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Basic Concepts of Epidemic Models Nikos Demiris Stats Dept @ Athens University of Economics and Business UCL Workshop on Infectious Disease Modelling in Public Health Policy: Current status and challenges, 4th July 2016 1 / 33 Outline of this lecture A bit of (local) Epidemic History Epidemic Models: Motivation and Basic Concepts Vaccination Multitype Models: Why and How Some results for Stochastic Epidemics (hopefully) Contemporary Epidemic Models (perhaps) 2 / 33 One the fathers of Epidemiology is 3 / 33 Not this guy 4 / 33 But the great John Snow Solved the Cholera problem in Soho 5 / 33 Soho, 1854 Soho was not connected to the London sewer system By inspection of the map Snow realised that the Broad Street pump was the focus of infection He removed the handle and the epidemic died out 6 / 33 A new theory Went beyond the miasma theory ’Bad air’ (= mala aria) Important note: None of the pub employees of the local brewery was affected Daily allowance of beer → they did not consume water from the nearby well. Boiling water when brewing → kills cholera bacteria 7 / 33 Why Epidemic Models Understanding the mechanism of disease spread is important for disease control (or eradication) Epidemic Models are a natural tool for preparing (scenario testing) for a forthcoming epidemic/pandemic, dealing with an emerging disease or bioterrorism Also, they provide the framework for control measures: calculating the critical vaccination coverage and the optimal vaccination policy Evaluation of the cost-effectiveness of different interventions 8 / 33 The structure of Epidemic Models They are mathematical models appropriate for describing the transmission of an infectious disease in a population of individuals such as humans, animals, computers, plants They have also been used for describing the transmission of rumours, financial crises, information... Here we shall focus on person-to-person transmission. Modifications are required for host-vector diseases and rumour models. 9 / 33 Epidemic Models: Challenges Strong dependencies are inherently present The chance (hazard) that an individual gets infected depends on the status of others in their vicinity. Makes model analysis and statistical inference hard. The epidemic process is never fully observed Models are defined in terms of who infected whom and when did this happen–although genetic info is changing that. We typically only observe the times that symptoms appear. Makes the statistical analysis non-trivial. 10 / 33 The deterministic General Epidemic (Kermack and McKendrick 1927) There are three possible states (compartments): Susceptible Infected S→I→R Removed Individuals move between these states based on the following system: dS = −λS(t)I (t) dt dI = λS(t)I (t) − γI (t) dt dR = γI (t) dt dI dR Note that dS dt + dt + dt = 0 so that the population size is closed (no demography), homogeneous and homogeneously mixing 11 / 33 Implications of this Model Dividing the 1st and 3rd equations we get: dS dR = −R0 S R0 = λγ is a threshold parameter that largely determines the system behaviour Integrating gives: S(t) = S(0)e −R0 (R(t)−R(0)) For a large population and t → ∞ we have 1 − τ = e −τ R0 . This transcendental equation has a non-trivial solution iff R0 > 1, whence τ ↑ with R0 . Otherwise τ = 0 dI Alternatively, I (t) is increasing ( dt > 0) when R0 > 1 S(0) Both suggest a threshold theorem: There will be a major (minor) epidemic iff R0 > 1 (R0 < 1) What is R0 ? 12 / 33 R0 values for some well-known diseases Disease Transmission R0 Measles Airborne 12-18 Pertussis Airborne droplet 12-17 Diphtheria Saliva 6-7 Smallpox Social contact 5-7 Polio Fecal-oral route 5-7 Rubella Airborne droplet 5-7 Mumps Airborne droplet 4-7 HIV/AIDS Sexual contact 2-5 SARS Airborne droplet 2-5 Influenza (1918 pandemic strain) Airborne droplet 2-3 13 / 33 Epidemic Control: Vaccination Disease control is, perhaps, the most important practical reason for using epidemic models Suppose that a (perfect) vaccine is available and we are interested in estimating the proportion, say ω, to be immunised in order to avoid a large outbreak The effective transmission rate is now λ(1 − ω) so R0 (ω) = λ(1−ω) = (1 − ω)R0 γ We wish to achieve R0 (ω) < 1 or ω > ωc = 1 − 1 R0 We call ωc as the critical vaccination coverage, e.g. ωc = {0.5, 0.8, 0.9} for R0 = {2, 5, 10} respectively. The state where ω > ωc is referred to as herd immunity 14 / 33 Imperfect Vaccine The