would follow. We construct suitable events B in section 5.

7 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES Upper bound: sketch of the proof. For A D let ρ T γ, A = inf γ A γ γ T. To prove the upper bound it is enough to show for any δ >, η > and s > P N ρt γ, Γs δ e Ns η for all large N see e.g. Freidlin and Wentzell 1998, Chap.3 3. Let {B ε } ε> be a collection of events and suppose there exists an ε s > such that 9 lim sup N for all < ε < ε s. Then 1 N log PN B c ε 2s 1 N log PN ρ T γ, Γs δ log2 { } 1 N + max N log PN{ } 1 ρ T γ, Γs δ Bε, N log PN Bε c and therefore it is enough to show the upper bound for P N{ ρ T γ, Γs δ } B ε. We choose convenient events Bε. For γ D let γ a = T a γ, where 1 T a γ = s a t, i a t = 1 as t, i t + a1/3, 1/3, so that rt a := 1 s a t i a t = 1 ar t + a/3. Then γ a γ T < a, and for a < δ P N ρ T γ, Γs δ P N ρ T γ a, Γs δ a. Following the strategy of Freidlin and Wentzell 1998, Chap.3 2 we construct polygonal approximations to random paths γ a, l = l a. Then P N ρ T γ a, Γ a s δ a P N ρ T l, Γs δ/2 a + P N γ a l T δ/2 for any allowed choice of λ, µ, ν for l and a small. Let < α < 1 then by Chebyshev s inequality P N Il > s + P N γ a l T δ/2 P N Sl λ, µ, ν > s + P N γ a l T δ/2 P N{ Sl λ, µ, ν s } B ε E N exp { αnsl λ, µ, ν } 1I{B ε } e αns. Next, we show that for all ε < ε E N exp { αnsl a λ, µ, ν } 1I{B ε } C α,ε exp { N 1 α + o ε 1 }

8 8 R. G. DOLGOARSHINNYKH for some constant C α,ε > and a convenient choice of λ, µ, ν. Taking α close to 1 and ε small the upper bound follows once we show that P N B ε and P N γ a l T δ/2 are suitably small. 4. The rate function. In this section we show that the function I defined by 3 in Section 3 is a good rate function. Recall that Γs = { γ D : Iγ s }. Lemma 1. For each s <, the elements of Γs are equicontinuous. Proof. Let γ Γs. Then there exist allowed λ, µ and ν such that Sγ λ, µ, ν 2s. Equicontinuity follows from Lemma 1 in Appendix C, since for t 1 < t 2 T such that t 2 t 1 < 1/θ t2 i t2 i t1 = i t2 tdt t2 λ t dt + µ t dt < 2 log 2s + 1 θt 2 t 1 and s t2 s t1 = t 1 t2 t 1 t 1 t 1 s t2 tdt t2 λ t dt + ν t dt < 2 t 1 t 1 log 2s + 1. θt 2 t 1 Lemma 2. Iγ defined by 3 is lower semicontinuous. Proof. For n N let γ n AC[, T ] be such that lim n γ n γ. It is enough to show that if for all n N Iγ n s for some s < then Iγ s. By Lemma 1 above functions γ n are equicontinuous, and therefore so is γ. Let δ >. It follows from Lemma 15 in Appendix C that for all a < a 1 Iγ IT a γ + δ/3. Lemma 16 in Appendix C implies that for all a < a 2 and all n N IT a γ n Iγ n + δ/3. Let a < min{a 1, a 2 }. The above and Lemma 15 imply that for all n > n a Iγ IT a γ + δ/3 IT a γ n + 2δ/3 Iγ n + δ. Since the above is true for all δ > the conclusion follows. Proposition 1. The functional I defined by 3 is a good rate function. Proof. Fix s <. From Lemma 1 above we have that the elements γ of Γs are equicontinuous and uniformly bounded since they take values in the set K. Also, Lemma 2 implies that Iγ is lower semicontinuous and therefore the level set Γs is closed. Compactness of Γs follows from the Arzelà-Ascoli theorem.

9 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 9 A more explicit form of the rate function I. Recall the definition of the action functional I; for γ AC[, T ] Iγ = inf Sγ λ, µ, ν = inf λ, µ, ν λ, µ, ν fλ t, θs t i t + fµ t, ρi t + fν t, r t dt, where fx, m = x logx/m x+m and the infimum is taken over all λ, µ, ν such that λ t, µ t, ν t and satisfy 11 ds t dt = ν t λ t di t dt = λ t µ t. As before we will call λ, µ, ν satisfying these conditions allowed for γ. In the following proposition we show that there exist unique λ, µ, ν at which the infimum is attained. We omit the proof which is based on straightforward calculus. Proposition 2. For every t [, T ] there exists a unique root of 12 as a function of λ t such that 13 λ t λ t + s tλ t i = θs t i t ρi t r t λ t, λ t + s t and λ t i. Let λ t be such root and let µ t = λ t i t and ν t = λ t + s t. Then Iγ = Sγ λ, µ, ν. 5. Lower bound. Fix N N. For a random path γ = γ N let M = the random number of transitions in the time interval [, T ], t k = the time of k th transition for k = 1, 2,..., M, and let index k refer to values at time t k, so that for example λ k = λ tk. For x, y {s, i, r} let I k x y be the indicator random variables of the event that the k th transition is from x to y. Let Q = Q N be the measures induced on D by continuous time Markov chains with transition rates given by 7. It is not hard to show that the likelihood ratio dp N /dq N is given by dp N M dq N γ = k=1 exp θsk i Ik s i Ik i r k ρik rk λ k { N µ k } ϕ t ψ t dt, ν k Ik r s where ψ t = λ t + µ t + ν t and ϕ t = θs t i t + ρi t + r t are the total rates at which transitions occur at time t under Q N and P N respectively.

