An epidemic in a dynamic population with importation of infectives

Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_0$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $ n \to \infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{ S(t);t\ge 0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the previous jump. Properties of the process $S$, including the jump size and stationary distributions, are determined.


Introduction
The mathematical theory for the spread of infectious diseases has a long history and is by now quite rich (e.g., [Diekmann et al. (2013)]). One of the more common type of disease models is called SIR (susceptible → infective → recovered) meaning that individuals are at first Susceptible. If infected (by someone) they immediately become Infectious (being able to spread the disease onwards). After some time an infectious individual Recovers, which also means that the individual is immune to further infection from the disease. Such models were originally studied for populations assuming homogeneous mixing, but during the last few decades considerable effort has been put into analysing epidemic models in communities which are not homogeneously mixing but instead may be described using some type of social structure, such as a community of households (e.g. [Ball et al. (1997)]) or a random network describing possible contacts (e.g. [Newman(2002)]). The vast majority of papers devoted to these type of problems assume a fixed community and community structure.
In the current paper we treat the situation where the population is dynamic in the sense that people die and new individuals are born, or more precisely immigrate into the population. Further, we assume that there is also importation of infectious individuals (randomly in time according to a homogeneous Poisson process), implying that the disease never vanishes forever. In order to facilitate analytical progress we consider only the case of a homogeneously mixing community, which in network terminology corresponds to treating the complete network.
Models for recurrent epidemics go back to the deterministic formulations of [Hamer(1906)] and [Soper (1929)]. A stochastic treatment was given first in the pioneering work of [Bartlett(1956)], who considered an SIR model with importation of both susceptibles and infectives, but without disease-unrelated deaths. An alternative model, with disease-unrelated deaths but no importation of infectives, has been studied extensively (e.g. [Nåsell(1999)] and the references therein). Interest often centres on the time to extinction of infection and the closely-related problem of the critical community size for an infection to persist in a population.
We consider a Markovian SIR epidemic with demography and importation of infectives, in which infectious individuals infect new individuals at constant rate and the infectious period is exponentially distributed. We study limit properties of the epidemic when the average population size n tends to infinity. Our focus lies on the case where the limit is taken keeping the basic reproduction number R 0 (i.e. the average number of susceptibles infected by a single infective in an otherwise fully susceptible population of size n) and the immigration rate of infectives fixed, whereas the quotient of the average infectious period and the average lifetime tends to 0 faster than 1/ log n. For many infectious diseases this quotient typically lies between 10 −4 and 10 −3 , hence supporting this asymptotic regime, but in the discussion we treat other asymptotic regimes briefly.
Under the above asymptotic regime, all epidemic outbreaks are short, having duration that tends to 0 in probability as n → ∞. Further, as n → ∞, epidemic outbreaks are either minor, having size of order o p (n), or major, having size of exact order Θ p (n). It follows that, as n → ∞, the behaviour of the three-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a one-dimensional regenerative process S = {S(t); t ≥ 0}, describing the limiting fraction of the population that are susceptible. During each cycle, the process S makes one down jump, corresponding to the occurrence of a major outbreak, and except for this increases deterministically, as minor outbreaks have no effect onS (n) in the limit as n → ∞. (Here, S (n) = {S (n) (t) : t ≥ 0}, where, for t ≥ 0,S (n) (t) = n −1 S (n) (t) with S (n) (t) being the number of susceptible individuals in the population at time t.) Note thatS (n) does not converge weakly to S in the Skorohod topology since the sample paths of S are almost surely discontinuous but those ofS (n) almost surely contain only jumps of size n −1 , so are close to being continuous. Thus to obtain rigorous convergence results, we consider two processes,S (n) − andS (n) + , which coincide withS (n) , except during major outbreaks during which they sandwichS (n) , and prove that bothS (n) − andS (n) + converge weakly to S in the Skorohod topology (Theorem 2.1). It then follows that certain functionals ofS (n) converge weakly to corresponding functionals of S (Corollary 2.1).