above calculations were made assuming that a vaccine offers perfect immunity. Suppose that the vaccine is imperfect with efficacy ψ. Then we have R0 (ω, ψ) = (1 − ω)R0 + ω(1 − ψ)R0 . The condition R0 (ω, ψ) < 1 now gives ω(ψ) > ωc (ψ) = So ψ < 1 − 1 R0 1− R1 0 ψ implies that herd immunity is impossible! Indeed, for a disease with R0 = 10 we need 90% coverage. Any vaccine with ψ < 90% cannot achieve that aim, even if everyone is vaccinated. A crude R0 estimate can be obtained from τ = 1 − e −τ R0 so ) R̂0 = − log(1−τ , with τ the proportion infected in a historical τ outbreak 15 / 33 What else matters in Epidemic Transmission? Realistic populations are not homogeneous, nor homogeneously mixing. Age is important, should we bother with age-grouping? Yes! A trivial – fictitious illustration for non-statisticians: Vaccine A Efficacy 78% = 273 350 Vaccine B 83% = 289 350 16 / 33 Age does matter: confounding Vaccine A Vaccine B Children 93% = 81 87 87% = 234 270 Adults 73% = 192 263 69% = 55 80 Efficacy 78% = 273 350 83% = 289 350 17 / 33 A solution: multitype models Split the population into k age groups → k 2 contact rates Can be over-parametrised. Marc’s talk will use Polymod data: contact survey, gives a rough idea of how age classes mix Disentangles the social and biological components of disease transmission R0 is now the largest eigenvalue of a matrix Vaccination more involved but follows from the same principles 18 / 33 Is everything changing with Big Data? In 2009 a group from Google published in Nature a study which used data from their Search Engine to predict/nowcast ILI activity A comparison of model estimates for the mid-Atlantic region (black) against CDC-reported ILI percentages (red), including points over which the model was fit and validated. J Ginsberg et al. Nature 000, 1-3 (2008) doi:10.1038/nature07634 Can such methods/data replace the current (more expensive) surveillance systems? 19 / 33 Not yet In 2013 GFT massively overestimated the peak of Flu activity But this was a fresh idea and can function to parametrise some components of the epidemic process. It represent a currently active research area. 20 / 33 Epidemic Models: Deterministic and Stochastic The former can be thought of as the ‘mean’ of the latter, when appropriately defined (Kurtz, LLN and CLT) Stochastic models are more natural since most relevant events like the disease transmission or extinction are defined in terms of their probability. Deterministic models are simpler (but not simple!) to analyse and dominated the previous century. Stochastic models are increasingly popular as they appear in many high-impact publications of this century. Both types are useful. 21 / 33 Stochastic SIR Epidemic Models The (Markov) stochastic version was introduced by Bartlett (1949) and is known as the ‘general stochastic epidemic’. Each individual, while infectious, makes contacts according to the points of a Poisson process with intensity λ/N. The Poisson contact processes are independent. The infectious periods, say Ij , are i.i.d. Exp(γ). This is a common (and often hidden) assumption in ODE models and in Markov Processes. We shall be concerned with the case where the Ij ’s are not necessarily exponential (usually Gamma). This model has been known as the generalised stochastic epidemic (GSE). 22 / 33 How does the model look? Initially the model is very much like a (BGW) branching process where Birth → new infection Death → removal This can be made fully rigorous using a coupling argument (Ball 1983) as follows: The life span of ancestor j is Ij , Ij ’s independent. Each ancestor gives birth during their lifetime at the points of a Poisson process with rate λ/N. Then, the two processes ‘agree’ until time C log(N) (Ball and Donnelly 1995). So we can use all the machinery of branching processes (Jagers 1975) to derive results for the initial behaviour of epidemics. 