RD processes on heterogeneous metapopulations: Continuous-time formulation and simulations wANPE08 – December 15-17, Udine Joan Saldaña Universitat de Girona

wANPE08 - Udine Outline of the talk 1. Introduction 1. SIS model with homogeneous mixing 2. Epidemic models on contact networks 1. Regular (homogeneous) random networks 2. Complex random networks 3. EM on complex metapopulations 1. Discrete-time diffusion 2. Continuous-time diffusion

wANPE08 - Udine SIS model The force of infection λ = rate at which susceptible individuals become infected  Proportional to the number of infective contacts µ = recovery rate

wANPE08 - Udine Homogeneous mixing Any infective is equally likely to transmit the disease to any susceptible λ = transmission rate across an infective contact x contact rate x proportion of infective contacts = β · c · I / N If c ≈ N → λ ≈ β · I (non-saturated) If c ≈ 1 → λ ≈ β · I / N (saturated) ( c is the average rate at which new contacts are made and can take into account other aspects like duration of a contact, etc.)

wANPE08 - Udine Basic reproductive number = Average number of infections produced by an infective individual in a wholly susceptible population = c ·β ·T = c ·β ·1/μ In a non-homogeneous mixing, c ~ structure of the contact network → Consider the probability of arriving at an infected individual across a contact instead of considering the fraction of infected individuals !!

wANPE08 - Udine Contact network epidemiology What are the implications of network topology for epidemic dynamics? (May 2001; Newmann 2002; Keeling et al. 1999, 2005; Cross et al. 2005, 2007; Pastor-Satorras & Vespignani 2001, …; Lloyd-Smith et al. 2005; etc. )

wANPE08 - Udine Contact network epidemiology (Meyers et al. JTB 2005)

Meyers et al. JTB 2005

wANPE08 - Udine Complex contact networks The contact structure in the population is given by  the degree (or connectivity) distribution P(k)  the conditional probability P(k’|k) If these two probabilities fully determine the contact structure → Markovian networks

wANPE08 - Udine Degree distributions Poisson: non-growing random networks Exponential: growing networks with new nodes randomly attached without preference Scale free (power law): preferential growing networks → existence of highly connected nodes (= superspreaders)

wANPE08 - Udine Network architectures Meyers et al. JTB 2005

wANPE08 - Udine A special degree distribution Distribution of the degrees of nodes reached by following a randomly chosen link: which has / as expected value. This is the value to be considered for c !!

wANPE08 - Udine for contact networks For this value of c, we have (Anderson & May 1991; Lloyd & May 2001; May & Lloyd 2001) (Pastor-Satorras & Vespignani 2001, Newmann 2002) For regular random networks, CV = 0 and hence Absence of epidemic threshold in SF networks!!

wANPE08 - Udine Epidemics on metapopulations Schematically (Colizza, Pastor-Satorras & Vespignani, Nature Physics 2007):

wANPE08 - Udine An example

wANPE08 - Udine More modern examples

wANPE08 - Udine A nice picture of the 1 st example

wANPE08 - Udine Modern examples (Colizza et al., PNAS 2006)

wANPE08 - Udine Global invasion threshold is not sufficient to predict the invasion success at the metapopulation level with small local population sizes (Ball et al. 1997; Cross et al. 2005, 2007)  Disease still needs to spread to different populations = number of subpopulations that become infected from a single initially infected population  Size of the local population ( N ),  Rate of diffusion among populations (D)  the length of the infectious period (1/μ ). ( Cross et al. 2005, 2007)

wANPE08 - Udine An alternative approach Consider a complex metapopulation as a structured population of nodes classified by their connectivity (degree) Include local population dynamics in each node Forget about the geographical location of nodes and consider only the topological aspects of the network

wANPE08 - Udine Global invasion threshold

wANPE08 - Udine Global invasion threshold In a regular random network with Similar expressions can be derived for complex metapopulations. For instance, if D = const, (Colizza & Vespignani Phys.Rev.Lett. 2007, JTB 2008)

wANPE08 - Udine A discrete-time model

wANPE08 - Udine Assumptions The spread of a disease is assumed to be two sequential (alternate) processes: 1) Reaction (to become infected or to recover) Homogeneous mixing at the population level 2) Diffusion: A fixed fraction of individuals migrate at the end of each time interval (after react !!)