10 1 R. G. DOLGOARSHINNYKH To prove the local lower bound 8 it is enough to consider γ AC[, T ] because when I γ = the bound is trivial. We apply the technical results of Appendix C to show that it is enough to consider only certain nice γ and allowed choices of λ, µ, ν. For a > let 14 R a = {γ AC[, T ] : s t, i t, r t a, t [, T ]}. Let ε > be such that T/ε N. Let γ ε be a polygonal approximation to γ connecting the points 15, γ ε, γ ε..., γ... T, γ T. Lemma 3. Let γ R a for some a >. For ε > let γ ε be the polygonal approximations to γ defined by 15. Suppose that for all η >, all suitably small δ > and some ε P N γ γ ε T < δ exp { N S γ ε λ, µ, ν + η } for all λ, µ, ν allowed for γ ε such that λ t, µ t, ν t L for some L >. Then lower bound holds for all γ AC[, T ]. Proof. Fix δ, η >. Let γ AC[, T ] be such that I γ <. Then by Lemma 11 there exists an a η such that for all a < a η there exists γ a R a such that γ γ a T < a and I γ a < I γ + η/4. Since I γ a < Lemma 12 implies that there exists an L > and γ a,l R a/2 such that γ a γ a,l T < a/2 and S γ a,l λ, µ, ν < I γ a + η/4 for some allowed choice of λ, µ, ν for γ a,l such that λ t, µ t, ν t L. Finally, it follows from Lemma 13 that for all small ε the polygonal approximation γ a,l,ε to γ a,l is such that γ a,l γ a,l,ε T < a/4 and S γ a,l,ε λ, µ, ν < S γ a,l λ, µ, ν + η/4 for some λ, µ, ν allowed for γ a,l,ε such that λ t, µ t, ν t L. Then P N γ γ T < δ + 2a P N γ γ a T < δ + a P N γ γ a,l T < δ + a/2 P N γ γ a,l,ε T < δ exp { N S γ a,l,ε λ, µ, ν + η/4 } exp { N S γ a,l λ, µ, ν + η/2 } exp { N I γ a + 3η/4 } exp { N I γ + η }. In what follows we assume that γ R a for some a >, γ is linear over the intervals [, k + 1ε and consider only the allowed choices of λ, µ, ν that are constant over these time intervals. We also assume that all λ, µ, ν are bounded above by some constant L >.

11 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 11 We introduce convenient events B 1, B 2, B 3 for controlling the likelihood ratio. For ξ > let { M θsk i k R } θ sl ī l B 1 = I k s i log N λ l log ε Nξ k=1 λ l where l = t k /ε and index l refers to values at time lε. Similarly let { M ρik R } ρīl B 2 = I k i r log N µ l log ε Nξ and k=1 { M B 3 = I k r s log k=1 µ l rk ν l N l=1 l=1 µ l λ l R } rl ν l log ε Nξ. Recall that ϕ = θsi+ρi+r and ψ = λ+µ+ν. Then on U δ γ B 1 B 2 B 3 { dp R dq exp λl µl νl N λ l log ε + µ l log ε + ν l log ε θ s l ī l ρī l r l +N l=1 l=1 ψ t ϕ t dt N 2T θ + ρ + 1δ 3ξ } since ϕ t ϕ t < 2θ +ρ+1δ on U δ γ. Because the integrand is continuous the Riemann sums converge as ε and { dp T [ dq exp λt µt νt N λ t log + µ t log + ν t log θ s t ī t ρī t r t ] } ψ t ϕ t dt N Oδ + ξ = e NS γ λ, µ, ν N Oδ+ξ. It is left to show that Q N U δ γ B 1 B 2 B 3 > c > for some constant c and all large N. The next Lemma concludes the proof. Lemma 4. lim N Q N U δ γ = 1 and lim N Q N U δ γ B c j = for j = 1, 2, 3. Proof. The first limit is the law of large numbers for processes under Q N. We will show Q N U δ γ B1 c as N and the limits for j = 2, 3 follow similarly. On U δ γ, sup k s k s k < δ, and by choosing ε small enough we can ensure that sup k s k s tk /ε ε < δ, so that sup k s k s tk /ε ε < 2δ and sk log = log 1 + s k s tk /ε ε log1 + 2δ/a 2δ/a s tk /ε ε s tk /ε ε ν l