23 / 33 Branching Process Approximation Let {Y (t); t ≥ 0} be the number of individuals alive at time t and D the number of offspring of a given individual. If the initial population size is m then there will be on average mE (D)ν individuals in the ν-th generation Clearly the process will become extinct iff E (D) ≤ 1 Using standard results from (Jagers 1975) one can show that when E (D) > 1 there will be extinction w.p. q m and explosion otherwise. q is the smallest solution of θ = E (θD ). Summary: E (D) ≤ 1 → extinction w .p. 1 extinction w .p. q m E (D) > 1 → explosion w .p. 1 − q m Note that given I = i the number of children D ∼ Po(λi) Which (after some work) gives E (D) = λE (I ) = R0 ! Hence, E (D) ≤ 1 (E (D) > 1) → R0 ≤ 1 (→ R0 > 1) 24 / 33 Thresholds and Phase Transitions We have seen that R0 determines whether or not an epidemic can occur. This dramatic change in the system behaviour has been observed in many processes in nature (0 and 100 degrees temperature, . . . ) and some connections will be given, particularly with random graphs and percolation models. Random graphs have also been used as population models. Social (and other) networks are often described with random graph models (p ∗ , powerlaw-type, . . . ). The analysis of such systems is largely similar to the analysis of stochastic epidemics, particularly the final outcome. 25 / 33 The final size distribution of the GSE Let pk be the probability that k individuals are ultimately infected, 0 ≤ k ≤ N. Then n−k l X n l−k pk , 0 ≤ l ≤ N, h ik+m = l λ(n−l) k=0 φ n where φ(θ) is the Laplace transform of I and m the number of initial infectives. This triangular system of equations was derived in Ball (1986) using Laplace transforms and a Wald-type identity. It is exact and, in principle, straightforward to solve numerically. In practice, however, numerical problems arise even for moderate population sizes, say N = 50, due to the nature of the solution. The likelihood is analytically and numerically intractable: 26 / 33 The final size distribution of the GSE 27 / 33 A (complementary) solution: Asymptotics For the deterministic model we have seen that when R0 > 1 the proportion infected is given by the non-trivial solution of τ = 1 − e −τ R0 . This holds true for the GSE in the supercritical case. Scalia-Tomba (1985) presented an elegant way of doing this using the Sellke construction and an imbedding representation Consider the infectious pressure process An (t) An (t) → λE (I )t in probability, uniformly on compact sets √ D n (An (t) − λE (I )t) −→ Wiener process, var=λ2 Var (I )t )+λ2 Var (I )τ (1−τ )2 Then, w.p. 1 − q m , `n → N τ, τ (1−τ(1−(1−τ as 2 )R0 ) n → ∞. Provides an approximate way to evaluate L(` | λ) 28 / 33 Epidemics upon Random Graphs Thus far, we have completely ignored any social structure that may be present in the population. This is unrealistic. We shall explore how some of the previous results transfer to epidemics defined in structured populations. These populations may be fixed or random (or both). A natural way to start is by using a random graph as a model of social network → huge area in sociology. More pragmatic to assume closer contacts locally as opposed to identical contacts with everyone. All individuals involved are assumed identical. Multitype models will be considered later. 29 / 33 Structured Epidemics: Two levels of mixing Small social groups like households, schools and workplaces are particularly important for disease transmission. A general framework for epidemic local and global epidemic spread was described in Ball et al 1997. The population is partitioned into groups (households, farms. . . ) and infected individuals can transmit the disease: globally, with everyone, at the points of a Poisson process with rate λG /N locally, within group, at the points of a Poisson process with rate λL The ‘great circle’ variant is the first small world model (200 vs 11000 citations). 30 / 33 A very simple illustration Two groups of 2 and 3 individuals respectively 31 / 33 Model Analysis Ball et al 1997 derive a branching process approximation for the early stages of the epidemic. Groups are (super)individuals and Each group has mean offspring (# infections) λG E (`), where ` = number infected in a ‘typical’ group. A threshold parameter as the number of groups →P∞ is given sµ π by R∗ = λG E (`) = λG E (I )ν(λL ), where ν(λL ) = s v s s is the average local final size if only local infections permitted The process may explode if R∗ > 1. The authors also derive a CLT for the final size and final severity, should a ‘large’ outbreak occur. Can be extended to other (appropriate) functionals like the total epidemic cost. 32 / 33 Notes on Additional Structure The framework with local and global transmission on a fixed population is very general. Several extensions are being developed where the local or the global population may be represented by a random network. Threshold parameters are derived using similar techniques Overlapping groups remain a challenge Can also consider multitype epidemics. 33 / 33