with multiple populations Gianluca Campanella Imperial College London Isabel C. M. Nat´ ario Ricardo A. M. Pereira Universidade Nova de Lisboa Universit` a degli Studi di Trento VI International Workshop on Spatio-Temporal Modelling Guimar˜ aes, Portugal · September 12, 2012
model of Kermack and McKendrick considers a closed population whose members are subdivided into three states: Susceptible Infectious Recovered S I R β γ β Infection rate γ Recovery rate (can be estimated clinically)
states (susceptible, infectious and recovered) 3. Two parameters (infection rate β and recovery rate γ) Meta-population models 1. Straightforward extension of compartmental models to multiple populations 2. Each population is modelled (almost) independently 3. Weak coupling provided by transportation network
belong to the same population 2. the disease reaches other populations by means of travels along the transportation network Main idea The time of arrival of the ﬁrst infectious individual at each population depends on the parameters β and γ.
t ≥ 0} is said to be a non-homogeneous Poisson process with time-dependent rate λ(t) ≥ 0, t ≥ 0, if the following conditions hold, 1. N(0) = 0, 2. {N(t), t ≥ 0} has independent increments, 3. Pr[N(t + h) − N(t) = 1] = λ(t) h + o(h), 4. Pr[N(t + h) − N(t) ≥ 2] = o(h), where o(h) is such that limh→ 0 o(h)/h = 0.
representing the time of the ﬁrst arrival in a non-homogeneous Poisson process {N(t), t ≥ 0} with time-dependent rate λ(t), and deﬁne Λ(t) = t 0 λ(τ) dτ. Distribution of the time of the ﬁrst arrival F(t) = Pr[N(t) ≥ 1] = 1 − Pr[N(t) = 0] = 1 − e−Λ(t) f(t) = dF dt = λ(t) e−Λ(t)
The arrival process of infectious individuals at population i is a non-homogeneous Poisson process, since: 1. initially, no infectious individuals are present; 2. travels (and thus increments) to i are independent; 3. the probability of arrival during h depends on the number of infectious individuals in neighbouring populations multiplied by the corresponding travel probabilities; 4. the probability of more than one arrival during h is negligible. Density of the time of the ﬁrst arrival at i fi (t | β, γ) = λi (t) e−Λi(t), λi (t) = j∈ N(i) pji Ij (t | β, γ)
2. Times of arrival of ﬁrst infectious individuals depend on model parameters β and γ Formalization 1. Arrivals at each population are non-homogeneous Poisson processes 2. The number of infectious individuals in neighbouring populations can be simulated, or approximated using the corresponding deterministic formulation
Portugal subregions (NUTS 3 classiﬁcation) • Resident and commuting population information provided by INE • Infection starts in Ave Norte Centro Lisboa Alentejo Algarve 0 50 100 150 200 250
continental Portugal subregions (NUTS 3 classiﬁcation) • Resident and commuting population information provided by INE • Infection starts in Grande Lisboa Norte Centro Lisboa Alentejo Algarve 0 100 200 300
multiple populations 2. Estimation of model parameters (especially the infection rate β) is still crucial Estimation method 1. Times of arrival of ﬁrst infectious individuals depend on model parameters β and γ 2. Our method seems to work well in a range of simulations using synthetic and real-world transportation networks 3. Only a few arrival times are needed for estimation, which can thus be carried out as the disease spreads
the 2009 pandemic ﬂu outbreak in Portugal (provided by Instituto Nacional de Sa´ ude P´ ublica) 2. Assessment of the eﬀect of travel restrictions 3. Sensitivity analysis for: • primary case in neighbouring populations • diﬀerent recovery rate γ • delays in recording of arrival times 4. Determination of the most probable transmission paths
Ave Minho−Lima Cávado Ave Grande Porto Tâmega Entre Douro e Vouga Douro Alto Trás−os−Montes Baixo Vouga Baixo Mondego Pinhal Litoral Pinhal Interior Norte Dão−Lafões Pinhal Interior Sul Serra da Estrela Beira Interior Norte Beira Interior Sul Cova da Beira Oeste Médio Tejo Grande Lisboa Península de Setúbal Alentejo Litoral Alto Alentejo Alentejo Central Baixo Alentejo Lezíria do Tejo Algarve 0.00 0.05 0.10 0.15 0.20
Let Tij be the time of travel of the ﬁrst infective individual from population i to population j. Distribution of the time of the ﬁrst arrival at i Fi (t) = Pr[Ti ≤ t] = Pr min j∈ N(i) Tji ≤ t = 1 − Pr j∈ N(i) Tji > t ≈ 1 − j∈ N(i) [1 − Fji (t)] = 1 − j∈ N(i) e−Λji(t) = 1 − exp − j∈ N(i) Λji (t)
induce a directed acyclic graph of possible transmissions between populations. Time-constrained neighbourhood of population i Nt (i) = N(i) ∩ {j | tj < ti } In a complete graph, the time-constrained neighbourhood of the ith infected population will have cardinality at most i − 1. Thus, the number of paths in this graph is bounded above by n i=2 (i − 1) = (n − 1)! = Γ(n).
Lisboa Minho−Lima Cávado Ave Grande Porto Tâmega Entre Douro e Vouga Douro Alto Trás−os−Montes Baixo Vouga Baixo Mondego Pinhal Litoral Pinhal Interior Norte Dão−Lafões Pinhal Interior Sul Serra da Estrela Beira Interior Norte Beira Interior Sul Cova da Beira Oeste Médio Tejo Grande Lisboa Península de Setúbal Alentejo Litoral Alto Alentejo Alentejo Central Baixo Alentejo Lezíria do Tejo Algarve 0.0 0.1 0.2 0.3 0.4