12 12 R. G. DOLGOARSHINNYKH uniformly in s. Similar estimates can be obtained for i. Then M θsk i k M θ sl ī l I k s i log I k s i log 4Mδ/a, k=1 λ l k=1 where l = t k /ε and index l refers to the value at time lε. Letting m l be the number of transitions in the time interval [l 1ε, lε, we get M θsk i k R θ sl ī l I k s i log N λ l log ε λ l λ l k=1 l=1 R θ sl ī l m l log I k s i Nλ l ε + 4Mδ/a. l=1 λ l λ l Recall that transition rates under Q are constant during the time period [l 1ε, lε and therefore m l k=1 I ks i, the number of transitions from susceptible to infected category in the time interval, is a Poisson random variable with mean Nλ l ε. Therefore, Chebyshev s inequality implies that m Q N θ sl ī l l log I k s i Nλ l ε > Nξ 2R Since s t, i t a and λ t L 4R 2 sup l R sup l R k=1 k=1 log 2 θ sl ī l λ l log 2 θsl i l λ l Nλ l ε N 2 ξ 2. λ l CL, a for some constant CL, a >. Combining the estimates, we get 16 Q N U δ γ B c 1 4T 2 CL, a Nξ 2 ε λ l + Q N 4Mδ/a Nξ/2. The total number of transitions prior to time T under Q is a sum of Poisson random variables with means Nψ l ε. Since the choice of ξ was arbitrary, we take ξ = 16δ/a R l=1 ψ lε we can ensure that δ/a is small. Therefore, as long as R l=1 ψ lε >, the law of large numbers for Poisson random variables implies Q N 4Mδ/a Nξ/2 = Q N M R N 2 ψ l ε as N and the conclusion follows from Upper bound. Lemma 5. For any δ, η, s > there exists N N such that P N ρ T γ, Γs δ exp { Ns η } l=1

13 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 13 whenever N N. Proof. Let γ a = s a t, i a t t T = T a γ defined by 1. Then γ a γ < a and for a < δ/3 P N ρ T γ, Γs δ P N ρ T γ a, Γs 2δ/3. Similarly to the strategy employed by Freidlin and Wentzell 1998, chap.3 2 for the proof of large deviations for Wiener processes, we approximate random paths γ by smoother paths. Let ε > be such that R := T/ε N. We construct a polygonal approximation to γ a, l = l a,ε = u t, v t t T by letting u = s a, v = i a, for k =, 1,..., R; and connecting these points with straight lines to get the rest of the path. Let w t = 1 v t u t. The event { γ a l T < δ/3 } { ρ T γ a, Γs 2δ/3 } is contained in { } ρ T l, Γs δ/3 and P N ρ T γ a, Γs 2δ/3 P N ρ T l, Γs δ/ P N γ a l T δ/3 P N Il > s + P N γ a l T δ/3. We first bound P N Il > s. For any particular choice of λ, µ, ν allowed for l satisfying 5, we have Il Sl λ, µ, ν and P N Il > s P N Sl λ, µ, ν > s. Recall that we can construct a version of γ satisfying 6. Let γ be constructed this way and let λ, µ, ν be constant on time intervals [ k 1ε, and equal to 18 where λ t = 1 a ε µ t = 1 a ε ν t = 1 a ε [ [ [ y 1 y 2 y 3 ] k 1ε β 1 γ u du y 1 β 1 γ u du ] k 1ε β 2 γ u du y 2 β 2 γ u du k 1ε β 3 γ u du y 3 β 3 γ u du β 1 γ = θsi, β 2 γ = ρi, β 3 γ = 1 s i. Since l is piecewise linear, for t k 1ε, du dt = 1 as + a/3 1 as k 1ε a/3 ε dv dt = 1 ai + a/3 1 ai k 1ε a/3 ε ] := 1 a ks ε := 1 a ki. ε,

14 14 R. G. DOLGOARSHINNYKH Note that λ, µ, ν given by 18 satisfy conditions 5 for the piecewise linear path l. To control the change in l over time intervals of length ε let gε = K log 1 ε 1, where K > is fixed, and define a collection of events B = {B ε } ε> { } B ε = sup x t1 x t2 gε, for x = s, i, r and 1 k R k 1ε t 1,t 2 It follows from Lemma 8 below that there exist ε >, N N and K > such that P N Sl λ, µ, ν > s P N{ Sl λ, µ, ν > s } B ε + P N Bε c P N{ Sl λ, µ, ν > s } B ε + 6T/ε e 2sN for all ε < ε and N > N. From Chebyshev inequality we have that for all < α < 1 19 P N{ Sl λ, µ, ν > s } E N exp { αnsl λ, µ, ν } 1I{B ε } B ε e αns. We need to show that the expectation above is appropriately small for α arbitrarily close to 1. Recall that Sl λ, µ, ν = fλ t, θu t v t dt + fµ t, ρv t dt + fν t, w t dt and we can split the integrals since fx, m = x logx/m x + m is nonnegative for x, m. We do this in two steps. First we give a bound on { } k+1ε 2 E exp N αn fλ t, θu t v t dt 1I{B ε } F k, where F k is the σ-algebra generated by events up until time. Similar bounds hold for µ and ν. This will lead to the sought bound on the expectation in 19. Second, we prove Lemma 8 to justify restriction to B ε. The next Lemma gives a bound on the expectation in 2. Lemma 6. For all < α < 1 there exist ε α and C α such that { } k+1ε E exp N αn fλ t, θu t v t dt 1I{B ε } F k for all ε < ε α. 2C α exp { Nεθ 1 α + 1g 1/4 ε } Proof. We let index k refer to the respective value at time. On B ε, for all ε such that gε < 1 and t [, k + 1ε ] θs t i t θs k i k < θs k gε + θi k gε < 2θgε.