The paper is structured as follows. In Section 2, we define the model and the limiting regenerative process, give an intuitive explanation of why S approximatesS (n) for large n and present the main convergence results. In Section 3, we derive some properties of the limiting regenerative process: the jump size distribution, the associated renewal time distribution and the stationary distribution. In Section 4, we present simulations supporting the convergence result and illustrating various features of the limiting process. In Section 5, we prove the main results. We end in Section 6 with a Discussion summarising our results and also exploring briefly additional questions, such as other asymptotic regimes.

The Markovian SIR epidemic with demography and importation of infectives
We now define the Markovian SIR epidemic with demography and importation of infectives (SIR-D-I). We consider the process to be indexed by a target population size n, which we assume is a strictly positive constant. The population model is an immigration-death process with constant immigration rate and linear death rate. For t ≥ 0, let N (n) (t) denote the population size at time t. Then N (n) (t) increases at constant rate µn and decreases at rate µN (n) (t). The population size hence fluctuates around n, which is assumed to be large. The Markovian SIR-epidemic on this population is defined as follows. For t ≥ 0, let S (n) (t), I (n) (t) and R (n) (t) denote the number of susceptibles, infectives and recovered, respectively, at time t, so S (n) (t) + I (n) (t) + R (n) (t) = N (n) (t). We assume that I (n) (0) = 0 and thatS (n) (0) → s 0 as n → ∞, where s 0 ∈ (0, 1] is constant. (The value of R (n) (0) has no effect on the ensuing epidemic.) . A fraction κ n of all births (i.e. immigrants) are infectives and the remaining births are all susceptibles, so births of infectives occur at rate µnκ n and births of susceptibles occur at rate µn(1 − κ n ). While infectious, any given infective infects any given susceptible at rate n −1 λ n , independently between each distinct pair of individuals. Thus, approximately, each infective makes infectious contacts at the points of a homogeneous Poisson process having rate λ n , with contacts being with individuals chosen independently and uniformly from the whole population; a contact with a susceptible individual results in that individual becoming infected, while a contact with an infectious or removed individual has no effect. Each infectious individual recovers and becomes immune at rate γ n , implying that the infectious period is exponentially distributed with rate parameter γ n .
More formally, the process S (n) (t), I (n) (t), R (n) (t) : t ≥ 0 is a continuous-time Markov chain, with state space Z 3 + and transition intensities given by corresponding to birth of a susceptible, birth of an infective, death of a susceptible, death of an infective, death of a recovered, infection of a susceptible and recovery of an infective, respectively.
We study specifically the case where the average population size n tends to infinity in such a way that (a) the total importation rate µnκ n of infectives tends to a strictly positive constant µκ, so κ n n → κ as n → ∞; and (b) the infection and recovery rates satisfy λ n /γ n → R 0 > 1 and λ n / log n → ∞ as n → ∞.
For ease of exposition, we assume that n is an integer, so sequences of epidemic processes are indexed by the natural numbers. However, all of the results of the paper are easily generalised to the case of a family of epidemic processes indexed by the positive real numbers. To conclude, the parameters of the model are: n, the average population size; µ, where 1/µ is the average lifetime and µn is the population birth rate; λ n , the infection rate; γ n , where 1/γ n is the average length of the infectious period; and κ n , the fraction of births which are infectious, so µnκ n is the birth (or importation) rate of infectives.

