Compartmental Models and Hawkes Processes: Equivalence and Computational Advantages in Epidemiological Modelling A Theoretical and Empirical Investigation of Infectious Disease Dynamics

Sarah Masri
Supervised by Dr. Daniel J. McDonald

Introduction

Motivation & Problem Statement
Theoretical Background
Methodology
Results
Discussion

Motivation

Problem Statement

Demonstrate the theoretical and empirical equivalence, in expectation, between the SIR and SEIR compartmental models and the Hawkes process.

\[\mathbb{E}[\lambda_{\text{SIR or SEIR}}(t)] = \lambda_{\text{Hawkes}}(t)\]

λ(t) denotes the rate of change of new cases over time.

Previous Work

Champredon et al. (2018):
Showed a theoretical connection between renewal equations and compartmental models.
Riziou et al. (2018):
Established the equivalence between the SIR model and the Hawkes process (in expectation).

Theoretical Background

Compartmental Models

Compartmental Models

Compartmental Models: Deterministic SIR & SEIR

SIR model
\[ \frac{dS}{dt} = - \beta \frac{IS}{N}, \quad \frac{dI}{dt} = \beta \frac{IS}{N} - \gamma I, \quad \frac{dR}{dt} = \gamma I \]

SEIR model
\[ \frac{dS}{dt} = - \beta \frac{IS}{N}, \quad \frac{dE}{dt} = \beta \frac{IS}{N} - \alpha E, \quad \frac{dI}{dt} = \alpha E - \gamma I, \quad \frac{dR}{dt} = \gamma I \]

\(\beta\): transmission rate,
\(\alpha\): rate from exposed to infectious,
\(\gamma\): recovery rate.

Compartmental Models: Stochastic SIR

SIR model
\[\begin{aligned} \Delta B_{t+h} &= \text{Binom} \Big (S_t, \frac{\beta I}{N} h \Big ), \\ \Delta D_{t+h} &= \text{Binom} (I_t, \gamma h), \\ S_{t+h} &= S_t - \Delta B_{t+h}, \\ I_{t+h} &= I_t + \Delta B_{t+h} - \Delta D_{t+h},\\ R_{t+h} &= R_t + \Delta D_{t+h}. \end{aligned}\]

\(h\): time-step
\(S_t\), \(I_t\), \(R_t\): discrete, integer-valued random variables corresponding to number of individuals in the susceptible, infected, and recovered compartments in the deterministic formulation, respectively, at time \(t\)
\(\Delta B_{t+h}\), \(\Delta D_{t+h}\): the number of new infection and recovery events that occur between time \(t\) and \(t+h\), respectively

Compartmental Models: Stochastic SEIR

SEIR model
\[\begin{aligned} \Delta B_{t+h} &= \text{Binom} \Big ( S_t, \frac{\beta I_t}{N} h \Big ), \\ \Delta A_{t+h} &= \text{Binom}(E_t, \alpha h), \\ \Delta D_{t+h} &= \text{Binom}(I_t, \gamma h), \\ S_{t+h} &= S_t - \Delta B_{t+h}, \\ E_{t+h} &= E_t + \Delta B_{t+h} - \Delta A_{t+h}, \\ I_{t+h} &= I_t + \Delta A_{t+h} - \Delta D_{t+h}, \\ R_{t+h} &= R_t + \Delta D_{t+h}. \end{aligned}\]

\(\Delta B_{t+h}\), \(\Delta A_{t+h}\), \(\Delta D_{t+h}\): the number of new infection, exposed, and recovery events that occur between time \(t\) and \(t+h\), respectively

Hawkes Process

The conditional intensity of the Hawkes process is given by:

\[ \lambda(t \mid \mathcal{H}_t) = \mu + K \sum_{t_i < t} \phi ( t - t_i). \]

The conditional intensity of the HawkesN process is given by:

\[ \lambda(t \mid \mathcal{H}_t) = \Big (1 - \frac{ C_t }{N} \Big ) \Big [ \mu + K \sum_{t_i < t} \phi ( t - t_i) \Big ]. \]