15 so that SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 15 N k+1ε β 1 γ u du Nεθs k i k Nε 2θgε. Let Z 1, Z 2 be Poisson random variables with means Nεµ k = Nεθ s k i k 2gε and Nεµ k = Nεθ s k i k +2gε respectively. Since λ, µ, ν satisfy 18, we can construct a version of random SIRS processes in such a way that given F k 1 az 1 /εn λ k 1 az 2 /εn. Indeed, we can take k 1ε Z 1 = Ny 1 β 1 γ u du + εθ u k v k 2gε k 1ε Ny 1 β 1 γ u du k 1ε Z 2 = Ny 1 β 1 γ u du + εθ u k v k + 2gε k 1ε Ny 1 β 1 γ u du. Recalling the definition of l, it is not hard to see that on B ε max { u t u k, v t v k } < 1 agε < gε for t [, k + 1ε ] and hence θu t v t θ u k v k 2gε := µ a k for all a < 1/2. Since λ t = λ k is constant over the interval [, k + 1ε we have on B ε { } exp αn k+1ε fλ t, θu t v t dt exp { αnεfλ k, µ a k + 2αNθgεε} and together with convexity of fx, m in x this implies that { } k+1ε 21 E exp N αn fλ t, θu t v t dt 1I{B ε } F k e 2αNθgεε E N exp { αnεf 1 az 1 /εn, µ a } k F k + E N exp { αnεf 1 az 2 /εn, µ a k } F k. The next proposition gives a bound for the right hand side of 21. Proposition 3. Let a = g 1/4 ε. For all < α < 1 there exist ε α and C α such that max EN exp { αnεf 1 az i /εn, µ a } k Fk i=1,2 C α exp { Nεθ 1 α + 8g 1/4 ε } for all ε < ε α.

16 16 R. G. DOLGOARSHINNYKH Proof. We first prove the bound for Z 1. Conditional on F k, Z 1 is a Poisson random variable with mean Nεµ k and is measurable with respect to F k. Recall that by definition max { µ a k µ k, µ a k µ k } θ u k v k s k i k + 4gε θ 2a + 4gε. To simplify expressions let ε = ε/1 a, and α = 1 aα. Then { } E exp N Z1 / εn αnε Z 1 / εn log µ a Z 1 / εn + µ a k F k k = { } k/ εn exp αnε k/ εn log k/ εn + µ a k k µ a k 22 Nεµ k k exp{ Nεµ k } k! e Nεθ3a+4gε k αk e αk k! k µk µ a k k Nεµ a k k1 α exp { Nεµ a k 1 α}. Function hx = x k1 α e 2x1 α reaches its maximum at x = k/2 so that for all x. In particular and it follows that x k1 α e nx1 α Nεµ a k k1 α e 2Nεµ k 1 α k αk e αk k k! µk µ a k k k1 α e k1 α 2 k k1 α e k1 α 2 k Nεµ a k k1 α exp { Nεµ a k 1 α} exp { Nεµ a k 1 α} k k e k k k! µk µ a k / 2 1 α k. Recall that µ k = θs k i k 2gε and µ a k = θu kv k + 2gε. Therefore µ k µ a k θs k i k θu k v k 2gε 1 a gε/θa 2. If we let a = g 1/4 ε then clearly we can make µ k /µ a k arbitrarily close to 1 by taking ε close to. Let ε α be so small that µ k /µ a k < 21 α/2 < 2 1 α/2

17 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 17 for all ε < ε α. Then for all small ε exp { Nεµ a k 1 α} k k e k / k µk k! µ a 2 1 α k k 23 e Nεθ1 α k k e k 1 k k! 2 1 α/2/2 := e Nεθ1 α C α k since the series above converge. The conclusion for Z 1 follows from 23 and 22. Indeed, we have for all small ε E N exp { αnεfz 1 /εn, µ a k } F k Cα exp { Nεθ 1 α + 8g 1/4 ε }. A bound for Z 2 follows similarly, it is only left to note that µ k µ a k = θs ki k + 2gε θu k v k 2gε 1 1 2g 1/2 ε/θ + 2gε g 1/2 ε 2gε 1 as ε and we can take ε α be so small that max { µ k /µ a k, µ k k } /µa < 2 1 α/2 for all ε < ε α. The conclusion of Proposition 3 and Lemma 6 follow. We state similar bounds for µ and ν for completeness. The proofs are similar to the proof of Lemma 6. Lemma 7. that Let a = g 1/4 ε. For all < α < 1 there exist ε α and C α such E N exp { αnfµ k, ρv k } 1I{B η } Fk 2Cα exp { Nερ 1 α + 5g 1/4 ε } and E N exp { αnfν k, w k } 1I{B η } F k 2Cα exp { Nε 1 α + 5g 1/4 ε } for all ε < ε α. We are now well equipped to bound E N e αnsla λ, µ, ν 1I{B ε }. First, note that λ k, µ k, ν k are independent given F k. Taking iterative conditional expectations with respect to F R 1, F R 2,..., F 1, we get that for all < α < 1 and ε < ε α E N e αnsl λ, µ, ν 1I{B ε } R 1 2C α 3 e 2Nεθ+ρ+11 α+1g1/4 ε k= The next lemma justifies restriction to B. = 2C α 3T/ε e 2NT θ+ρ+11 α+1g1/4 ε. Lemma 8. Let ε x = sup k R 1 x k+1 x k and gε = K log 1 ε. There exist ε >, N N and K N such that P N max ε x > gε < 6T/ε e 2sN x=u,v,w

