Stochastic ling of the Spatial Spread of Influenza in Germany, Leonhard Held Department of Statistics Ludwig-Maximilians-University Munich Financial support by the German Research Foundation (DFG), SFB 386 Hannover, November 2005
Influenza Worldwide one of the most common and severe infectious diseases. Major epidemics and pandemics of the 20th century: Spanish Flu (1918-20), Asian Flu (1957-58), Hong Kong Flu (1969) Annual number of deaths caused by influenza in Germany is twice as high as those caused by road accidents, nevertheless low vaccination rates. Steadily new antigen mutants of the influenza virus coming up. In the opinion of experts the next pandemic is just a question of time (caused e.g. by avian flu). Emergency plans are essential.
Motivation Mathematical modelling of the spread of epidemics dates bac to 1760 (Daniel Bernoulli) and has been well developed since then. Globalization has induced new era of epidemiology: People travel more frequently, faster and further than in former times. Hufnagel et al. (2004) have successfully simulated the world-wide spatial spread of the SARS epidemic in 2003 using a spatial SIR model based on global air traffic data.
Outline 1 2 3 4 5
Standard SIR Divide population of size N into susceptible (S), infected (I ), and removed (R) individuals. Transitions: S + I α 2I, I β R α: contact rate of an infectious individual sufficient to spread the disease, β: reciprocal average infectious period. Infection dynamics: ds/dt = αsj, dj/dt = αsj βj, (1) s = S/N, j = I /N, r = R/N = 1 s j.
Standard SIR (cont.) Crucial parameter: the basic reproduction number ρ = α/β, the average number of persons directly infected by an infectious case during its entire infectious period after entering a totally susceptible population. ρ 1 > s(0): no epidemic will occur ρ 1 > s(t): epidemic decays, i.e. major epidemic will occur when, in the early stages of an outbrea, each infective on average produces more than one further infective.
Standard SIR (cont.) Evolution of proportions of susceptible (solid), infected (dashed), and removed (dotted) individuals in the standard SIR model; ρ = 1.5.
Stochastic SIR Infection and recovery processes are of rather stochastic than deterministic character. Write (1) in terms of Langevin equations: ds = αsj + 1 αsj ξ1 (t) dt N dj = αsj βj 1 αsj ξ1 (t) + 1 βj ξ2 (t), dt N N where ξ 1 (t) and ξ 2 (t) are independent Gaussian white noise forces, modelling fluctuations in disease transmission and recovery (Hufnagel et al., 2004).
Spatial SIR Problem: Assumption of homogeneous mixing is not given in our fully connected world anymore! Idea: Introduce networ of subregions i = 1,..., n of sizes N i. Local infection dynamics within a subregion is given by stochastic SIR model as before: S i + I i α β 2I i, I i R i. Global dispersal between nodes of networ is rated in a connectivity matrix γ = (γ ij ) ij : S i γ ij S j, I i γ ij I j.
Spatial SIR model (cont.) The system of stochastic differential equations now changes to ds i dt = αs i j i X γ i s i + X s + p 1 X Ni γ i s + p 1 p αsi j i ξ (i) 1 (t) Ni s γ i s i ξ (i) 4 (t) p 1 X Ni γ i s ξ (i) 5 (t) dj i dt = αs i j i βj i X γ i j i + X γ i j p 1 p αsi j i ξ (i) 1 (t) + 1 p p βji ξ (i) 2 (t) Ni Ni s + p 1 X Ni s γ i j i ξ (i) 4 (t) p 1 X Ni γ i j ξ (i) 5 (t) dr i dt = βj i p 1 p βji ξ (i) 2 (t). Ni for i = 1,..., n, where ξ 1 (t), ξ 2 (t), ξ 4 (t), and ξ 5 (t) denote independent vector-valued white noise forces standing for fluctuations in transmission, recovery, and outbound and inbound traffic, respectively.
Keeping the System Closed Area of n regions is assumed to be closed. i.e. we have to require n i=1 ( dsi dt + dj i dt + dr ) i = 0. dt Introduce a wea form of dependence to the white noise forces such that the above equality holds almost surely.
