The discrete-time Kermack–McKendrick model: A versatile and computationally attractive framework for modeling epidemics

2021 ◽  
Vol 118 (39) ◽  
pp. e2106332118
Author(s):  
Odo Diekmann ◽  
Hans G. Othmer ◽  
Robert Planqué ◽  
Martin C. J. Bootsma
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Generation Time ◽  
Time Distribution ◽  
Reproduction Number ◽  
Compartmental Models ◽  
Infectious Period ◽  
Initial Growth ◽  
Fixed Duration ◽  
Spread Of Infection

The COVID-19 pandemic has led to numerous mathematical models for the spread of infection, the majority of which are large compartmental models that implicitly constrain the generation-time distribution. On the other hand, the continuous-time Kermack–McKendrick epidemic model of 1927 (KM27) allows an arbitrary generation-time distribution, but it suffers from the drawback that its numerical implementation is rather cumbersome. Here, we introduce a discrete-time version of KM27 that is as general and flexible, and yet is very easy to implement computationally. Thus, it promises to become a very powerful tool for exploring control scenarios for specific infectious diseases such as COVID-19. To demonstrate this potential, we investigate numerically how the incidence-peak size depends on model ingredients. We find that, with the same reproduction number and the same initial growth rate, compartmental models systematically predict lower peak sizes than models in which the latent and the infectious period have fixed duration.

scholarly journals On discrete time epidemic models in Kermack-McKendrick form

2021 ◽  
Author(s):  
Odo Diekmann ◽  
Hans G. Othmer ◽  
Robert Planque ◽  
Martin CJ Bootsma
Keyword(s):  
Infectious Diseases ◽  
Discrete Time ◽  
Epidemic Model ◽  
Continuous Time ◽  
Reproduction Number ◽  
Compartmental Models ◽  
Initial Speed ◽  
Fixed Duration ◽  
Epidemic Spread ◽  
Special Cases

Surprisingly, the discrete-time version of the general 1927 Kermack-McKendrick epidemic model has, to our knowledge, not been formulated in the literature, and we rectify this omission here. The discrete time version is as general and flexible as its continuous-time counterpart, and contains numerous compartmental models as special cases. In contrast to the continuous time version, the discrete time version of the model is very easy to implement computationally, and thus promises to become a powerful tool for exploring control scenarios for specific infectious diseases. To demonstrate the potential, we investigate numerically how the incidence-peak size depends on model ingredients. We find that, with the same reproduction number and initial speed of epidemic spread, compartmental models systematically predict lower peak sizes than models that use a fixed duration for the latent and infectious periods.

scholarly journals Realized generation times: contraction and impact of infectious period, reproduction number and population size

2019 ◽  
Author(s):  
Andrea Torneri ◽  
Amin Azmon ◽  
Christel Faes ◽  
Eben Kenah ◽  
Gianpaolo Scalia Tomba ◽  
...  
Keyword(s):  
Infectious Diseases ◽  
Generation Time ◽  
Time Distribution ◽  
Reproduction Number ◽  
Mean Value ◽  
Infectious Period ◽  
Calendar Time ◽  
Generation Times ◽  
The Mean ◽  
Over Time

AbstractOne of the key characteristics of the transmission dynamics of infectious diseases is the generation time which refers to the time interval between the infection of a secondary case and the infection of its infector. The generation time distribution together with the reproduction number determines the rate at which an infection spreads in a population. When defining the generation time distribution at a calendar time t two definitions are plausible according whether we regard t as the infection time of the infector or the infection time of the infectee. The resulting measurements are respectively called forward generation time and backward generation time. It has been observed that the mean forward generation time contracts around the peak of an epidemic. This contraction effect has previously been attributed to either competition among potential infectors or depletion of susceptibles in the population. The first explanation requires many infectives for contraction to occur whereas the latter explanation suggests that contraction occurs even when there are few infectives. With a simulation study we show that both competition and depletion cause the mean forward generation time to contract. Our results also reveal that the distribution of the infectious period and the reproduction number have a strong effect on the size and timing of the contraction, as well as on the mean value of the generation time in both forward and backward scheme.Author summaryInfectious diseases remain one of the greatest threats to human health and commerce, and the analysis of epidemic data is one of the most important applications of statistics in public health. Thus, having reliable estimates of fundamental infectious diseases parameters is critical for public health decision-makers in order to take appropriate actions for the global prevention and management of outbreaks and other health emergencies. A key example is given by the prediction models of the reproduction numbers: these rely on the generation time distribution that is usually estimated from contact tracing data collected at a precise calendar time. The forward scheme is used in such a prediction model and the knowledge of its evolution over time is crucial to correctly estimate the parameters of interest. It is therefore important to characterize the causes that lead to the contraction of the mean forward generation time during the course of an outbreak.In this paper, we firstly identify the impact of the epidemiological quantities as reproduction number, infectious period and population size on the mean forward and backward generation time. Moreover, we analyze the phenomena of competition among infectives and depletion of susceptible individuals highlighting their effects on the contraction of the mean forward generation time. The upshot of this investigation is that the variance of the infectious period distribution and the reproduction number have a strong impact on the generation times affecting both the mean value and the evolution over time. Furthermore, competition and depletion can both cause contraction even for small values of the reproduction number suggesting that, in epidemic models where the generation time is considered time-inhomogeneous, estimators accounting for both depletion and competing risks are to be preferred in the inference of the generation interval distributions.