The limiting process S
is the "fraction" of the population that is susceptible at time t. The process S = {S(t); t ≥ 0} can be viewed as the limit ofS (n) as n → ∞ under the above asymptotic regime. It is a Markovian regenerative process (e.g. [Asmussen(1987)], Chapter V), with renewals occurring whenever S(t) = 1/R 0 . Between each renewal S(t) increases deterministically according to the differential equation (2.1) except for one down jump (from above 1/R 0 to below 1/R 0 ). This implies that before the jump (if u denotes the time from the last renewal). The random time T from a renewal to the jump has distribution specified by The size of the jump is specified by the value S(T −) of the process just prior to the jump. More precisely, S(T ) = S(T −)(1 − τ (S(T −))), where for s > R −1 0 , τ (s) is the unique strictly positive solution to the equation (cf. [Diekmann et al. (2013)], equation (3.15)) In epidemic theory τ (s) is known as the relative fraction infected among the initially susceptible of an SIR epidemic outbreak in which a fraction s are initially susceptible and the rest immune. Hence, the size of the down jump is S(T −)τ (S(T −)). After the down jump, S(t) increases deterministically according to the same differential equation (2.1) until the next renewal point, so and the inter-renewal time is T + µ −1 log[(1 − S(T ))/(1 − 1/R 0 )]. Illustrations of S are given in Section 4.

Main results and heuristics
We first explain heuristically why S can be viewed as the limit ofS (n) as n → ∞ under that asymptotic regime described in Section 2.1. Suppose that n is large. Then when no infective is present, all that happens is that individuals die and new ones are born at approximately the same rate µn. Recovered (immune) individuals that die are replaced by susceptible individuals, so the fraction of susceptibles increases at rate µ(1 −S (n) (t)) which explains the deterministic growth rate of S. After an exponentially distributed holding time, with rate parameter µnκ n ≈ µκ, an infective is born into the community. If the fraction susceptibleS (n) (t) is below 1/R 0 , then the effective reproduction number R e = R 0S (n) (t) is strictly less than one, implying that, with probability tending to one as n → ∞, a large outbreak will not occur, soS (n) (t) continues to grow approximately deterministically. IfS (n) (t) > 1/R 0 when a new born infective enters the community, then with approximate probability 1 − 1/(R 0S (n) (t)) that infective gives rise to a major outbreak that infects order Θ(n) susceptibles (cf. [Diekmann et al. (2013)], pages 53 and 376), otherwise only a minor outbreak, which infects order o(n) susceptibles, occurs andS (n) (t) continues to grow approximately deterministically. This explains the distribution for T , the time from a renewal until a down jump in S, which has time varying intensity given by µκ multiplied by the limiting major outbreak probability (cf. [Bartlett(1956)]).
If a major outbreak takes place, the size of the outbreak among the susceptibles is given approximately by τ (S(T −))S (n) (T −) where S(T −) denotes the limiting (as n → ∞) fraction susceptible just prior to the outbreak and τ (s) is defined above (cf. [Diekmann et al. (2013)], page 60). The duration of such a major outbreak is of order Θ(log n/λ n ) (cf. [Barbour(1997)]) which tends to 0 by assumption. Thus, if there is a major outbreak it happens momentarily and, in the limit as n → ∞, the fraction susceptible after the outbreak, S(T ), satisfies S(T ) = S(T −)(1 − τ (S(T −)).
Although the above heuristic argument makes it plausible that the normalised susceptible processS (n) converges to the regenerative process S, there are two complicating factors in making the argument fully rigorous. First, as explained in Section 1, it is not true that S (n) ⇒ S as n → ∞, where ⇒ denotes weak convergence in the space D[0, ∞) of rightcontinuous functions f : [0, ∞) → R having limits from the left (i.e. càdlàg functions), endowed with the Skorohod metric (e.g. [Ethier and Kurtz(1986)], Chapter 3). As explained also in Section 1, we overcome this problem by considering two processes,S (n) − andS (n) + , which coincide withS (n) except during major outbreaks, when they sandwichS (n) , and show that S (n) − ⇒ S andS (n) + ⇒ S(·) as n → ∞; see Theorem 2.1. The second complicating factor is that the results referred to above concerning the probability, size and duration of a major outbreak are for an epidemic in a static population, whereas our population is dynamic. The results carry over to our setting because, in the limit as n → ∞, the time scale of an epidemic outbreak is infinitely faster than that of demographic change, but proofs need to be adapted accordingly.
