Equivalence of the Erlang-Distributed SEIR Epidemic Model and the Renewal Equation
Abstract
Most compartmental epidemic models can be represented using the renewal equation. The value of the renewal equation is not widely appreciated in the epidemiological modelling community, perhaps because its equivalence to standard models has not been presented rigorously in nontrivial cases. Here, we provide analytical expressions for the intrinsic generation-interval distribution that must be used in the renewal equation in order to yield epidemic dynamics that are identical to those of the susceptible-exposed-infectious-recovered (SEIR) compartmental model with Erlang-distributed latent and infectious periods. This class of models includes the standard (exponentially distributed) SIR and SEIR models as special cases.
Keywords
MSC codes
Formats available
You can view the full content in the following formats:
Supplementary Material
PLEASE NOTE: These supplementary files have not been peer-reviewed.
Index of Supplementary Materials
Title of paper: Equivalence of the Erlang-Distributed SEIR Epidemic Model and the Renewal Equation
Authors: David Champredon, Jonathan Dushoff, and David J. D. Earn
File: M118641SupMat.zip
Type: compressed files
Contents: R scripts and makefile
References
1.
D. Anderson and R. Watson, On the spread of a disease with gamma distributed latent and infectious periods, Biometrika, 67 (1980), pp. 191--198.
2.
R. M. Anderson and R. M. May, Infectious Diseases of Humans - Dynamics and Control, Oxford University Press, New York, 1991.
3.
J. Arino and P. van den Driessche, Time delays in epidemic models - modeling and numerical considerations, in Delay Differential Equations and Applications, Springer, New York, 2006, pp. 539--578.
4.
N. T. J. Bailey, Some stochastic models for small epidemics in large populations, Appl. Statist., 13 (1964), pp. 9--19.
5.
D. Breda, O. Diekmann, W. F. de Graaf, A. Pugliese, and R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dyn., 6 (2012), pp. 103--117.
6.
S. Butler and P. Karasik, A note on nested sums, J. Integer Seq., 13 (2010), 10.4.4.
7.
D. Champredon and J. Dushoff, Intrinsic and realized generation intervals in infectious-disease transmission, Roy. Soc. Lond. Proc. Ser. Biol. Sci., 282 (2015), 20152026.
8.
D. Champredon, M. Li, B. Bolker, and J. Dushoff, Two approaches to forecast Ebola synthetic epidemics, Epidemics, 22 (2018), pp. 36--42.
9.
O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Anal., 1 (1977), pp. 459--470.
10.
O. Diekmann, J. Heesterbeek, and J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), pp. 365--382.
11.
L. Euler, Recherches générales sur la mortalité et la multiplication du genre humain, Mém. Acad. R. Sci. Belles Lett., 16 (1760), pp. 144--164.
12.
D. Fargue, Réducibilité des systèmes héréditaires, Int. J. Nonlinear Mech., 9 (1974), pp. 331--338.
13.
Z. Feng and H. Thieme, Endemic models with arbitrarily distributed periods of infection. I: Fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), pp. 803--833, https://doi.org/10.1137/S0036139998347834.
14.
Z. Feng, D. Xu, and H. Zhao, Epidemiological models with non-exponentially distributed disease stages and applications to disease control, Bull. Math. Biol., 69 (2007), pp. 1511--1536.
15.
P. E. M. Fine, The interval between successive cases of an infectious disease, Am. J. Epidemiology, 158 (2003), pp. 1039--1047.
16.
J. L. W. Gielen, A stochastic model for epidemics based on the renewal equation, J. Biol. Syst., 08 (2000), pp. 1--20.
17.
H. B. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases, Math. Biosci. Eng., 3 (2006), pp. 513--525.
18.
P. J. Hurtado and A. S. Kirosingh, Generalizations of the “Linear Chain Trick”: Incorporating More Flexible Dwell Time Distributions into Mean Field ODE Models, preprint, https://arxiv.org/abs/1808.07571, 2018.
19.
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London. Ser. A., 115 (1927), pp. 700--721.
20.