18 18 R. G. DOLGOARSHINNYKH for all ε < ε and N > N. Proof. Let L = θ + ρ + 1 and let Z 1, Z 2 be independent Poisson random variables with means NLε. For i = 1, 2, 3, β i γ u < L so that β 1 γ u du < k 1ε β 1 γ u du + Lε. Using 6, we can construct a version of the random SIRS process so that k+1ε u k+1 u k = s k+1 s k y 1 β 1 γ u du y 1 β 1 γ u du k+1ε + y 3 β 3 γ u du y 3 β 3 γ u du Then N 1 Z 1 + Z 2. P N u k+1 u k > gε 2 P N 1 Z 1 > gε/2 and it follows from Lemma 17 that there exist a constant K N and ε >, N N such that P N u k+1 u k > gε < e 2sN for all ε < ε and N > N. A similar argument leads to corresponding bounds for v and w. Combining the estimates we get P N max ε x > K log 1 ε < 6Re 2sN. x=u,v,w It is left to bound P N γ a l T δ/3 in 17. Lemma 9. For any δ > there exist ε δ >, N N such that for all ε < ε δ and N > N. Proof. For a < δ/2 P N γ a l T δ 2T/ε e 2sN P N γ a l T δ P N γ l T δ/2, and it is enough to show the inequality for γ. Once again we use representation 6 to write that for t [, k + 1ε k+1ε u t s t 2 y 1 β 1 γ u du y 1 β 1 γ u du + 24 k+1ε + 2 y 3 β 3 γ u du y 3 β 3 γ u du,

19 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 19 and similarly 25 k+1ε v t i t 2 y 2 β 2 γ u du y 2 k+1ε + 2 y 3 β 3 γ u du y 3 β 2 γ u du + β 3 γ u du. Let ε 1 be the maximal ε such that δ/4 > K log 1 ε 1. Let Z be distributed like a Poisson random variable with mean Nθ + ρ + 1 ε. It follows from Lemma 17 that for all ε < ε δ := min{ε, ε 1 } and N > N P N γ l T δ P N s t u t > δ/2 or i t v t > δ/2 for some t [, T ] R max PN s t u t > δ/2 or i t v t > δ/2 1 k R 1 for some t [, k + 1ε 2R P N 1 Z > δ/4 2Re 2sN, where the next to last inequality follows since 2Z is stochastically greater than the right hand sides of 24 and 25. At this point we have for all δ >, < α < 1, ε < min{ε, ε δ/3, ε α } and a = g 1/4 ε P N ρ T γ, Γs δ P N Sl a λ, µ, ν > s η/3 + P N γ l T δ/3 EN exp{αnsl a λ, µ, ν} 1I{B ε } e αns η/3 + P N B c η + P N γ l T δ/3 2C α 3T/ε exp { 2NT θ + ρ α + 7g 1/4 ε } e Nαs+η/3 + 8T/ε e 2sN. Taking α and ε small enough to ensure that 2T θ+ρ+1 1 α+7g 1/4 ε < η/6 and 1 αs < η/6 and N large enough so that 3T log2c α /εn < η/6 and log16t/ε/n < η/6, we get P N ρ T γ, Γs δ e Ns 5η/6 + 8T/ε e 2sN 16T/ε max { e Ns 5η/6, e 2sN} e Nη/6 e Ns 5η/6 = e Ns η for all large N and the conclusion of Lemma 5 follows. APPENDIX A

20 2 R. G. DOLGOARSHINNYKH Assumption C of Ventsel 1976b does not hold for SIRS processes. Assumption C requires that Hδ := Hy ; u Hy; u sup as δ. y y <δ 1 + Hy; u Hy;u< Let y = s, i, r = 1 s i, and u = u 1, u 2 then for SIRS processes Hy; u = ν logν/r + µ logµ/ρi + λ logλ/θsi ψ ϕ where ν = λ + u 1, µ = λ u 2 and λ solves subject to λ >, ν >, and µ >. λλ + u 1 λ u 2 = θsi ρi r Consider the situation when s = s = 1/2, i = 1/2 δ, i = 1/2 δ 1/δ, so that r = δ and r = δ 1/δ. Also, assume that u 2 > u 1 >. Then the following is not hard to verify that Hδ logδ/δ + logδ 1 logδ APPENDIX B 1/δ + as δ. Properties of function I. The following lemma is used in the proof that I is a rate function in Section 4. Lemma 1. If Sγ λ, µ, ν s then for all t 1 < t 2 T such that t 2 t 1 < 1/θ { t2 t2 t2 } s + 1 max λ t dt, µ t dt, ν t dt t 1 t 1 t 1 log θt 2 t 1 Proof. Since fx, m defined by 4 is nonnegative for x, m, we have that fλ t, θs t i t dt Sγ λ, µ, ν s For x let hx = x logx/θ x. Then h is a convex function of x so that for t 1 < t 2 T 1 t2 h λ t dt 1 t2 hλ t dt t 2 t 1 t 1 t 2 t 1 t 1 1 t2 λ t log λ 26 t λ t + θs t i t dt t 2 t 1 t 1 θs t i t s, t 2 t 1