scholarly journals Likelihood-based estimation of continuous-time epidemic models from time-series data: application to measles transmission in London

2008 ◽  
Vol 5 (25) ◽  
pp. 885-897 ◽  
Author(s):  
Simon Cauchemez ◽  
Neil M Ferguson
Keyword(s):  
Time Series ◽  
Discrete Time ◽  
Continuous Time ◽  
Generation Time ◽  
Data Augmentation ◽  
Time Series Data ◽  
Observation Interval ◽  
Series Data ◽  
Time Model ◽  
Similar Time

We present a new statistical approach to analyse epidemic time-series data. A major difficulty for inference is that (i) the latent transmission process is partially observed and (ii) observed quantities are further aggregated temporally. We develop a data augmentation strategy to tackle these problems and introduce a diffusion process that mimicks the susceptible–infectious–removed (SIR) epidemic process, but that is more tractable analytically. While methods based on discrete-time models require epidemic and data collection processes to have similar time scales, our approach, based on a continuous-time model, is free of such constraint. Using simulated data, we found that all parameters of the SIR model, including the generation time, were estimated accurately if the observation interval was less than 2.5 times the generation time of the disease. Previous discrete-time TSIR models have been unable to estimate generation times, given that they assume the generation time is equal to the observation interval. However, we were unable to estimate the generation time of measles accurately from historical data. This indicates that simple models assuming homogenous mixing (even with age structure) of the type which are standard in mathematical epidemiology miss key features of epidemics in large populations.

scholarly journals The SIR model towards the Data One year of Covid-19 pandemic in Italy case study and plausible "real" numbers.

2021 ◽  
Author(s):  
Ignazio Lazzizzera
Keyword(s):  
Generation Time ◽  
Reproduction Number ◽  
Removal Rate ◽  
Infectious Period ◽  
Third Wave ◽  
Real Numbers ◽  
Study Case ◽  
One Year ◽  
Good Agreement

Abstract In this work, the SIR epidemiological model is reformulated so to highlight the important effective reproduction number, as well as to account for the generation time, the inverse of the incidence rate, and the infectious period (or removal period), the inverse of the removal rate. The aim is to check whether the relationships the model poses among the various observables are actually found in the data. The study case of the second through the third wave of the Covid-19 pandemic in Italy is taken. Given its scale invariance, initially the model is tested with reference to the curve of swab-confirmed infectious individuals only. It is found to match the data if the curve of the removed (that is healed or deceased) individuals is assumed underestimated by a factor of about 3 together with other related curves. Contextually, the generation time and the removal period, as well as the effective reproduction number, are obtained fitting the SIR equations to the data; the outcomes prove to be in good agreement with those of other works. Then, using knowledge of the proportion of Covid-19 transmissions likely occurring from individuals who didn't develop symptoms, thus mainly undetected, an estimate of the "real numbers'' of the epidemic is obtained, looking also in good agreement with results from other, completely different works. The line of this work is new and the procedures, computationally really inexpensive, can be applied to any other national or regional case besides Italy's study case here.

scholarly journals The SIR model towards the data

2021 ◽  
Vol 136 (8) ◽  
Author(s):  
Ignazio Lazzizzera
Keyword(s):  
Generation Time ◽  
Reproduction Number ◽  
Removal Rate ◽  
Sir Model ◽  
Infectious Period ◽  
Third Wave ◽  
Real Numbers ◽  
The Third ◽  
Study Case ◽  
Good Agreement

AbstractIn this work, the SIR epidemiological model is reformulated so to highlight the important effective reproduction number, as well as to account for the generation time, the inverse of the incidence rate, and the infectious period (or removal period), the inverse of the removal rate. The aim is to check whether the relationships the model poses among the various observables are actually found in the data. The study case of the second through the third wave of the Covid-19 pandemic in Italy is taken. Given its scale invariance, initially the model is tested with reference to the curve of swab-confirmed infectious individuals only. It is found to match the data, if the curve of the removed (that is healed or deceased) individuals is assumed underestimated by a factor of about 3 together with other related curves. Contextually, the generation time and the removal period, as well as the effective reproduction number, are obtained fitting the SIR equations to the data; the outcomes prove to be in good agreement with those of other works. Then, using knowledge of the proportion of Covid-19 transmissions likely occurring from individuals who didn’t develop symptoms, thus mainly undetected, an estimate of the real numbers of the epidemic is obtained, looking also in good agreement with results from other, completely different works. The line of this work is new, and the procedures, computationally really inexpensive, can be applied to any other national or regional case besides Italy’s study case here.