Before stating our main theorem, some more notation is required. Recall that I (n) (t) is the number of infectives at time t in the SIR-D-I epidemic with average population size n and that we consider epidemics with no infective at time 0, i.e. with I (n) (0) = 0. Let t : I (n) (t) = 0}. Thus, provided n is sufficiently large, the kth major outbreak starts at approximately time t (n) k and ends at time u (n) k . (The choice of log n to delineate major outbreaks is essentially arbitrary. Our proofs work equally well if log n is replaced by any function g(n) which satisfies g(n) → ∞ and n − 1 2 g(n) → 0 as n → ∞.) For t ≥ 0, let The following theorem is proved in Section 5.1. An immediate consequence of Theorem 2.1 is that suitable functionals ofS (n) converge weakly to corresponding functionals of S.
The following corollary, which can clearly be generalised to suitable non-realvalued functionals, follows immediately from Theorem 2.1 by using the continuous mapping theorem (e.g. [Billingsley(1968) One functional which satisfies the conditions of Corollary 2.1 is the first passage time functional H a , defined for given a ∈ (0, 1) by The functional H a is clearly monotone and P (S ∈ C Ha ) = 1, cf. [Pollard(1984)], page 124. Another functional satisfying the conditions of Corollary 2.1 is the occupancy time functional H a t * , defined for any given t * > 0 and a ∈ (0, 1) by This functional is again clearly monotone. The proof that P S ∈ C H a t * = 1 is given at the end of Section 5.1.

Properties of the limiting process S
We now outline some properties of the regenerative process S which can be obtained from renewal and regenerative process theory (e.g. [Asmussen(1987)], Chapters IV and V). As described in Section 2.2 the stochastic part of the regenerative process is completely specified by the waiting time T until the down jump, but it can be specified equivalently by the jump which can be used to obtain the distribution of the jump size X. The jump size is strictly less than τ (1), as S(t) < 1 for all t ≥ 0. Hence, for 0 < x < τ (1), (3.1) The lifetime distribution for the renewal process describing successive visits of S to 1/R 0 may be derived as follows. During a cycle, the regenerative process S starts at 1/R 0 and grows deterministically, according to (2.1), until the time T of the down jump. After this down jump it again grows deterministically, according to (2.1), until it reaches 1/R 0 , when the next renewal occurs. If we change the order of these two parts, the process starts at S(T ) and grows deterministically until it reaches S(T −). The lifetime T * hence equals the time it takes for the deterministic curve defined by (2.1) to travel from S(T ) to S(T −) This time equals This is a monotonic increasing function of X, so the renewal time distribution can be obtained numerically using the expression F X (x) given by (3.1). The stationary distribution of S can be obtained using regenerative process theory (e.g. [Asmussen(1987)], Chapter V, Section 3). During a regenerative cycle, the process S traverses s if and only if s lies between S(T ) and S(T −). If it does, the density for the time spent there is inversely proportional to the derivative µ(1 − s). Consequently, if we let f S * (s) denote the density of the stationary distribution of S, we have . It then follows using (2.3) and (3.1) that, withs = g −1 (s), In the next section the density f S * (s) is calculated numerically and shown to agree with corresponding empirical values from simulations.