21 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 21 where the second inequality follows from the fact that s t, i t 1. It is not hard to see that for all α > 27 hx αx θe α. Combining 26 and 27, we get for t 1 < t 2 T and α > 28 t2 t 1 λ t dt 1 α s + t 2 t 1 θe α. Recall that α was an arbitrary positive number, so if t 2 t 1 < 1/θ and we let α = log θt 2 t 1 then t2 t 1 λ t dt 1 log s + 1. θt 2 t 1 Similar arguments lead to the inequalities for µ and ν. We need the next three technical lemmas for the proof of the lower bound in Section 5. Lemma 11. Let γ be such that Iγ <. For < a < 1/3 let γ a = T a γ, where T a is defined by 1. Then γ γ a T < a and lim a Iγ a Iγ. Proof. It is clear that γ γ a T < 2 2a/3 2 < a and that γ a is absolutely continuous. By the definition of Iγ, for any η > there exists an allowed choice of λ, µ, ν for γ such that Sγ λ, µ, ν < Iγ + η <. Let λ a, µ a, ν a = 1 aλ, µ, ν so that λ a, µ a, ν a is an allowed choice for γ a. It is enough to show that Sγ a λ a, µ a, ν a Sγ λ, µ, ν as a. Convexity of fx, m in x and λ a t λ t imply that fλ a t, θs a t i a t f, θs a t i a t + fλ t, θs a t i a t θ + fλ t, θs a t i a t. Also, by construction 1 a 2 θs t i t θs a t i a t θ and convexity of fx, m in m implies fλ t, θs a t i a t fλ t, 1 a 2 θs t i t + fλ t, θ 2fλ t, θs t i t 2λ t log1 a + θ. It follows from Lemma 1 that λ t is integrable, so we have bounded fλ a t, θs a t i a t for a 1/3 by an integrable function. Since fλ a t, θs a t i a t fλ t, θs t i t the dominated convergence theorem implies that fλ a t, θs a t i a t dt fλ t, θs t i t dt as a. Applying the same argument to fµ a t, ρi a t and fν a t, r a t gives the result. For a > let R a be defined by 14. Lemma 12. Fix a > and let γ R a such that Iγ <. For any η > there exist L > and γ L R a/2 such that γ γ L T < a/2 and Sγ L λ L, µ L, ν L < Iγ + η for some allowed choice of λ L, µ L, ν L for γ L such that λ L t, µ L t, νt L L.

22 22 R. G. DOLGOARSHINNYKH Proof. Let η > and let λ, µ, ν be an allowed choice for γ such that Sγ λ, µ, ν < Iγ + η/2. For L > let λ L t = λ t L, µ L t = µ t L, and νt L = ν t L. Let γ L solve 5 with λ, µ, ν replaced by λ L, µ L, ν L, and let r L = 1 s L i L. We first show that for L sufficiently large γ L is close to γ in supnorm. Since λ t is integrable over [, T ] and λ L t λ t, the monotone convergence theorem implies that λ L s ds λ s ds < a/8 for all L L a. Similar statements are true for µ and ν, and taking L a to be the maximum of the three, we have s L T t s t λ s λ L T s ds + ν s νs L ds < a/8 + a/8 = a/4 and similarly i L t i t < a/4, r L t r t < s L t s t + i L t i t < a/4 for all t T and therefore γ L γ T < a/2 for all L L a. Since γ R a the above also ensures that γ L R a/2. To show that Sγ L λ L, µ L, ν L converges to Sγ λ, µ, ν recall that fx, m is convex in x and therefore fλ L t, θs L t i L t f, θs L t i L t + fλ t, θs L t i L t θ + fλ t, θs L t i L t. Since γ R a we have θ θs t i t θa 2 and, similarly, θ θs L t i L t θa 2 /4 for all L > L a. Notice that fx, m m = x m + 1 and therefore on the interval [a 2 /4, θ] fλ t, θs L t i L t fλ t, θs t i t < Cλ t + 1 for some constant C >. Since λ t and fλ t, θs t i t are integrable the dominated convergence theorem implies that fλ L t, θs L t i L t dt Similar arguments for µ and ν imply the result. fλ t, θs t i t dt. For ε > such that T/ε N let γ ε be defined by 15. Lemma 13. Fix η >. Let a, 1 and γ R a be such that Iγ <. Suppose that λ, µ, ν are allowed for γ and are such that λ t, µ t, ν t L for some L > and Sγ λ, µ, ν <. Then for all a < a η there exists an