\(\mathcal{H}_t\): history of events | \(\mu\): background intensity | \(K\): productivity | \(\phi\): triggering kernel
\(t_i\): infection times | \(C_t\): cumulative cases | \(N\): population size

Methodology

Key Theoretical Results: Linking SIR and HawkesN

\[ \mathbb{E}_{\tau_{IR}} [ \lambda^c (t)] = \lambda^h (t). \]

For triggering kernel \(\phi(\tau) = \theta e^{-\theta \tau}\),

\(\mu = 0\) | \(\beta = K \theta\) | \(\gamma = \theta\) | \(\tau_{IR}\): recovery times.

Key Theoretical Results: Linking SEIR and HawkesN

\[ \mathbb{E}_{\tau} [\lambda^c(t)] = \lambda^h(t). \]

For triggering kernel \(\phi(\tau) = \begin{cases} \frac{\alpha \theta}{\alpha - \theta} \big( e^{-\theta \tau} - e^{-\alpha \tau} \big), & \text{if } \alpha \neq \theta, \\ \theta^2 \tau e^{-\theta \tau}, & \text{if } \alpha = \theta \end{cases}\),

\(\mu = 0\) | \(\beta = K \theta\) | \(\gamma = \theta\) | \(\tau = \{ \tau_{EI}, \tau_{IR}\}\): wait times from E to I, and I to R.

Results

Simulation Study

Four independent simulation experiments using different generated datasets (SIR, SEIR, HawkesN with different triggering kernel)
Horizon of 150 days with incidence reported twice daily
Results in each experiment are comparable/promising

Real-World Application: 1918 Flu

Daily incidence of influenza onset during the 1918 H1N1 pandemic
Baltimore, Maryland over 92 days in late 1918
28,977 individuals from 6,753 households in a house-to-house canvass

\[ \begin{array}{l l | c c c | c c c} \textbf{Model} & \textbf{Metric} & K & \theta & \alpha & \beta = K\theta & \gamma = \theta & \alpha \\ \hline \text{Hawkes} & \text{Median} & 1.15 & 2.26 & 2.01 & {\bf 2.60} & {\bf 2.26} & {\bf 2.01} \\ \text{Hawkes} & \text{Std. Dev.} & 0.02 & 0.45 & 0.55 & \text{--} & 0.45 & 0.55 \\ \text{SEIR} & \text{Median} & \text{--} & \text{--} & \text{--} & {\bf 2.49} & {\bf 2.53} & {\bf 2.05} \\ \text{SEIR} & \text{Std. Dev.} & \text{--} & \text{--} & \text{--} & 1.47 & 1.93 & 0.81 \\ \end{array} \]

Discussion

stochastic SIR and SEIR models can be viewed as special cases of the HawkesN process
unified framework for finite population dynamics
HawkesN does not rely on latent measurements
Real-World Example: The model was successfully applied to the 1918 H1N1 influenza pandemic in Baltimore, demonstrating practical use
Future studies could explore combining HawkesN with other epidemic models

Thank you!

References

Alan G. Hawkes. Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1):83–90, 1971.

Alex Reinhart. A review of self-exciting spatio-temporal point processes and their applications. Statistical Science, 33(3):299–318, 2018.

David Champredon, Jonathan Dushoff, and David J. D. Earn. Equivalence of the Erlang-distributed SEIR epidemic model and the renewal equation. SIAM Journal on Applied Mathematics, 78(6):3258–3278, 2018.

Marian-Andrei Rizoiu, Swapnil Mishra, Quyu Kong, Mark Carman, and Lexing Xie. SIR-Hawkes: Linking epidemic models and Hawkes processes to model diffusions in finite populations. In Proceedings of the 2018 World Wide Web Conference (WWW ’18), pages 419–428, Republic and Canton of Geneva, Switzerland, 2018. International World Wide Web Conferences Steering Committee.

W. O. Kermack and A. G. McKendrick. Contributions to the mathematical theory of epidemics—I. Bulletin of Mathematical Biology, 53(1–2):33–55, 1927.