Numerical illustrations
We now present briefly some numerical and simulation results, which illustrate convergence of the epidemic process as well as properties of the limiting stationary distribution of the fraction susceptible S * . In Figure 1 the epidemic is simulated for 100 years in a population of n = 10, 000 individuals. In all figures, R 0 = 2 implying that the effective reproduction number R e = R 0S (n) (t) is supercritical as soon as the population fraction susceptible exceeds 1/R 0 = 0.5. The average lifetime is 1/µ = 75 years and γ = 50, so the average length of the infectious period is about 1 week. In the left panels of Figure 1, κ = 20, so the rate at which new infectives enter the population (µκ) equals 1 per 3.75 years, and in the right panels κ = 200, so new infectives enter the population at rate 2 2 3 per year. The upper panels show the fraction of the population that is susceptible over the 100 period and the lower panels show the corresponding fraction that is infective. Observe that when κ = 20 major outbreaks are less frequent but larger than when κ = 200, and that there are appreciably more minor outbreaks when κ = 100. Note also that epidemics are rarer than the importation rate of infectives suggests, for two reasons. First, major outbreaks can occur only when S (n) (t) > 1/R 0 = 0.5, and secondly, whenS (n) (t) is above this threshold, major outbreaks do not occur each time an infective enters the community. In the lower left panel of Figure  1 some minor outbreaks caused by importation of infectives can also be seen.
In Figure 2 realisations of the corresponding limiting processes are plotted. The same parameter values are used in both figures. The stochastic features of the epidemic and the limiting process are in agreement, suggesting that the limiting behaviour has kicked in when n = 10, 000. Note that, unlike in Figure 1, there are no near-vertical lines as outbreaks are now instantaneous.
We now illustrate properties of the stationary distribution of the fraction susceptible S * , both for the epidemic with n = 1, 000 and n = 10, 000, as well as for the limiting process. For the three processes, and for three different values of κ, we simulate the epidemic and limiting processes for 10,000 years and in Figure 3 we plot bar charts of the relative time spent with specified fraction susceptible. The processes are simulated over a very long time span so that the empirical distribution of the fraction susceptible is close to the corresponding stationary distribution. (Recalling the functional H a t * defined at the end of Section 2.3, note that by standard regenerative process theory, for any fixed a ∈ (0, 1), 1 The values of µ, γ and R 0 are the same as in Figure 1. (Note that the value of γ, and hence also λ (= R 0 γ), is the same for both values of n.) The chosen values of κ are κ = 1, 3 and 100, corresponding to importation of infectious individuals on average one every 75, 25 and 0.75 years, respectively. In the plots we have also computed f S * (s), the stationary distribution of the limiting process, numerically as described in Section 3.
It is seen that the bar charts from the epidemics resemble the limiting stationary distri-  bution f S * (s), except when n = 1, 000 and κ = 100. When κ is small, few outbreaks take place, so even if the outbreaks are large, the population fraction of susceptibles is close to 1 most of the time, which explains why the stationary distribution S * is concentrated at values close to 1. For moderate values of κ, the stationary distribution has positive mass for nearly all s values between 1 − τ (1) = 0.2032 (the fraction susceptible after a major outbreak starting with the entire population being susceptible) and 1. The stationary distribution is seen to be concentrated around 1/R 0 when κ is large, owing to the fact that a new major outbreak occurs quite soon after the population fraction of susceptibles exceeds 1/R 0 , with the effect that the size of major outbreaks is generally small. These observations imply that the stationary distribution is not stochastically decreasing (nor increasing) in κ.

Proof of Theorem 2.1
Let (Ω, F, P ) be a probability space on which is defined a homogeneous Poisson process η on (0, ∞) having rate µκ and let 0 < r 1 < r 2 < · · · denote the times of the points in η. For n = 1, 2, · · · , let η (n) denote the point process with points at 0 < r . Let E (n) denote the epidemic process indexed by n. Then η (n) gives the points in time when infectives immigrate into the population in E (n) . We construct E (n) (n = 1, 2, · · · ) and S by first conditioning on η.
The process S is constructed as follows. Recall the definition of τ (s) at 2.4. Between the points of η, S(t) increases deterministically according to the differential equation (2.1). For k = 1, 2, · · · , S has a down jump to S(r k −)[1 − τ (S(r k −))] at time r k with probability max(1 − (R 0 S(r k −)) −1 , 0) (independently for successive k), otherwise S continues to grow according to (2.1). Thus, S can be described as follows. Let t 1 < t 2 < · · · be the times of the down jumps of S, so these form a subset of the points of η. Let so, for fixed x, the solution of (2.1) with S(0) = x is f (x, t). Let t 0 = 0 and suppose that s 0 = S(0) is given. Then, for k = 0, 1, · · · , where, for k = 1, 2, · · · , the initial value . The precise definition of the construction of E (n) (n = 1, 2, · · · ) is not relevant at this stage. We prove Theorem 2.1 by first proving the corresponding result for processes conditioned on η.