23 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 23 ε a > such that for all ε < ε a, the polygonal approximation γ ε R a/2 and γ ε γ T < a/2. Moreover, there exist λ ε, µ ε, ν ε allowed for γ ε such that λ ε t, µ ε t, ν ε t L and Sγ ε λ ε, µ ε, ν ε < Sγ λ, µ, ν + η. Proof. Since γ is uniformly continuous on [, T ] there exists an ε a such that for all ε < ε a sup γ t γ t < a 3 /2 t t <2ε and therefore γ γ ε T < a/2 and γ ε R a/2. For t, k + 1ε ds ε dt = s k+1ε s ε di ε dt = i k+1ε i ε = 1 ε = 1 ε Therefore for t [, k + 1ε λ ε t := 1 ε k+1ε λ t dt, µ ε t := 1 ε k+1ε k+1ε k+1ε λ t dt + 1 ε λ t dt 1 ε k+1ε k+1ε ν t dt, µ t dt. µ t dt, and ν ε t := 1 ε are allowed for γ ε and constant over intervals [, k + 1ε. λ ε t, µ ε t, νt ε L. If < x L and m θa 2 /4 > then fx, m m = x m + 1 L θa 2 / Then for t [, k + 1ε and C = 4L/θ + 1 fλ ε t, θs ε ti ε t fλ ε t, θs i Cθa, fλ t, θs t i t fλ t, θs i Cθa. The above and the Jensen s inequality imply that k+1ε fλ ε t, θs ε ti ε tdt k+1ε and and k+1ε fλ ε t, θs ε iε dt + Cθaε = εfλ ε, s i + Cθaε k+1ε k+1ε fλ t, θs i dt + Cθaε fλ t, θs t i t dt + 2Cθaε. Similar arguments lead to bounds for ν and µ. Therefore Sγ ε λ ε, µ ε, ν ε Sγ λ, µ, ν + 2Cθa and choosing a < η/2cθ gives the result. APPENDIX C ν t dt Note that

24 24 R. G. DOLGOARSHINNYKH Lower semicontinuity of the rate function. Lemma 14. Fix a >. For all n N let γ n R a be such that Iγ n s for some s >. Suppose that Then Iγ s. lim γ n γn T = a.s. Proof. The strategy of the proof is similar to that of the proof of Theorem 5.1 partb Ventsel 1976b. Once again, since γ n are absolutely equicontinuous it follows that γ is absolutely continuous. Let δ > and let λ n, µ n, ν n be such that Sγ n λ n, µ n, ν n Iγ n + δ. Then inequality 28 of section 4 implies that for all n λ n t dt 2s + T e < if we take δ s. Noting that θs n i n θ we have that λ fλ n t, θs n t i n t λ n n t log t λ n t := gλ n t. θ Since gx/x as x + and gλn t dt s it follows that λ n are uniformly integrable on [, T ]. Therefore there exists λ t integrable on [, T ] such that for any t 1 t 2 T t2 t 1 λ n t dt t2 t 1 λ t dt as n see e.g. Meyer 1966, theorem II.T23. Then λ t dt = lim n λ n t dt 2s + T e <. Similarly, there exist µ t and ν t with the corresponding properties. It is not hard to see that λ t, µ t, ν t chosen this way are allowed for γ. For x, m fx, m 1 m = x x m m + 1, and therefore if m, m C for some C > and m m < ε then 29 fx, m fx, m εc 1 x + 1. Partition = t < t 1 <... t J = T so that the oscillations of γ n and γ on intervals [t j, t j+1 ] are less than ε. Then since max{θs n t i n t, ρi n t, r n t }

25 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 25 min{θ, ρ, 1}a 2 := C >, we have from 29 s + δ Sγ n λ n, µ n, ν n = fλ n t, θs n t i n t + fµ n t, ρi n t + fν n t, r n t dt J 1 tj+1 j= t j εc2s + T e. fλ n t, θs n t j i n t j + fµ n t, ρi n t j + fν n t, r n t j dt Since fx, m is convex in x Jensen s inequality implies that J 1 tj+1 j= t j fλ n t, θs n t j i n t j + fµ n t, ρi n t j + fν n t, r n t j dt J 1 tj+1 t t j+1 t j f j λ n t dt, θs n t t j= j+1 t j i n t j + f j tj+1 t +f j νt n dt, r n t t j+1 t j. j µ n t dt, ρi n t t j+1 t j j tj+1 t j Let ε be such that εc2s + T e < δ. Taking limit in n, we get by the continuity of f J 1 tj+1 tj+1 t s + 2δ t j+1 t j f j λ t dt t, θs tj i tj + f j µ t dt, ρi tj t j= j+1 t j t j+1 t j tj+1 t +f j ν t dt, r tj. t j+1 t j Noting that γ R a we apply 29 one more time to get J 1 tj+1 tj+1 tj+1 t s + 3δ f j λ t dt t, θs t i t + f j µ t dt, ρi t j= t j t j+1 t j t j+1 t j tj+1 t + f j ν t dt, r t dt. t j+1 t j Let lt J = / t j+1 t j λ t dt t j+1 t j, m J t = / t j+1 t j λ t dt t j+1 t j and n J t = / tj+1 λ t dt t j+1 t j for t j t < t j+1, j =,..., J 1. If we consider t j finer and finer partitions, then lt J λ t, m J t µ t and n J t ν t almost everywhere. The function fx, m and is continuous in x; therefore