scholarly journals Estimating effective reproduction number using generation time versus serial interval, with application to COVID-19 in the Greater Toronto Area, Canada

2020 ◽  
Author(s):  
Jesse Knight ◽  
Sharmistha Mishra
Keyword(s):  
Time Series ◽  
Generation Time ◽  
Time Distribution ◽  
Reproduction Number ◽  
Reproductive Number ◽  
Transmission Potential ◽  
Greater Toronto Area ◽  
Critical Measure ◽  
Serial Interval ◽  
Interval Distribution

AbstractBackgroundThe effective reproductive number Re(t) is a critical measure of epidemic potential. Re(t) can be calculated in near real time using an incidence time series and the generation time distribution—the time between infection events in an infector-infectee pair. In calculating Re(t), the generation time distribution is often approximated by the serial interval distribution—the time between symptom onset in an infector-infectee pair. However, while generation time must be positive by definition, serial interval can be negative if transmission can occur before symptoms, such as in covid-19, rendering such an approximation improper in some contexts.MethodsWe developed a method to infer the generation time distribution from parametric definitions of the serial interval and incubation period distributions. We then compared estimates of Re(t) for covid-19 in the Greater Toronto Area of Canada using: negative-permitting versus non-negative serial interval distributions, versus the inferred generation time distribution.ResultsWe estimated the generation time of covid-19 to be Gamma-distributed with mean 3.99 and standard deviation 2.96 days. Relative to the generation time distribution, non-negative serial interval distribution caused overestimation of Re(t) due to larger mean, while negative-permitting serial interval distribution caused underestimation of Re(t) due to larger variance.ImplicationsApproximation of the generation time distribution of covid-19 with non-negative or negative-permitting serial interval distributions when calculating Re(t) may result in over or underestimation of transmission potential, respectively.

scholarly journals The ideal reporting interval for an epidemic to objectively interpret the epidemiological time course

2009 ◽  
Vol 7 (43) ◽  
pp. 297-307 ◽  
Author(s):  
Hiroshi Nishiura ◽  
Gerardo Chowell ◽  
Hans Heesterbeek ◽  
Jacco Wallinga
Keyword(s):  
Infectious Diseases ◽  
Generation Time ◽  
Time Distribution ◽  
Time Course ◽  
Practical Purpose ◽  
Reproduction Number ◽  
The Mean ◽  
Corrected Expression ◽  
Epidemiological Characteristics ◽  
The Ideal

The reporting interval of infectious diseases is often determined as a time unit in the calendar regardless of the epidemiological characteristics of the disease. No guidelines have been proposed to choose the reporting interval of infectious diseases. The present study aims at translating coarsely reported epidemic data into the reproduction number and clarifying the ideal reporting interval to offer detailed insights into the time course of an epidemic. We briefly revisit the dispersibility ratio, i.e. ratio of cases in successive reporting intervals, proposed by Clare Oswald Stallybrass, detecting technical flaws in the historical studies. We derive a corrected expression for this quantity and propose simple algorithms to estimate the effective reproduction number as a function of time, adjusting the reporting interval to the generation time of a disease and demonstrating a clear relationship among the generation-time distribution, reporting interval and growth rate of an epidemic. Our exercise suggests that an ideal reporting interval is the mean generation time, so that the ratio of cases in successive intervals can yield the reproduction number. When it is impractical to report observations every mean generation time, we also present an alternative method that enables us to obtain straightforward estimates of the reproduction number for any reporting interval that suits the practical purpose of infection control.

Design of Programmable Wideband Low Pass Filter with Continuous-Time/Discrete-Time Hybrid Architecture

2017 ◽  
Vol E100.C (10) ◽  
pp. 858-865 ◽  
Author(s):  
Yohei MORISHITA ◽  
Koichi MIZUNO ◽  
Junji SATO ◽  
Koji TAKINAMI ◽  
Kazuaki TAKAHASHI
Keyword(s):  
Discrete Time ◽  
Continuous Time ◽  
Hybrid Architecture ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass

scholarly journals Time to Intervene: A Continuous-Time Approach to Network Analysis and Centrality

Psychometrika ◽  
2021 ◽  
Author(s):  
Oisín Ryan ◽  
Ellen L. Hamaker
Keyword(s):  
Network Analysis ◽  
Discrete Time ◽  
Continuous Time ◽  
Centrality Measures ◽  
Time Interval ◽  
Direct Effects ◽  
Network Approach ◽  
Var Models ◽  
Auto Regressive ◽  
Conceptual Shift

AbstractNetwork analysis of ESM data has become popular in clinical psychology. In this approach, discrete-time (DT) vector auto-regressive (VAR) models define the network structure with centrality measures used to identify intervention targets. However, VAR models suffer from time-interval dependency. Continuous-time (CT) models have been suggested as an alternative but require a conceptual shift, implying that DT-VAR parameters reflect total rather than direct effects. In this paper, we propose and illustrate a CT network approach using CT-VAR models. We define a new network representation and develop centrality measures which inform intervention targeting. This methodology is illustrated with an ESM dataset.

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