(i) For k = 1, 2, · · · , u Proof. See Section 5.2. 2 Proof of Lemma 5.1. First note that since results concerning convergence in distribution in the Euclidean space R k carry over in all essential respects to convergence in distribution in R ∞ (see [Billingsley(1968)], page 19), the Skorohod representation theorem implies that we may assume that the convergence in Lemma 5.2 holds almost surely. Let A ∈ F be the set ω ∈ Ω such that (i) for k = 1, 2 · · · , ( 5.4) and (iii) t k (ω) → ∞ as k → ∞. Then P(A|η) = 1 for P-almost all η.
Note thatS k+1 ), so (5.4) implies that B(n, k) also converges to 0 as n → ∞, whence (5.12) Combining (5.9) and (5.12) yields that, A similar argument to the derivation of (5.12) yields which together with (5.13) yields (5.7), as required. The proof of (5.8) is similar to that of (5.7) and hence omitted. 2 Proof of Theorem 2.1. We prove the result forS  [Ethier and Kurtz(1986)], Chapter 3, Theorem 3.1). Let f : D[0, ∞) → R be any such function. Then Lemma 5.1 implies that, for P-almost all η, Hence, by the dominated convergence theorem, This holds for all bounded, uniformly continuous f : D[0, ∞) → R, soS (n) − ⇒ S as n → ∞, as required. 2 We end this subsection by showing that the occupancy time functional H a t * , defined at (2.5), satisfies P S ∈ C H a t * = 1. Recall that t 1 < t 2 < · · · denote the jump times of S.

Proof of Lemma 5.2
We prove Lemma 5.2 by splitting the SIR-D-I epidemic process E (n) into cycles, where now a cycle begins at the end of a major outbreak and finishes at the end of the following major outbreak. Thus a cycle consists of two stages: stage 1, during which the susceptible population grows approximately deterministically until there are at least log n infectives present; and stage 2, comprising the major outbreak caused by these log n infectives, during which the susceptible population crashes. Recall that, as n → ∞, the point process η (n) , describing immigration times of infectives in E (n) converges almost surely to the point process η governing times when down jumps may occur in the limiting process S. Lemma 5.4 considers the initial stage 1 and shows, using birth-and-death processes that sandwich the process of infectives, that for P-almost all η, as n → ∞, for successive importations of infectives until a major outbreak occurs, the probability a given importation triggers a major outbreak converges to the probability that the corresponding importation results in a down jump in the limiting process S. Consequently, the time until there are at least log n infectives in E (n) converges weakly to the time of the first down jump in S, since η (n) converges almost surely to η. Further, application of the law of large numbers for density dependent population processes ( [Ethier and Kurtz(1986)], Chapter 11) shows that up until the first down jump of S, the scaled process of susceptibles, S (n) = n −1 S, converges weakly in the uniform metric to S, since minor epidemics infect order o p (n) individuals.
Lemmas 5.5 and 5.6 concern the limiting size and duration of a typical major outbreak. Lemma 5.5 considers outbreaks in which the initial number of infectives is of exact order n, for which the above-mentioned law of large numbers is applicable. This is then used to prove Lemma 5.6, which considers major outbreaks triggered by log n infectives. Finally, Lemma 5.2 follows easily by induction using Lemmas 5.4 and 5.6, since E (n) is Markov.
The proof involves extensive use of birth-and-death processes that bound the process of infectives in the epidemic model (cf. [Whittle(1955)]). We first give some notation concerning birth-and-death processes and then state a lemma, proved in Appendix A, concerning properties of sequences of such processes.
