Introduction & Tutorial to Network Epidemiology

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INTRODUCTION & TUTORIAL TO NETWORK EPIDEMIOLOGY. Laurent H´ ebert-Dufresne laurent.hebert-dufresne@uvm.edu :: @LHDnets Vermont Complex Systems Center & Department of Computer Science [Title page illustration by Joerael Numina based on work by LHD] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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LESSONS 1 A model is not better because it is a network model. 2 Networks can consider heterogeneity and randomness. 3 Networks are useful for interventions and forecasts. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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OUTLINE 1 Compartmental epidemiological models • Susceptible-Infectious-Susceptible dynamics • Susceptible-Infectious-Recovered dynamics + Kermack-McKendrick 2 Deterministic network models • Pair approximation + heterogeneity & adaptive network • Heterogeneous mean-ﬁeld 3 Probabilistic network models • Probability Generating Functions + Generalized Kermack-McKendrick 4 Agent-based models and simulations Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Compartmental epidemiological models time % With immunity Without immunity time % S I R S I [Anderson & May, Infectious Diseases of Humans (1992)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Mass-action SIS model Population of N people, all potentially connected. Infectious individuals transmit at rate β and recover at rate α. We follow the numbers I of infectious individuals. . . d dt I = βSI − αI = β(N − I)I − αI Previous animation is from the Washington Post, Why outbreaks like coronavirus spread exponentially, and how to “ﬂatten the curve” (Mar 14 2020) Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Mass-action SIS model We have a logistic growth of the epidemic following. . . d dt I = ˙ I = β(N − I)I − αI Let’s calculate the infections caused per infectious individuals when I is small. . . lim I→0 ˙ I I = lim I→0 β(N − I) − α = βN − α We ﬁnd an epidemic threshold! Outbreak occurs if β > βc . . . βc N = α or R0 = βN α And when R0 > 1, we can ask what is the equilibrium. . . ˙ I = β(N − I)I − αI = 0 → I∗ = 1 − 1/R0 Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Compartmental epidemiological models time % With immunity Without immunity time % S I R S I transmission R* transmission I* [Anderson & May, Infectious Diseases of Humans (1992)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Mass-action SIR model, the Kermack-McKendrick solution R∗ = − 1 R0 ln (1 − R∗) [Kermack & McKendrick, Contributions to the mathematical theory of epidemics I–III (1927, 1932, 1933)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Assumptions of Kermack-McKendrick Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts occur according to the law of mass-action, 6 the population is large enough to justify a deterministic analysis. [Diekmann, Heesterbeek & Metz, The legacy of Kermack and McKendrick (1995)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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First network model: Pair approximations Not everyone is interconnected! We now follow the state of individuals I, and of pairwise interactions [SI]. . . d dt S = αI − β[SI] d dt [SS] = α [SI] − β [SSI] Previous animation is from https://www.complexity-explorables.org/, “I herd you!” (Aug 6 2018) Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Note on moment closures In Erd˝ os-R´ enyi random graph where all edges exist with probability p, we would approximate edges with [SI] = pSI and go back to the mass-action system with β = pβ. We can also follow pairs [SI] and approximate triplets, e.g.[ISI]: d dt [SI] = −α [SI]+2α [II] − β [SI] + β [SSI] − β [ISI] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Note on moment closures [ISI] ∝ [SI] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Note on moment closures [ISI] ∝ [SI] · kex Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Note on moment closures [ISI] ∝ [SI] · kex [SI] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Note on moment closures [ISI] = [SI] · kex [SI] /S k Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Is kex = k ? First network epiphany: Friendship Paradox Your friends have more friends than you do.* Explanation: 1 You: Random person drawn from a degree distribution pk . 2 Your friends: Random neighbor drawn proportionally to kpk . *on average. If you ignore the friendship paradox, you fall back on the mass-action model. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Pair approximations, cool. What’s the big picture? Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts occur according to the law of mass-action, 6 the population is large enough to justify a deterministic analysis. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Pair approximations, cool. What’s the big picture? Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts k follow a distribution approximated by < kex > / < k >, 6 infection status is not correlated with contacts k, 7 the population is large enough to justify a deterministic analysis. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Second network epiphany: Importance of heterogeneity [SI] decreases with β [SI] but increases with β [SSI] ∝ β . Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Second network epiphany: Importance of heterogeneity [SI] decreases with β [SI] but increases with β [SSI] ∝ β . Degree distribution % transmission Epidemic size degree % Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Second network epiphany: Importance of heterogeneity [SI] decreases with β [SI] but increases with β [SSI] ∝ β . Degree distribution % transmission Epidemic size degree % In the limit of extreme heterogeneity, there is no epidemic threshold. E.g., scale-free networks with power-law degree distribution pk ∝ k−γ if γ ≤ 3. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Third network epiphany: Importance of behaviour Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Adaptive networks: Structure co-evolves with epidemics 1 2 Patient #1 Patient #2 Nurse Substitute Substitute Nurse Nurse Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Adaptive networks with pair approximations We introduce a rate γ at which [XI] edges are rewired to [XS]. ˙ [S] = α[I]−β [SI] ˙ [SS] = (α + γ) [SI]−2β [SI] [SS] [S] ˙ [II] = β [SI] 1 + [SI] [S] −2 (α + γ) [II] [Scarpino, Allard & H´ ebert-Dufresne, The eﬀect of a prudent adaptive behaviour on disease transmission (2016)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Adaptive networks with pair approximations [Scarpino, Allard & H´ ebert-Dufresne, The eﬀect of a prudent adaptive behaviour on disease transmission (2016)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Assumptions of pair approximations Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts k follow a distribution approximated by < kex > / < k >, 6 infection status is not correlated with contacts k, 7 the population is large enough to justify a deterministic analysis. