Introduction Model Implementation and Initialization Simulation Conclusion References Stochastic Modelling of the Spatial Spread of Influenza in Germany Christiane Dargatz, Leonhard Held Department of Statistics Ludwig-Maximilians-University Munich Financial support by the German Research Foundation (DFG), SFB 386 Hannover, November 2005 Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References 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. Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Motivation Mathematical modelling of the spread of epidemics dates back 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. Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Outline Introduction 1 Model 2 Implementation and Initialization 3 Simulation 4 Conclusion 5 Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Standard SIR Model Divide population of size N into susceptible ( S ), infected ( I ), and removed ( R ) individuals. Transitions: β α S + I → 2 I , − I − → R α : contact rate of an infectious individual sufficient to spread the disease, β : reciprocal average infectious period. Infection dynamics: ds / dt = − α sj , dj / dt = α sj − β j , (1) s = S / N , j = I / N , r = R / N = 1 − s − j . Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Standard SIR Model (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 outbreak, each infective on average produces more than one further infective. Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Standard SIR Model (cont.) Evolution of proportions of susceptible (solid), infected (dashed), and removed (dotted) individuals in the standard SIR model; ρ = 1 . 5. Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Stochastic SIR Model Infection and recovery processes are of rather stochastic than deterministic character. ⇒ Write (1) in terms of Langevin equations: 1 ds � dt = − α sj + √ α sj ξ 1 ( t ) N 1 1 dj � � dt = α sj − β j − √ α sj ξ 1 ( t ) + √ β j ξ 2 ( t ), 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). Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Spatial SIR Model Problem: Assumption of homogeneous mixing is not given in our fully connected world anymore! Idea: Introduce network of subregions i = 1 , . . . , n of sizes N i . Local infection dynamics within a subregion is given by stochastic SIR model as before: β α S i + I i → 2 I i , → R i . − I i − Global dispersal between nodes of network is rated in a connectivity matrix γ = ( γ ij ) ij : γ ij γ ij S i − → S j , I i − → I j . Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Spatial SIR model (cont.) The system of stochastic differential equations now changes to ds i 1 α s i j i ξ ( i ) ❳ ❳ ♣ = − α s i j i − γ ik s i + γ ki s k + 1 ( t ) ♣ dt N i k k 1 1 s❳ s❳ γ ik s i ξ ( i ) γ ki s k ξ ( i ) + 4 ( t ) − 5 ( t ) ♣ ♣ N i N i k k dj i 1 1 α s i j i ξ ( i ) β j i ξ ( i ) ❳ ❳ ♣ ♣ = α s i j i − β j i − γ ik j i + γ ki j k − 1 ( t ) + 2 ( t ) ♣ ♣ dt N i N i k k 1 1 s❳ s❳ γ ik j i ξ ( i ) γ ki j k ξ ( i ) + 4 ( t ) − 5 ( t ) ♣ ♣ N i N i k k dr i 1 β j i ξ ( i ) ♣ = β j i − 2 ( t ). ♣ dt N i 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. Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Keeping the System Closed Area of n regions is assumed to be closed. i.e. we have to require n � ds i � dt + dj i dt + dr i � = 0 . dt i =1 ⇒ Introduce a weak form of dependence to the white noise forces such that the above equality holds almost surely. Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Numerical Scheme Define functions a j and b jk , 1 ≤ j ≤ 3, 1 ≤ k ≤ 5, such that 5 � � � � � � � dW ( i ) ds i t = a 1 t , s i ( t ) dt + b 1 k t , s i ( t ) k ( t ) k =1 5 � � � � � � � dW ( i ) = a 2 t , j i ( t ) dt + t , j i ( t ) k ( t ) dj i t b 2 k k =1 5 � � � � � � � dW ( i ) dr i t = a 3 t , r i ( t ) dt + b 3 k t , r i ( t ) k ( t ). k =1 Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Numerical Scheme (cont.) For example, for i = 1 , . . . , n : � � 1 � � � ξ ( i ) ds i = − α s i j i − γ ik s i + γ ki s k dt + √ N i α s i j i 1 ( t ) dt k k � �� � � �� � := b 11 ( t , s i ( t )) := a 1 ( t , s i ( t )) �� �� 1 4 ( t ) dt − 1 ξ ( i ) ξ ( i ) + √ N i γ ik s i √ N i γ ki s k 5 ( t ) dt , k k � �� � � �� � := b 14 ( t , s i ( t )) := b 15 ( t , s i ( t )) b 12 ( t , s i ( t )) = b 13 ( t , s i ( t )) = 0, ξ ( i ) k ( t ) dt = dW ( i ) k ( t ). Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Numerical Scheme (cont.) Apply Euler-Maruyama approximation scheme to numerically solve the system of SDEs at discrete, equidistant instants 0 , δ, 2 δ, . . . in the time domain: 5 △ W ( i ) ❳ � ✁ � ✁ � ✁ � ✁ s i m δ = s i ( m − 1) δ + a 1 ( m − 1) δ, s i (( m − 1) δ ) δ + b 1 k ( m − 1) δ, s i (( m − 1) δ ) k ( m ) k =1 5 △ W ( i ) � ✁ � ✁ � ✁ ❳ � ✁ j i m δ = j i ( m − 1) δ + a 2 ( m − 1) δ, j i (( m − 1) δ ) δ + b 2 k ( m − 1) δ, j i (( m − 1) δ ) k ( m ) k =1 5 △ W ( i ) ❳ � ✁ = r i � ( m − 1) δ ✁ + a 3 � ( m − 1) δ, r i (( m − 1) δ ) ✁ δ + � ( m − 1) δ, r i (( m − 1) δ ) ✁ k ( m ) r i m δ b 3 k k =1 for m ≥ 1, i = 1 , . . . , n and △ W ( i ) k ( m ) := W ( i ) k ( m δ ) − W ( i ) k (( m − 1) δ ). Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Application: Influenza in Germany Consider the 438 districts of Germany. Data about incidences of influenza taken 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. Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References 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 likely 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 . Christiane Dargatz
Introduction Model Implementation and Initialization Simulation Conclusion References Simulation 1 Simulation 1: Starting values based on surveillance data from week 5/2005. Animation shows proportion of infectives and time trend. Surprisingly good agreement with actual course of the influenza epidemic 2005. 2 Simulation 2: Artificial starting values in three districts. Simulation based on three different connectivity matrices: local, train and air, 1 local and train, 2 only local. 3 Christiane Dargatz
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