In order to briefly demonstrate the dynamics of the multitype SIR model, the course of an epidemic is simulated for network matrices
γN =γS =γI = 1−(n−1)a a · · · a a 1−(n−1)a · · · a ... ... ... ... a a · · · 1−(n−1)a ∈Rn×n (5.34)
with 0 ≤ a ≤(n−1)−1 describing the strength of contacts between clusters. Figure 5.2
shows the evolution of the fractions of infectives during an epidemic with n = 5 clusters
which agree in all parameters but the initial numbers of infectives. In the graphic on the
(a) no migration (a= 0) (b) weak migration (a= 0.0025) (c) strong migration (a= 0.25)
Figure 5.2: Evolution of the fractions of infectives in n = 5 clusters. These agree in all
parameters but the initial fractions of infectives. In particular,αj = 0.5, βj = 0.25 and Nj = 1000
forj∈ {1, . . . ,5}. The initial numbers of infectives vary from one to five percent of the population. Contacts between clusters occur according to the network matrix (5.34). There is no connection (a = 0) between clusters in Figure (a), weak contact (a = 0.0025) in Figure (b), and strong influence (a= 0.25) in Figure (c). The thick lines show the deterministic evolution, the thin lines are simulations of the diffusion process. All paths have been obtained by application of the Euler scheme with time step 0.025, see Section3.3.2.
Figure 5.3: Evolution of the fractions of infectives in n = 5 clusters between which people have contacts according to the network matrix (5.34) with a = 0.0025. The clusters agree in all parameters but the initial numbers of infectives, which vary from one to five percent of the population. As in Figure 5.2, one has αj = 0.5, βj = 0.25 and Nj = 1000 for j ∈ {1, . . . ,5}.
The thick curves show the deterministic evolution, the thin lines are stochastic simulations of the diffusion process. The dotted vertical lines indicate the instants at which the fractions of susceptibles in the deterministic course fall belowR−01. The dashed vertical lines mark the actual turning points of the deterministic course of the epidemic, that are the time instants where the fractions of infectives reach their maximums. Without contacts between clusters, these lines would agree within each community. The sample paths have been obtained by application of the Euler scheme with time step 0.025, introduced in Section 3.3.2.
very left there is no contact between clusters (a = 0), while there is strong exchange on
the right (a= 0.25). Apparently, with increasing contacts of individuals between clusters,
the courses of the epidemics synchronise. This fact is again illustrated in Figure 5.3, where the dotted vertical lines mark the instants at which the fractions of susceptibles in the deterministic course fall belowR−1
0 , while the dashed lines indicate the actual turning points
of the deterministic course of the epidemics, defined as the instants where the maximum amounts of infectives are reached. For clusters with initially high fractions of infectives, the actual turning point lies before the one that is valid for the model without exchange; for clusters with relatively few cases, the opposite situation applies.
The definition of a multitype counterpart to the basic reproductive ratioR0 in the standard
SIR model with one homogeneous population is for example discussed by Andersson
and Britton (2000) and Isham (2004). Moreover, M. Roberts and Heesterbeek (2003)
and Heesterbeek and Roberts (2007) define and analyse a type-reproduction number as an
alternative threshold quantity. This number coincides withR0 for homogeneous populations.
A possible modification of the multitype SIR model in this section is to consider movement of individuals between clusters instead of cross-infection. That means, individuals can change the cluster which they are associated with, and infection occurs only within clusters. This
5.3 Existence and Uniqueness of Solutions 109
case is for example investigated by Dargatz, Georgescu, and Held (2006). A disadvantage of that approach, however, is that population sizes of the distinct clusters do not remain constant, and the model is not immediately applicable to, for example, the case where clusters represent age groups.
In Chapter 8, the multitype SIR model is applied for modelling the spatial spread of influenza in Germany. Other models involving local and global infection dynamics are developed inHufnagel et al.(2004),Germann et al.(2006),Débarre, Bonhoeffer, and Regoes
(2007),Dybiec et al. (2009) and Ball et al.(2010). Watts et al.(2005) consider mixing on
even more than two scales.
5.3
Existence and Uniqueness of Solutions
When considering an SDE as a model for some natural phenomenon, one implicitly assumes the existence of a solution of this SDE. Section 3.2.3 specified the Lipschitz condition (3.10) under which a strong solution of an SDE exists pathwise uniquely. This solution is non- explosive when it satisfies the growth condition (3.11).