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Second network model: Heterogeneous mean-ﬁeld We now follow the state of individuals of degree k, i.e. Ik d dt Ik = Sk × kΘβ − αIk where Θ is a “mean-ﬁeld” ≈ the probability of an infectious neighbour: Θ = k kIk kpk Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Third network model: Approximate Master Equations Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Third network model: Approximate Master Equations Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Third network model: Approximate Master Equations [St-Onge, Thibeault, Allard, Dub´ e & H´ ebert-Dufresne, arXiv 2003.05924 (preprint, 2020)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Assumptions of heterogeneous mean-ﬁeld Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts k follow a distribution approximated by < kex > / < k >, 6 infection status is not correlated with contacts k, 7 the population is large enough to justify a deterministic analysis. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Assumptions of heterogeneous mean-ﬁeld Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts follow some statistical distribution, 6 the population is large enough to justify a deterministic analysis. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Assumptions of heterogeneous mean-ﬁeld Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts follow some statistical distribution, 6 the population is large enough to justify a deterministic analysis. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Outline 1 Compartmental epidemiological models • Susceptible-Infectious-Susceptible dynamics • Susceptible-Infectious-Recovered dynamics + Kermack-McKendrick 2 Deterministic network models • Pair approximation + heterogeneity & adaptive network • Heterogeneous mean-ﬁeld 3 Probabilistic network models • Probability Generating Functions + Generalized Kermack-McKendrick 4 Agent-based models and simulations Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Where do we get degree distribution data? 1 By educated guess. 2 Social media or proximity data. 3 Contact tracing data, i.e. who infected whom. 4 Viral genome sequences to reconstruct transmission trees. 5 Analogies with past outbreaks. 6 Fitting agent-based simulations to incidence data. [H´ ebert-Dufresne, Althouse, Scarpino & Allard, Beyond R0 (preprint, 2020)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Contact tracing for COVID-19 [Meili Li et al., Transmission characteristics of the COVID-19 outbreak in China: a study driven by data (preprint, 2020)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Probabilistic transmission trees Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Transmission networks, not contact networks • Edges are contacts that would transmit if given the chance, i.e., infections not just contacts. • Average excess degree kex is R0 , i.e., the average number of secondary infections caused by a case. • Next model considers the full distribution of secondary infections around R0 . Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Fourth model: Probability Generating Functions (PGFs) PGFs use polynomials to store discrete stochastic events and their probabilities. Example: PGFs of a randomly chosen patient zero ... G0 (x) = k pk xk where pk is the degree distribution of the network and powers of x count future infections. [Newman, Strogatz & Watts, Random graphs with arbitrary degree distributions and their applications (2001)] [Newman, The spread of epidemic disease on networks (2002)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Fourth model: Probability Generating Functions (PGFs) PGFs are useful to manipulate probability distributions. Example: PGFs of a secondary infection of our patient zero. Friendship paradox! ... G1 (x) ∝ k kpk xk−1 G1 (x) = 1 k dG0(x) dx Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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PGFs: Never stop never stopping What if the infection tree never ends? Example: Probability u that infections following an edge does eventually stop. u = G1 (u) What is the probability the outbreak never stops from patient zero? R(∞) = 1 − G0 (u) R(∞) is both the probability and relative size of that inﬁnite outbreak. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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What is the ﬁnite outbreak size if it does stop? How many infections will occur downstream of the previous one? Example: PGFs of a secondary infection of our patient zero. H1 (x) = x (G1 (H1 (x))) How many infections will occur from patient zero? H0 (x) = xG0 (H1 (x)) Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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This is a generalized Kermack-McKendrick analysis If we assume a Poisson distribution (mass-action), G0 (x) = exp [ k (x − 1)] and kex = k = R0 for G0 (x) = G1 (x) , the network R(∞) is equal to the mass-action R∗ from Kermack-McKendrick. [H´ ebert-Dufresne, Althouse, Scarpino & Allard, Beyond R0 (preprint, 2020)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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But the forecast is probabilistic. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Importance of heterogeneity for COVID-19 Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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If R0 > 1, we shouldn’t be arguing about whether it’s actually 2.5 or 3.5, we should be ﬁguring out how much heterogeneity there is behind it. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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And other diseases? 0.0 0.2 0.4 0.6 0.8 1.0 Proportion of susceptible individuals infected [R(∞)] MERS (2013) SARS (2003) COVID-19 (2020) Inﬂuenza (2009) Inﬂuenza (1918) Smallpox (1958-1973) Reported Kermack-McKendrick Network model Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Assumptions of probabilistic/random network forecasts Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts follow some statistical distribution, 6 the population is large enough to justify a deterministic analysis. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Assumptions of probabilistic/random network forecasts Epidemiological 1 if SIR, the disease results either in complete immunity or death, 2 if SIS, the disease is not fatal and conveys no immunity, 3 the disease is transmitted in a closed population, 4 all individuals are equally susceptible, Structural 5 contacts follow some statistical distribution, 6 the population is large enough. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Global Epidemic and Mobility Model https://www.gleamproject.org/covid-19 [Chinazzi et al., The eﬀect of travel restrictions on the spread of the 2019 novel coronavirus (COVID-19) outbreak (2020)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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Metapopulation model framework [Colizza & Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations (2008)] Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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For other simulations Epidemics on Networks (EoN) is python package for simulations and solving of epidemiological models built on the NetworkX package. https://EpidemicSonnetWorks.readthedocs.io/ https://networkx.github.io/ Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne

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For a more complete overview This textbook and other references from slides. Introduction & tutorial to network epidemiology Laurent H´ ebert-Dufresne