wANPE08 - Udine Transmission rates In type-I (non-saturated) spreading: In type-II (saturated) spreading:

wANPE08 - Udine The discrete equations Susceptible individuals: Infected individuals: Diffusion at the end of the time interval

wANPE08 - Udine The continuous equations Taking the approximation dρ/dt ≈ ρ(t + 1) – ρ(t) it follows: (Colizza et al., Nature Physics 2007)

wANPE08 - Udine For sequential Type-I processes (Colizza et al., Nature Physics 2007) The number of infectives and susceptibles are linear in the node degree k → Diffusion effect Constant prevalence across the metapopulation Lack of epidemic threshold in SF networks

wANPE08 - Udine For sequential Type-II processes (Colizza et al., Nature Physics 2007) The number of infectives and susceptibles are linear in the node degree k → Strong diffusion effect Constant prevalence across the metapopulation

wANPE08 - Udine The continuous-time model The limit of the discrete model as τ → 0 is not defined !!! → The previous equations are not the continuous time limit of the discrete equations !! Assuming uniform diffusion during each time interval (with probability τ ·Di ), the limit as τ → 0 becomes well-defined and one obtains …

wANPE08 - Udine The discrete equations Susceptible individuals: Infected individuals:

wANPE08 - Udine The limit equations Susceptible individuals: Infected individuals: (Saldaña, Phys. Rev. E 78 (2008))

wANPE08 - Udine Conserv. of number of particles Consistency relation between P(k) and P(k’|k) : Mean number of particles: Conservation of the number of particles:

wANPE08 - Udine Equilibrium equations

wANPE08 - Udine Uncorrelated networks In uncorrelated networks: = Degree distribution of nodes that we arrive at by following a randomly chosen link

wANPE08 - Udine Equilibrium equations in U.N.

wANPE08 - Udine Disease-free equilibrium In this case, and → the number of individuals is linear in the node degree k → Diffusion effect

wANPE08 - Udine Endemic equilibrium in type-II Saturation in the transmission of the infection → all the local populations have the equal prevalence of the disease: All are linear in the degree k

wANPE08 - Udine Endemic equilibrium in type-II Therefore, the condition for its existence at the metapopulation level is the same as the one for each subpopulation: There is no implication of the network topology for the spread and prevalence of the disease

wANPE08 - Udine Endemic equilibrium in type-I Increase of the prevalence with node degree (being almost linear for large k) Absence of epidemic threshold in networks with unbounded maximum degree There is an implication of the network topology for the spread and prevalence of the disease When D A = D B, the size a each population is linear with k, as in type-II

wANPE08 - Udine A sufficient condition in type I The disease-free equilibrium will be unstable whenever the following condition holds: This condition follows from the localization of the roots of the Jacobian matrix J of the linearized system around the disease-free equilibrium

wANPE08 - Udine A sufficient condition in type I Precisely, with The roots of being simple and satisfying

wANPE08 - Udine A remark on the suff. condition For regular random networks, k = and the condition reads as which is more restrictive than the n. & s. condition that follows directly from the model, namely,

wANPE08 - Udine Simulations under type-I trans. (Saldaña, Phys.Rev. E 2008)

wANPE08 - Udine Monte Carlo simulations (Baronchelli et al., Phys.Rev. E 78 (2008)) Not when D and R occur simultaneously !!

wANPE08 - Udine Monte Carlo simulations - 2 The length τ of the time interval must be small enough to guarantee that events are disjoint  The diffusing prob. of susceptibles and infectives are τ·D A and τ·D B, respectively  The prob. of becoming infected after all the infectious contacts is σ = σ(τ, k )  τμ is the recovering probability

wANPE08 - Udine Monte Carlo simulations - 3 For infective individuals For susceptible individuals with

wANPE08 - Udine Monte Carlo simulations - 4 This last inequality can be rewritten as If we consider the minimum of these τ’ s over the network:, the value of τ we take for each time step is

wANPE08 - Udine MC simulations for type-II trans. (Juher, Ripoll, Saldaña, in preparation) The same output as with discrete-time diffusion

wANPE08 - Udine MC simulations for type-I trans. (Juher, Ripoll, Saldaña, in preparation) Prevalence is NOT constant with k !!

wANPE08 - Udine Future work Analytical study of the properties of the equilibrium in type-I transmission More general diffusion rates (for instance, depending on the population degree) Impact of degree-degree correlations Introduction of local contact patterns