Joan Saldaña Universitat de Girona

1. RD processes on heterogeneous metapopulations: Continuous-time formulation and simulationswANPE08 – December 15-17, Udine Joan Saldaña Universitat de Girona

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

3. SIS model • The force of infectionλ =rate at which susceptible individuals become infected • Proportional to the number of infective contacts • µ = recovery rate wANPE08 - Udine

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

5. 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

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

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

8. Meyers et al. JTB 2005

9. 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

10. 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

11. Network architectures Meyers et al. JTB 2005 wANPE08 - Udine

12. A special degree distribution • Distribution of the degrees of nodes reached by following a randomly chosen link: which has < k² > / <k > as expected value. This is the value to be considered for c!! wANPE08 - Udine

13. Absence of epidemic threshold in SF networks!! 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 wANPE08 - Udine

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

15. An example wANPE08 - Udine

16. More modern examples wANPE08 - Udine

17. A nice picture of the 1st example wANPE08 - Udine

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

19. 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

20. 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

21. Global invasion threshold wANPE08 - Udine

22. 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

23. A discrete-time model wANPE08 - Udine

24. 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

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

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

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

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

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

30. 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

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

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

33. 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

34. Equilibrium equations wANPE08 - Udine

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

36. Equilibrium equations in U.N. wANPE08 - Udine

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

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

39. 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

40. 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 DA = DB, the size a each population is linear with k,as in type-II wANPE08 - Udine

41. 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

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

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

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

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

46. 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·DA and τ·DB ,respectively • The prob. of becoming infected after all the infectious contacts is σ = σ(τ,k) • τμ is the recovering probability wANPE08 - Udine

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

48. 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 τwe take for each time step is wANPE08 - Udine

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

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