Metapopulation models of disease spread

Transcription

1 Metapopulation models of disease spread Julien Arino Department of Mathematics University of Manitoba Winnipeg, Manitoba, Canada Centre for Inner City Health Saint Michael s Hospital Toronto, Ontario, Canada arinoj@cc.umanitoba.ca

2 What is different between these diseases? Countries Reporting Wild Polio in 2006 and Routes of Viral Spread NEJM 2006;355:2508

3 Train travel (from Paris)

4 Air travel (from YVR in 2007)

5 Metapopulations A metapopulation is a population of populations. Constituting populations are often called patches (here cities). A patch is a unit within which the population is considered homogeneous, or a geographical location: neighborhood city region country Patches may or may not overlap, may or may not be adjacent. They are connected through movement (travel) of species between them

6 Theoretical setting n cities, vertices in a graph G, where individuals live Edges of G represent the possibility for individuals to move between two cities Edges are oriented (=arcs): movement is not always symmetric Deduce a network: arcs are valued by a measure of the rate of flow along them In each city: system describing the dynamics of the disease in the population of that city. For example, SI, SIR, SEIR, deterministic (ODE, DDE, PDE), stochastic, hybrid

7 Spatial spread of SARS We have information on trans-border movements of SARS (WHO data dates from end of epidemic): investigation with each of the 27 countries that had reported SARS cases in the WHO data (26 have confirmed cases) obtain classification of cases as locally generated or imported in case of importation, POO, POE, DAE 143 imported cases to 41 airports outside of China By comparison, 9 cases in total crossed borders by land, 8 in Mongolia and 1 in Russia

8 Simplifying assumptions Start with SEIR in each city City population is N p Incidence is proportional (or standard), Φ p = β p S p I p N p Want to model short term effect of disease, and are interested by initial apparition of cases, so S p N p As a consequence, incidence is Φ p = β p I p Since S p N p and N p constant, consider S p as fixed Short term behavior wanted, so also neglect R p

9 Continuous-time Markov chain Process X (t) = (E 1 (t), I 1 (t),..., E p (t), I p (t)) is a continuous-time, discrete-space Markov chain Times to transition follow an exponential distribution with parameter f := p j=1 (ε j e j + (β j + γ j )i j ) + p j,k=1, k j ( ) mjk E e j + mjk I i j (1) when system is in state (e, i) := (e 1, i 1,..., e p, i p )

10 The process then jumps to state (e, i ) := (e 1, i 1,..., e p, i p) with P (e,i) (e,i ) = β ji j if (e, i ) = (e 1, i 1,..., e j + 1, i j,..., e p, i p ) f for some j P (e,i) (e,i ) = γ ji j if (e, i ) = (e 1, i 1,..., e j, i j 1,..., e p, i p ) f for some j P (e,i) (e,i ) = ε je j if, for some j, f (e, i ) = (e 1, i 1,..., e j 1, i j + 1,..., e p, i p ) P (e,i) (e,i ) = me jk e j if, for some j and some k, f (e, i ) = (e 1, i 1,..., e j 1, i j,..., e k + 1, i k,..., e p, i p ) P (e,i) (e,i ) = mi jk i j if, for some j and some k, f (e, i ) = (e 1, i 1,..., e j, i j 1,..., e k, i k + 1,..., e p, i p ) P (e,i) (e,i ) = 0 in any other case

11 Connection and flight intensity data Data can take the form depcity,arrcity,traveltime,minstops,directvolume YWG,PAR,10:15,1, YWG,XCH,31:35,4, YWG,YEA,2:04,0, YWG,YQT,1:25,0,76359 YWG,YTO,2:21,0, YWG,YVR,3:10,0, YWG,YYC,2:09,0, (PAR = CDG+ORY+LBG) (Christmas Island) (YEA = YEG+YXD) (Thunder Bay) (YTO = YTZ+YYZ) (Vancouver) (Calgary) and = 983, 065 more lines.. So we know if two cities are connected, and if they are, how many seats (here, per year) between the two cities.

12 In the following, we use The movement matrix m ij = s ij 992 s ij i=1 proportion of seats out of airport i that go to airport j, where s ij number of seats available for flights from airport i to airport j, 0 if there is no direct flight Take YWG to YYC, for example: m YWG,YYC = s YWG,YYC 992 s YWG,i i=1 =

13 Expected values E j := E(E j (t)) and I j := E(I j (t)) verify for all j I the deterministic model d dt E j = β j I j ε j E j + d dt I j = ε j E j γ j I j + p k=1,k j p k=1,k j ( ) mkj E E j mjk E E k ( ) mkj I I j mjk I I k (2a) (2b)