(i) For k = 1, 2, · · · , lim n→∞ P χ (ii) For k = 1, 2, · · · , as n → ∞, (iii) For k = 1, 2, · · · , as n → ∞, Proof. For ease of presentation we suppress explicit conditioning on η in the proof. First note that P S(r 1 −) = R −1 0 = 0, since r 1 is a realisation of a continuous random variable. Assume without loss of generality that there is no recovered individual at time t = 0. For t ≥ 0, let S (n) 0 (t) be the number of susceptibles at time t under the assumption that the immigration rate for susceptibles is µn(1 − κ n ) and the immigration rate for infectives is 0, and letS  n) 0 (t)/n. Then, for any t > 0, application of Theorem 11.2.1 of [Ethier and Kurtz(1986)] (using the more general definition of a density dependent family given by equation (11.1.13) of that book) yields that, for any > 0, Recall that E (n) denote the epidemic process with average population size n. Consider the epidemic initiated by the immigration of an infective at time r (n) 1 in E (n) and let s (n) 1 =S (n) (r (n) 1 ). For ease of exposition, translate the time axis of E (n) so that the origin corresponds to r (n) 1 . With this new time origin, I (n) (t) : t ≥ 0 can be approximated by a linear birth-and-death process Ĩ (n) (t) : t ≥ 0 having death rate γ n + µ and (random) timedependent birth rate given by λ nS (n) 0 (t). This approximation ignores depletion in the number of susceptibles owing to infection, so Ĩ (n) (t) : t ≥ 0 is an upper bound for I (n) (t) : t ≥ 0 .

Discussion
In the paper it is proved that for an SIR epidemic in a dynamic population (whose size fluctuates around n), in which there is importation of infectives at a constant rate, the normalised process of susceptibles converges to a regenerative process S as n → ∞. Further, properties of the limiting process S are derived. The asymptotic regime considered is for the situation when the rate of importation of infectives κµ and the basic reproduction number R 0 remain constant with n, whereas the average length of the infectious period 1/γ n converges to 0 faster than 1/ log n (in most real-life epidemics, the ratio of average infectious period and average lifetime lies between 10 −4 and 10 −3 ).
Other asymptotic regimes could of course also be considered. For example, if the importation rate of infectives grows with n, then there will always be infectives present in the population resembling an endemic situation. If the duration of an infectious period remains fixed (or at least grows slower than log n), then the duration of a single outbreak will be long and the typical time horizon will not go beyond the first outbreak. A more complicated and interesting scenario seems to be for the asymptotic situation treated in the current paper, but where the epidemic is initiated with a fraction 1/R 0 of the population susceptible and a large enough number of infectives. It then seems as if an endemic equilibrium will stabilize, but determining and proving this rigorously remains an open problem. For large but finite n, it is possible for the process to get stuck in an endemic situation near the end of a major outbreak (with states similar to those just described). Eventually the epidemic leaves this endemic state and returns to the behaviour of the limiting process. In Figure 4 such a simulation is presented. The parameter values are n = 100, 000, µ = 1/75, κ = 1 (so the importation rate of infectives is one per 75 years), R 0 = 2 and γ = 2 (so the average infectious period is 6 months). The left and right plots show the fraction of the population that are susceptible and infective, respectively, as functions of time. A quasi-endemic phase lasts roughly from years 1, 300 to 3, 000. Observe that major outbreaks become smaller prior to the process entering the quasi-endemic phase and fluctuations in the number of infectives increase in amplitude prior to the end of the quasi-endemic phase. Beside studying other asymptotic regimes, it could be of interest to increase realism in the model, for example, by relaxing exponential distributions of infectious periods and lifetimes and allowing for a latent state (cf. [Andersson and Britton(2000b)], who consider epidemics with importation of susceptibles only) or by having some population structure, such as network or households (see the challenges in [Pellis et al. (2015)] and ]).