# EpiInfer: A Non-Markovian Method and System to Forecast Infection Rates in Epidemics
## 1 Introduction  
[PERSON] et al. [1] introduced compartmental epidemiological models such as the Susceptible-Infected-Recovered model in which an agent is either Susceptible (healthy) or Infected (symptomatic and contagious) or Recovered (healthy and with partial immunity). By adding the Exposed compartment (asymptomatic and contagious), this model can be extended to the SEIRS model which characterizes most epidemics in which people can be contagious but not symptomatic. Understanding the evolution of an epidemic requiresknowing the transition probabilities between these states, a process called _calibration_. The relationship between the states and parameters can be seen in Figure 1.  
Various surveys have compared calibration methods. For instance, the survey conducted by [PERSON] et al. [2] used EpiPolicy [3] to compare these methods through simulations. The EpiPolicy simulator allows users to include features such as sub-population groups, facilities, population mobility, vaccination, hospitalization, mask-wearing, border closures, lockdowns, and school closures. The survey concluded that the best algorithm depends on the type of model and data, but there are certain algorithms such as Levenberg-Marquardt (least squares) that tend to perform well in many different cases.  
The overall process of estimating infection rates can be broken down into two tasks: (i) inferring transition parameters between states and then (ii) using those parameter values to infer infection rates.  
Some methods for both tasks make use of contact tracing information which gives how many people each person meets and whom they meet. Because such information is rarely available in quantity, EpiInfer only makes use of the distribution (and even only the mean of the distribution) of how many people each person meets (but not whom).  
This paper presents and evaluates a new algorithm called EpiInfer as well as some variants. The paper's contributions can be divided into a few parts:  
1. Our algorithm EpiInfer consists of two components _Epilnfer-core_ and an outer-loop algorithm _ContinuousCalibrate_ that iterates over various hyperparameter settings in calling _EpiInfer-core_.
2. The non-Markovian algorithm _EpiInfer-core_ projects future infection rates starting at any day \(d\) by considering (i) the number of newly infected patients in a few days before \(d\), (ii) some information about the meeting distribution of people, and (iii) transition probabilities between the Susceptible (S), Exposed (E), and Infected (I) states.
3. Because the transition probabilities are not known at the beginning of an epidemic, _ContinuousCalibrate_ recomputes the transition probabilities at each day by calling _Epilnfer-core_ on previous data and various parameter settings. This dynamic setting of parameters follows the spirit of [4].
4. We show how using multiple locales can help _ContinuousCalibrate_ improve the estimation of transition parameters and thereby improve infection prediction, especially early in an epidemic.
5. Though EpiInfer is most accurate when given a full meeting distribution, we show that using just the mean number of meetings per person is often enough.
6. Our experiments show that the two-component EpiInfer predicts better than (Markovian) Differential Equation models and LSTMs (as well as other neural network models) regardless of the forecast window. ARIMA is more accurate than EpiInfer  
Figure 1: SEIRS (Susceptible, Exposed, Infected, Recovered, Susceptible) Model Diagram: Susceptible means healthy and asymptomatic but not immune; Exposed means contagious but asymptomatic; Infected means symptomatic but assumed isolated so not contagious; and Recovered means healthy and temporarily immune. Differential equation models are Markovian in that the number of individuals in each state at time \(t+1\) depends on the parameter values \((\beta,\sigma,\gamma,\eta)\) and the number of individuals in each state at time \(t\), ignoring previous times.  
for short-term predictions, but EpiInfer is more accurate than ARIMA starting at 7 days.