26 26 R. G. DOLGOARSHINNYKH using Fatou s lemma we obtain Sγ λ, µ, ν = J 1 lim J fλ t, θs t i t + fµ t, ρi t + fν t, r t dt j= s + 3δ. tj+1 t j f l J t, θs t i t + f m J t, ρi t + f n J t, r t dt Therefore, for every δ > there exist allowed λ, µ, ν for γ such that Sγ λ, µ, ν s + 3δ and so Iγ = inf λ, µ, ν Sγ λ, µ, ν s. Lemma 15. Let γ D be absolutely continuous. For a > let T a be defined by 1. Then Iγ lim a IT a γ. Proof. In proposition 2 in Section 4 we show that there always exists a unique allowed choice of λ, µ, ν minimizing Iγ for any absolutely continuous γ and give a description of it. In particular, the minimizing λ solves the cubic equation 12 subject to constraints 13. The transformation T a is such that γ a = T a γ γ and ds a = 1 ads dt dt ds dt, and dia = 1 adi dt dt di dt as a. The coefficients in the cubic equation 12 and the constraints 13 depend only on γ and its derivatives. Therefore, solutions λ a, µ a, ν a of 12 subject to 13 for γ a converge to λ, µ, ν, solutions for γ, almost everywhere. Then as a, fλ a t, θs a t i a t +fµ a t, ρi a t +fν a t, r a t fλ t, θs t i t +fµ t, ρi t +fν t, r t a.s., and f. Since λ, µ, ν and λ a, µ a, ν a are the minimizing allowed choices for γ and γ a respectively Iγ = Sγ λ, µ, ν lim a Sγ a λ a, µ a, ν a = Iγ a, where the inequality follows from Fatou s lemma. Lemma 16. For n N let γ n be such that Iγ n s for some s >. Then for any δ > there exists an a δ such that for all a < a δ and all n N where T a is defined by 1. IT a γ n Iγ n + δ, Proof. Fix n N. Let λ n, µ n, ν n be such that Sγ n λ n, µ n, ν n < Iγ n + δ/2 and let T a λ n = 1 aλ n, T a µ n = 1 aµ n, T a ν n = 1 aν n. Note that T a λ n, µ n, ν n are allowed for T a γ n. Then ST a γ n T a λ n, T a µ n, T a ν n = ft a λ n t, θt a s n t T a i n t + ft a µ n t, ρt a i n t +ft a ν n t, T a r n t dt.

27 SAMPLE PATH LARGE DEVIATIONS FOR SIRS EPIDEMIC PROCESSES 27 We will show that for all n ft a λ n t, θt a s n t T a i n t fλ n t, θs n t i n t + Ca for some constant C > not dependent on n. The same argument would lead to similar inequalities for the integrands involving µ and ν. The conclusion of the lemma would then follow by taking a δ = δ/6ct. To simplify the notation we write λ a, µ a, ν a = T a λ n t, T a µ n t, T a νt n, λ, µ, ν = λ n t, µ n t, νt n, γ = γt n and γ a = T a γt n. We show that for all t and all n fλ a, θs a i a fλ, θsi + Ca for some constant C >. Recall that fλ a, θs a i a = λ a log λ a λ a logθs a i a λ a + θs a i a. We bound each of the four summands separately. First note that θsi1 a 2 θs a i a = θ 1 as + a 1 ai + a θsi + 2a Then λ a log λ a λ log λ aλ log1 a; λ a logθs a i a λ logθsi + aλ log θ 2λ log1 a; λ a 1 aλ = λ + aλ; θs a i a θsi + 2θa/3. Combining the inequalities we obtain fλ a, θs a i a fλ, θsi + 2θa/3 + λ a a log1 a + a log θ 2 log1 a. Inequality 28 in Appendix C implies that for all n λ n t dt C for some constant C >. The conclusion follows since a a log1 a + a log θ 2 log1 a = Oa. APPENDIX D A large deviations estimate for Poisson processes. Let Y 1, Y 2,... be independent Poisson random variables with mean µ. For all N N let y N = 1 N Y k. N Lemma 17. For any s > there exist a constant K N and µ >, N N such that P N y N > K log 1 µ 1 < e sn for all µ < µ and N > N. k=1

28 28 R. G. DOLGOARSHINNYKH Proof. This is a straightforward application of Cramér s theorem see e.g. Dembo and Zeitouni 1992, Chapter 2 Acknowledgment. The author is very grateful to her graduate advisor Steven P. Lalley for his guidance and a lot of valuable advise. References Bahr, B. V. and Martin-Löf, A Threshold limit theorems. Advances in Applied Probability Dembo, A. and Zeitouni, O Large Deviations Techniques and Applications. Jones and Bartlett, London. Ethier, S. N. and Kurtz, T. G Markov Processes. Characterization and Convergence. Wiley, New York. Freidlin, M. I. and Wentzell, A. D Random Perturbations of Dynamical Systems. 2nd ed. Springer, New York. Kermack, W. O. and McKendrick, A. G Contributions to the mathematical theory of epidemics, part i. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Kermack, W. O. and McKendrick, A. G Contributions to the mathematical theory of epidemics, ii - the problem of endemicity. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Kermack, W. O. and McKendrick, A. G Contributions to the mathematical theory of epidemics, iii - further studies of the problem of endemicity. Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Meyer, P. A. ed Probability and potentials. Waltham, Massachusetts. Nagaev, A. V. and Startsev, A. N The asymptotic analysis of a stochastic model of an epidemic. Theory of Probability and its Applications Sellke, T On the asymptotic distribution of the size of a stochastic epidemic. Journal of Applied Probability Ventsel, A. D. 1976a. Rough limit theorems on large deviations for markov stochastic processes, i. Theory of Probability and its Applications Ventsel, A. D. 1976b. Rough limit theorems on large deviations for markov stochastic processes, ii. Theory of Probability and its Applications