O. Krylova and D. J. D. Earn, Effects of the infectious period distribution on predicted transitions in childhood disease dynamics, J. Royal Soc. Interface, 10 (2013), 20130098.
21.
A. L. Lloyd, Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics, Theoret. Popul. Biol, 60 (2001), pp. 59--71.
22.
A. J. Lotka, Relation between birth rates and death rates, Science, 26 (1907), pp. 21--22.
23.
J. A. J. Metz, The epidemic in a closed population with all susceptible equally vulnerable; some results for large susceptible populations and small initial infections, Acta Biotheoretica, 27 (1978), pp. 75--123.
24.
NIST Handbook of Mathematical Functions, National Institute of Standards and Technology (NIST) and Cambridge University Press, New York, 2010, https://dlmf.nist.gov/.
25.
P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), pp. 29--48.
26.
J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers, Roy. Soc. Lond. Proc. Ser. Biol. Sci., 274 (2007), pp. 599--604.
27.
H. J. Wearing, P. Rohani, and M. J. Keeling, Appropriate models for the management of infectious diseases, PLoS Med., 2 (2005), e174.
28.
WHO Ebola Response Team, Ebola virus disease in West Africa --- The first \textup9 months of the epidemic and forward projections, N. Engl. J. Med., 371 (2014), pp. 1481--1495.
29.
A. Wörz-Busekros, Global stability in ecological systems with continuous time delay, SIAM J. Appl. Math., 35 (1978), pp. 123--134, https://doi.org/10.1137/0135011.
Information & Authors
Information
Published In
SIAM Journal on Applied Mathematics
Pages: 3258 - 3278
DOI: 10.1137/18M1186411
ISSN (online): 1095-712X
Copyright
© 2018, Society for Industrial and Applied Mathematics. This work is licensed under a Creative Commons Attribution 4.0 International License
History
Submitted: 10 May 2018
Accepted: 3 October 2018
Published online: 11 December 2018
Keywords
MSC codes
Authors
Funding Information
Canadian Institutes of Health Research https://doi.org/10.13039/501100000024
Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Multiphasic stochastic epidemic modelsJournal of the Royal Statistical Society Series C: Applied Statistics, Vol. 74, No. 2 | 21 November 2024
- Time-varying reproductive number estimation for practical application in structured populationsEpidemiologic Methods, Vol. 14, No. 1 | 6 January 2025
- Lyapunov stability tests for integral delay systemsAnnual Reviews in Control, Vol. 59 | 1 Jan 2025
- Validity of Markovian modeling for transient memory-dependent epidemic dynamicsCommunications Physics, Vol. 7, No. 1 | 8 March 2024
- Advancing Mathematical Epidemiology and Chemical Reaction Network Theory via Synergies Between ThemEntropy, Vol. 26, No. 11 | 31 October 2024
- Fitting Epidemic Models to Data: A Tutorial in Memory of Fred BrauerBulletin of Mathematical Biology, Vol. 86, No. 9 | 25 July 2024
- Uniform Asymptotic Approximations for the Phase Plane Trajectories of the SIR Model with Vital DynamicsSIAM Journal on Applied Mathematics, Vol. 84, No. 4 | 22 July 2024
- Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with ExamplesMathematics, Vol. 12, No. 15 | 4 August 2024
- Impact of infectious diseases on wild bovidae populations in Thailand: insights from population modelling and disease dynamicsJournal of The Royal Society Interface, Vol. 21, No. 216 | 3 July 2024
- ern: An R package to estimate the effective reproduction number using clinical and wastewater surveillance dataPLOS ONE, Vol. 19, No. 6 | 21 June 2024
- flepiMoP: The evolution of a flexible infectious disease modeling pipeline during the COVID-19 pandemicEpidemics, Vol. 47 | 1 Jun 2024
- Global analysis of an age-structured tuberculosis model with an application to Jiangsu, ChinaJournal of Mathematical Biology, Vol. 88, No. 5 | 2 April 2024
- Generative Bayesian modeling to nowcast the effective reproduction number from line list data with missing symptom onset datesPLOS Computational Biology, Vol. 20, No. 4 | 16 April 2024