For the standard and the multitype SIR models, the Lipschitz condition is actually not fulfilled as demonstrated in Section B.2 in the appendix. Importantly, conditions (3.10) and (3.11) are sufficient but not necessary for the unique existence and non-explosiveness. Some authors describe weaker conditions, see e. g. Kloeden & Platen (1999, pages 134–135). Further references include Kushner (1972), who studies the existence of a solution of an SDE when the drift function is not Lipschitz continuous,Abundo (1991), who considers the existence of solutions for a predator-prey model, and Kusuoka (2010), who investigates the existence of densities of solutions in case the Lipschitz condition is not fulfilled. Related to this general problem, Kaneko and Nakao (1988), Marion, Mao, and Renshaw (2002) and
Berkaoui, Bossy, and Diop (2005) deal with conditions under which numerical schemes
converge to the true but unknown solution in case the Lipschitz condition is violated. Alternatively, one could settle for weak instead of strong solutions as distinguished in Section3.2.3; this requires weaker assumptions.
In this thesis, the question of the existence of a strong solution for the considered SIR models on the entire state space is not completely answered as it is not the focus of this work. For our purposes, it suffices to consider the standard and multitype SIR models on a slightly restricted state space such that all fractions of susceptible and infectious individuals are bounded from below by an arbitrarily small but fixed positive constant ε. This does
not limit the practical applicability of the diffusion models. The original state spaces C
and C(n) from Equations (5.20) and (5.33) on pages 95and 107 then become
Cε= n (s, i)0 ∈[ε,1]2∩R2 0|s+i≤1 o and C(n) ε = n (s1, . . . , sn, i1, . . . , in)0 ∈[ε,1]2n∩R20n sj +ij ≤1 for all j = 1, . . . , n o ,
respectively. This modification has the effect that the drift vectors and diffusion coefficients fulfil the Lipschitz and growth bound conditions as shown in SectionB.2, i. e. there uniquely exist non-explosive strong solutions of the SDEs on the modified state spaces.
Independently of the investigation of the existence of a solution, diffusion approximations for the SIR model are considered problematic anyway when there are only few infectious individuals (e. g. Andersson & Britton, 2000). The above proposed restriction of the state spaces of the diffusion approximations does hence not impose a serious constraint. An alternative approximation of the general stochastic epidemic during the initial and final phase of an epidemic is for example provided by Barbour (1976) andAndersson & Britton
(2000, Chapter 3.3).
5.4
Conclusion
The description of the spread of infectious diseases in terms of diffusion processes enables convenient simulation of the random course of an epidemic even for large populations. In this chapter, diffusion approximations for the standard SIR model and a multitype extension were derived. On the one hand, these served as illustrations for the theoretical investigations in Chapter 4. On the other hand, the present chapter provides the basis for Chapter 8, where an influenza outbreak in a boarding school and the geographical spread of influenza in Germany are statistically analysed. Another application of diffusion approximations in life sciences is presented in Chapter 9. There, the in vivo binding behaviour of proteins is investigated as an example from molecular biology.
When applying the multitype SIR model in practice, several difficulties arise: First of all, one will typically want to prespecify the network matrices γN,γS and γI, or at least
supply some information on their structure. That requires knowledge about, for example, transportation or social networks, depending on the definition of the clusters. In Chapter 8, commuter data from Germany is taken in order to estimate the geographical dispersal of the population. References for further examples for the utilisation of transportation networks are given in that chapter. Social contact networks may, for example, be approximated by the evaluation of contact diaries of similar surveys (Edmunds et al., 1997,2006, Beutels et
al.,2006, Wallinga et al., 2006, Mossong et al., 2008).
Another issue concerns the data about disease counts which is most often incomplete as many cases are not reported. In general, one also does not know the exact times at which infections occurred, and data is aggregated over periods of time. This is, of course, also problematic in case of one homogeneous population, but worsens in case of multiple communities. For example, Uphoff et al. (2004) summarise several difficulties arising from data aggregation over large geographical areas, ranging from dissimilar consultation behaviour to differences in physicians’ opening hours, which limit the comparability of disease counts in distinct regions. These examples represent only some out of many challenges which epidemiologists are facing. Dealing with them is the subject of active research.
Chapter 6
Parametric Inference for
Discretely-observed Diffusions
As we have seen in Chapter 3, diffusion processes provide a widely-used and powerful modelling tool, and their mathematics is well understood. Chapter 4 described how to construct a diffusion approximation to a given stochastic phenomenon. This diffusion model is then known in parametric form. In practice, one usually wishes to furthermore estimate the parameters of this model. Statistical inference for diffusion processes, however, is a challenging problem. Difficulties arise from the fact that observations are typically discrete while the underlying diffusion model is continuous in time. In case of time-discrete observations, the likelihood function for the model parameters is generally unknown, and hence maximum likelihood estimation is not immediately possible.
