Studying the recovery procedure for the time-dependent transmission rate(s) in epidemic models

Abstract

Determining the time-dependent transmission function that exactly reproduces disease incidence data can yield useful information about disease outbreaks, including a range potential values for the recovery rate of the disease and could offer a method to test the "school year" hypothesis (seasonality) for disease transmission. Recently two procedures have been developed to recover the time-dependent transmission function, β(t), for classical disease models given the disease incidence data. We first review the β(t) recovery procedures and give the resulting formulas, using both methods, for the susceptible-infected-recovered (SIR) and susceptible-exposed-infected-recovered (SEIR) models. We present a modification of one procedure, which is then shown to be identical to the other. Second, we explore several technical issues that appear when implementing the procedure for the SIR model; these are important when generating the time-dependent transmission function for real-world disease data. Third, we extend the recovery method to heterogeneous populations modeled with a certain SIR-type model with multiple time-dependent transmission functions. Finally, we apply the β(t) recovery procedure to data from the 2002-2003 influenza season and for the six seasons from 2002-2003 through 2007-2008, for both one population class and for two age classes. We discuss the consequences of the technical conditions of the procedure applied to the influenza data. We show that the method is robust in the heterogeneous cases, producing comparable results under two different hypotheses. We perform a frequency analysis, which shows a dominant 1-year period for the multi-year influenza transmission function(s).

A Perspective on Multiple Waves of Influenza Pandemics

“…The transmission rate β t is determined using a new algorithm [25], [26], [27] that ensures the output I ( t ) of the model perfectly fits a smooth interpolation of the reported cases by the Centers for Disease Prevention and Control (CDC)[28]. There is no error between the model output of I ( t ) and the data, and only the initial value β 0 needs to be specified.…”

Time-Discrete Parameter Identification Algorithms for Two Deterministic Epidemiological Models applied to the Spread of COVID-19

“…Many approaches regarding parameter estimation techniques in differential equations and especially in epidemiological models are well known [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] . The works by Hadeler 45,46 and Mummert 52 use the idea of assuming all transfer rates to be time-varying in their models. However, they only apply this in a continuous setting.…”

Identifying the number of unreported cases in SIR epidemic models

“…One of the most important considerations of epidemic models is the identification of parameters needed for applications. The parameter identification problem for SIR model has been investigated by many researchers, including [2,3,8,9,10,12,13,14,15,16,18,20,21,23,24,25,27,29,30,31,33]. Our objective here is to continue the investigation in [26] of the parameter identification problem for the standard SIR ordinary differential equations model of an outbreak epidemic: S (t) = −τ S(t)I(t), I (t) = τ S(t)I(t) − νI(t).…”