Numerical Scheme Define functions a j and b j, 1 j 3, 1 5, such that ds i ( t ) = a1 ( t, si (t) ) dt + dj i ( t ) = a2 ( t, ji (t) ) dt + dr i ( t ) = a3 ( t, ri (t) ) dt + 5 =1 5 =1 5 =1 ( b 1 t, si (t) ) dw (i) (t) ( b 2 t, ji (t) ) dw (i) (t) ( b 3 t, ri (t) ) dw (i) (t).
Numerical Scheme (cont.) For example, for i = 1,..., n: ( ds i = αs i j i γ i s i + dt + 1 αsi j i ξ (i) 1 (t)dt Ni }{{}}{{} :=b :=a 1(t,s i (t)) 11(t,s i (t)) + 1 γ i s i Ni }{{} :=b 14(t,s i (t)) γ i s ) ξ (i) 4 (t)dt 1 Ni γ i s } {{ } :=b 15(t,s i (t)) b 12 (t, s i (t)) = b 13 (t, s i (t)) = 0, ξ (i) (i) (t)dt = dw (t). ξ (i) 5 (t)dt,
Numerical Scheme (cont.) Apply Euler-Maruyama approximation scheme to numerically solve the system of SDEs at discrete, equidistant instants 0, δ, 2δ,... in the time domain: s i mδ = s i (m 1)δ + a 1 (m 1)δ, s i ((m 1)δ) δ + j i mδ = j i (m 1)δ + a 2 (m 1)δ, j i ((m 1)δ) δ + r i mδ = r i (m 1)δ + a 3 (m 1)δ, r i ((m 1)δ) δ + 5X b 1 =1 5X b 2 =1 5X b 3 =1 (m 1)δ, s i ((m 1)δ) W (i) (m) (m 1)δ, j i ((m 1)δ) W (i) (m) (m 1)δ, r i ((m 1)δ) W (i) (m) for m 1, i = 1,..., n and W (i) (i) (i) (m) := W (mδ) W ((m 1)δ).
Application: Influenza in Germany Consider the 438 districts of Germany. Data about incidences of influenza taen from Robert Koch Institute. Build up connectivity matrix to describe the strength between parts of Germany, considering dispersal between adjacent regions, caused e.g. by commuters (info from Federal Statistical Office Germany), domestic train traffic (ICE), domestic air traffic (www.oagflights.com), where each of these components is provided with a weight regulating its influence.
Parameter choice We assume that the basic reproduction number ρ = α/β depends on population density: ρ(d i ) = 1.0179 + 10 5 d i, where d i is the population density of region i. (Disease is more liely to spread in areas with high population densities.) Infectious period of influenza: 4-5 days, hence we choose β = 2/9. We obtain the contact rates α i via β ρ(d i ), i = 1,..., n.
1 1: Starting values based on surveillance data from wee 5/2005. Animation shows proportion of infectives and time trend. Surprisingly good agreement with actual course of the influenza epidemic 2005. 2 2: Artificial starting values in three districts. based on three different connectivity matrices: 1 local, train and air, 2 local and train, 3 only local.
Spatial extension of classical SIR model. Implementation of certain sum-to-zero contraints and numerical approximation. Parameter choice based on external nowledge. of the spread of an influenza epidemic in Germany.
Construction Sites Prevalence data of influenza is highly underreported. Simulate underreporting. Other data sources (Sentinella). Consider different diseases. does not tae into account important factors lie weather/temperature, seasonal holiday travel, measures undertaen to lower transmission rate.
Future Wor Main purposes will be finding surveillance strategies in case of a sudden outbrea of an epidemic (isolation, vaccination, observation of migration), more formal statistical inference on model parameters based on available data from surveillance databases.
Dargatz, C., Georgescu, V. and Held, L. (2005) Stochastic ling of the Spatial Spread of Influenza in Germany. Technical Report, Munich University. Hufnagel, L., Brocmann, D. and Geisel, T. (2004). Forecast and Control of Epidemics in a Globalized World. Proceedings of the National Academy of Sciences, 101, 15124-15129.