This chapter provides a review on more sophisticated approaches to parametric inference for discretely-observed diffusion processes. The literature already provides a variety of different estimation techniques, but this subject is also still a highly developing research area. The present chapter concentrates on frequentist methodology and serves as an overview and introduction to statistical inference for diffusions. The emphasis of this thesis, however, lies on Bayesian techniques, which show even more attractive characteristics. These are presented and further developed in Chapter 7.
Throughout this chapter, we consider the time-homogeneous Itô diffusion X = (Xt)t≥0 sat-
isfying the stochastic differential equation
dXt=µ(Xt,θ)dt+σ(Xt,θ)dBt , Xt0 =x0, (6.1)
with state space X ⊆ Rd, starting value x
0 ∈ X at time t0 = 0 and m-dimensional
standard Brownian motion B= (Bt)t≥0. The drift function µ:X ×Θ→Rd and diffusion
coefficient σ : X ×Θ → Rd×m are assumed to be known in a parametric form. The
statistical estimation of the possibly vector-valued parameter θ from an open set Θ⊆Rp is
We assume that µ, σ and the diffusion matrix Σ = σσ0 fulfil the regularity conditions
stated in Section 3.4 for all θ ∈ Θ; in particular, it is provided that an almost surely
pathwise unique solution of the differential equation (6.1) exists for all parameters on a respective filtered probability space (Ω,F∗,F,
Pθ), cf. Section 3.2.3. The state space X
is the same for all values of θ. The true parameter value is denoted by θ0 ∈Θ, and Eθ
and Varθ stand for the expectation and variance with respect to Pθ, respectively. For some
estimation approaches it is furthermore required that the diffusion process is ergodic. Such assumptions are indicated in the respective sections. Observations of the diffusion path are always considered to be measured without error.
This chapter is organised as follows: In order to provide the theoretical background, Section 6.1 starts with the formulation of the estimation problem for continuous-time observations and then goes over to discrete time under the assumption that the likelihood function of the parameter is known. Both scenarios are not directly applicable in practice. Section 6.2 hence presents a first attempt to obtain a feasible approximate maximum likelihood estimator. This approach, however, leads to asymptotically biased estimators. The remaining techniques covered in this chapter are more elaborate. They are grouped into three categories, in particular into approximations of the likelihood function in Section 6.3, alternatives to maximum likelihood estimation in Section 6.4 and a recent approach called the Exact Algorithm in Section 6.5. A comparison of the presented estimation techniques by means of a simulation study is beyond the scope of this thesis. However, a discussion follows in Section 6.6 including a summary and references to evaluation studies from the literature.
Other surveys on inference for discretely-observed diffusion processes are given by Prakasa
Rao (1999), Nielsen, Madsen, and Young (2000), H. Sørensen (2004), Jimenez, Biscay, and
Ozaki (2006), Hurn, Jeisman, and Lindsay (2007) andIacus(2008). None of these, however,
covers all approaches described in this chapter. Furthermore, whenever an estimation technique is formulated for multi-dimensional diffusion processes in the original work, or the extension to multi-dimensional diffusions is obvious, this chapter presents the more general multi-dimensional case. Observation times are assumed non-equidistant even though the simpler equidistant setting is common in the original literature. The present review is thus more general with respect to these two points than the above mentioned surveys (apart
from Prakasa Rao, 1999, Nielsen et al.,2000, and Jimenez et al., 2006, who consider multi-
dimensional processes as well). Overall, the emphasis of this chapter is on the presentation of ideas and not on technical detail. For the latter, the reader is referred to the references given along the way.
The present review omits nonparametric inference. References for this topic include
Florens-Zmirou (1993), Aït-Sahalia (1996), Jiang and Knight(1997),Soulier (1998),Jacod
(2000), Hurn, Lindsay, and Martin(2003),Nicolau (2003) andComte, Genon-Catalot, and
Rozenholc(2007). An introduction to the subject is given in Iacus(2008, Chapter 4.2), a
6.1 Preliminaries 113
6.1
Preliminaries
Crucially different situations occur depending on whether a diffusion process is observed continuously or discretely in time. Time-continuous observation is obviously impossible in practical applications. Still, the corresponding well-established theory is discussed in Section 6.1.1 for the sake of completeness and further understanding of subsequent asymptotic considerations. It forms the basis for the investigations in Section7.3in the next chapter, for example. In real data situations, one naturally has to deal with time-discrete observations. Section 6.1.2 briefly presents the challenges of parameter estimation for this setting. This is the starting point for the remainder of this chapter. Finally, Section 6.1.3
specifies the data situation which is considered in subsequent sections.