Web-based Supplementary Materials for A space-time conditional intensity model for invasive meningococcal disease occurence by Sebastian Meyer 1,2, Johannes Elias 3, and Michael Höhle 4,2 1 Department of Psychiatry and Psychotherapy, Ludwig-Maximilians-Universität, München, Germany 2 Department of Statistics, Ludwig-Maximilians-Universität, München, Germany 3 German Reference Centre for Meningococci, University of Würzburg, Würzburg, Germany 4 Department for Infectious Disease Epidemiology, Robert Koch Institute, Berlin, Germany August 3, 211
Web Appendix A: Calculus of the Score Function Let ϑ denote any subvector of θ. Then, the partial derivative of the log-likelihood with respect to ϑ is s ϑ (θ) := ϑ l(θ) = n i=1 ϑ λ θ(t i, s i, κ i ) T λ θ (t i, s i, κ i ) W ϑ λ θ(t, s, κ) dt ds, (1) and the score function is s(θ) = l(θ) = ( ) s θ β, s β, s γ, s σ, s α (θ). The necessary partial derivatives of the CIF with their respective time-space-mark integrals are given in the following subsections and can be plugged into the equation (1). The analytic derivatives of the interaction function f and g with respect to σ and α, respectively, have to be determined for the specific model at hand. For instance, a type-specific spatial Gaussian kernel f σ (s κ) = exp ( s 2 2σ 2 κ ) with σ = (σ 1,..., σ K ) has partial derivatives ( ) f σ (s κ) = 1 k=κ (κ) exp s 2 s 2, for any k K. σ k 2σk 2 σk 3 The type-specific temporal interaction function g α (t κ) = e ακt with α = (α 1,..., α K ) has partial derivatives integral of α k g α (t κ) = 1 k=κ (κ) ( t e α k t ), for any k K. While the σ κ f σ (s κ) over the region R j will be approximated by numerical integration, the temporal function α κ g α (t κ) is assumed to permit analytical integration. 1
Endemic intercept(s) β : Let β,k, k {1,..., K} be one of the type-specific intercepts in β. Then, ) λ β θ(t, s, κ) = 1 k=κ (κ) exp (β,k + o ξ(s) + β z τ(t),ξ(s),k since the parameter β,k appears in the endemic component h θ (t, s, κ) if and only if κ = k. The corresponding integrated value is T W β,k λ θ(t, s, κ) dt ds = e β,k D M C τ A ξ exp (o ξ + β z τ,ξ ), τ=1 ξ=1 cf. the integral of the endemic component in equation (6) of the paper. If the model assumes a type-invariant endemic intercept β = β, then ) λ β θ(t, s, κ) = exp (β + o ξ(s) + β z τ(t),ξ(s) with integrated value T W β λ θ(t, s, κ) dt ds = K e β D M C τ A ξ exp (o ξ + β z τ,ξ ). τ=1 ξ=1 Endemic covariate effects β: β λ θ(t, s, κ) = exp (β (κ) ) + o ξ(s) + β z τ(t),ξ(s) z τ(t),ξ(s) with corresponding integral vector (element-wise integral values) ( ) D M exp (β (κ)) C τ A ξ exp(o ξ + β z τ,ξ ) z τ,ξ. τ=1 ξ=1 2
Epidemic effects γ: γ λ θ(t, s, κ) = j I (t,s,κ) e γ m j g α (t t j κ j ) f σ (s s j κ j ) m j, and the corresponding integral can be deduced similar to equation (7) of the paper as [ n ][ min{t tj ;ε} ] q κj, e γ m j g α (t κ j ) dt f σ (s κ j ) ds m j. j=1 R j Parameters σ and α of the interaction functions: For a general spatial kernel f σ (s κ), σ λ θ(t, s, κ) = with corresponding integral j I (t,s,κ) e η j g α (t t j κ j ) [ n ][ min{t q κj, e η tj ;ε} j g α (t κ j ) dt j=1 R j [ ] σ f σ(s s j κ j ) ] σ f σ(s κ j ) ds. Similarly, for a general temporal kernel g α (t κ), α λ θ(t, s, κ) = j I (t,s,κ) e η j [ ] α g α(t t j κ j ) f σ (s s j κ j ) with corresponding integral [ n ][ min{t q κj, e η tj ;ε} ] j j=1 α g α(t κ j ) dt f σ (s κ j ) ds. R j 3
Web Appendix B: Fisher Information Matrix The inverse of the Fisher information matrix at the maximum likelihood estimate (MLE) ˆθ ML is in general likelihood theory used as an estimate of the variance of ˆθ ML. The precise conditions under which asymptotic properties of MLEs hold for spatio-temporal point processes have been established by Rathbun (1996). Specifically, the conditions for existence, consistence and asymptotic normality of a local maximum ˆθ ML as T for a fixed observation region W are discussed in Meyer (29, Section 4.2.3). The expected Fisher information I(θ) can be estimated by the optional variation process adapted to the marked spatio-temporal setting (Rathbun, 1996, equation (4.7)) T W K ( ) 2 θ log λ θ(t, s, κ) dn(t, s, κ) through its observed realisation ( ) n Î(θ) = i=1 θ log λ θ(t 2 n i, s i, κ i ) = θ λ θ (t 2 i, s i, κ i ) θ=θ i=1 λ θ(t i, s i, κ i ) θ=θ, where a 2 := aa for a vector a. Uncertainty of the parameter estimates is thus deduced from the diagonal of Î 1/2 (ˆθ ML ), which contains their standard errors. References Meyer, S. (29). Spatio-temporal infectious disease epidemiology based on point processes. Master s thesis, Department of Statistics, Ludwig-Maximilians-Universität, München. Available as http://epub.ub.uni-muenchen.de/1173/. Rathbun, S. L. (1996). Asymptotic properties of the maximum likelihood estimator for spatiotemporal point processes. Journal of Statistical Planning and Inference 51, 55 74. 4
Web Appendix C: Simulation Algorithm This appendix provides a more implementational view on the simulation algorithm described in Section 4 of the paper. In addition to the notation of the paper, let L(t) be the next time point after time t where any endemic covariate in any tile changes its value, or a previously infected individual stops spreading the disease, i.e.: L(t) = min { t > t ( ξ {1,..., M} : z τ(t ),ξ z τ(t),ξ ) ( j {1,..., N g (t)} : t = t j + ε) }. An implementational perspective on the simulation algorithm is then: 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 Algorithm 1: Ogata s modified thinning adapted for twinstim Given current time t, update L(t) and calculate local upper bound λ g(t); Generate proposed waiting time Exp(λ g(t)); if t + > L(t) then Let t = L(t); else Let t = t + ; Accept t with probability λ g(t)/λ g(t), otherwise goto step 1; Draw the source of infection (infective individual j or endemic) with weights equal to the respective components of λ g(t) (cf. equation (8) of the paper); if endemic source of infection then Draw the type κ of the new event with weights exp(β (κ)), κ K; Draw the tile A ξ of the new event with weights A ξ ρ τ(t),ξ e β z τ(t),ξ, ξ {1,..., M}; Draw the location s of the new event uniform within the sampled tile A ξ ; else Draw the type κ of the new event at random out of the types which can be triggered by the source individual j, i.e. draw from U({κ K : q κj,κ = 1}); Draw the relative location v of the new event (relative to the source j) from the density f(s κ j )/ R j f(s κ j ) ds on R j, i.e. s = s j + v; Draw additional marks according to the pre-specified distribution; Update the event history; Goto step 1; An R implementation of the algorithm can be found as the function simepidatacs in the popular surveillance package. 5