- Incorporating testing volume into estimation of effective reproduction number dynamicsJournal of the Royal Statistical Society Series A: Statistics in Society, Vol. 187, No. 2 | 13 December 2023
- On the almost periodic and almost automorphic solution for linear renewal equations with infinite delay via reduction principleChaos, Solitons & Fractals, Vol. 181 | 1 Apr 2024
- Robust Stability Analysis for Integral Delay Systems: A Complete Type Functional ApproachIEEE Transactions on Automatic Control, Vol. 69, No. 4 | 1 Apr 2024
- Analysis of optimal lockdown in integral economic–epidemic modelEconomic Theory, Vol. 77, No. 1-2 | 16 November 2022
- The probability of epidemic burnout in the stochastic SIR model with vital dynamicsProceedings of the National Academy of Sciences, Vol. 121, No. 5 | 26 January 2024
- Analysis and prediction of omicron infection cases daily in the United Kingdom in November and December 2021Biostatistics & Epidemiology, Vol. 8, No. 1 | 25 May 2024
- Semi-mechanistic Bayesian modelling of COVID-19 with renewal processesJournal of the Royal Statistical Society Series A: Statistics in Society, Vol. 186, No. 4 | 28 February 2023
- Proposer of the vote of thanks and contribution to the Discussion of ‘The Second Discussion Meeting on Statistical aspects of the Covid-19 Pandemic’Journal of the Royal Statistical Society Series A: Statistics in Society, Vol. 186, No. 4 | 9 June 2023
- When to lock, not whom: Managing epidemics using time-based restrictionsReview of Economic Dynamics, Vol. 51 | 1 Dec 2023
- Scale-free dynamics of COVID-19 in a Brazilian cityApplied Mathematical Modelling, Vol. 121 | 1 Sep 2023
- Unifying incidence and prevalence under a time-varying general branching processJournal of Mathematical Biology, Vol. 87, No. 2 | 1 August 2023
- Improved time-varying reproduction numbers using the generation interval for COVID-19Frontiers in Public Health, Vol. 11 | 30 June 2023
- Effect of sojourn time distributions on the early dynamics of COVID-19 outbreakNonlinear Dynamics, Vol. 111, No. 12 | 19 April 2023
- A hospital demand and capacity intervention approach for COVID-19PLOS ONE, Vol. 18, No. 5 | 3 May 2023
- Heterogeneity is a key factor describing the initial outbreak of COVID-19Applied Mathematical Modelling, Vol. 117 | 1 May 2023
- Erlang-Distributed SEIR Epidemic Models with Cross-DiffusionMathematics, Vol. 11, No. 9 | 5 May 2023
- A discrete model for the evaluation of public policies: The case of Colombia during the COVID-19 pandemicPLOS ONE, Vol. 18, No. 2 | 14 February 2023
- United Arab Emirates (UAE) leadership in controlling COVID-19: practical policy steps and outcomesJournal of Global Health Economics and Policy, Vol. 3 | 22 March 2023
- Numerical methods and hypoexponential approximations for gamma distributed delay differential equationsIMA Journal of Applied Mathematics, Vol. 87, No. 6 | 13 December 2022
- Extended transit compartment model to describe tumor delay using Coxian distributionScientific Reports, Vol. 12, No. 1 | 16 June 2022
- Stability of Integral Delay Equations and Their Application in Infectious Disease Modeling2022 10th International Conference on Systems and Control (ICSC) | 23 Nov 2022
- Fractional transit compartment model for describing drug delayed response to tumors using Mittag-Leffler distribution on age-structured PKPD modelPLOS ONE, Vol. 17, No. 11 | 4 November 2022
- Model with transmission delays for COVID‐19 control: Theory and empirical assessmentJournal of Public Economic Theory, Vol. 24, No. 5 | 23 November 2021
- Filtering and improved Uncertainty Quantification in the dynamic estimation of effective reproduction numbersEpidemics, Vol. 40 | 1 Sep 2022
- Estimating the effective reproduction number for heterogeneous models using incidence dataRoyal Society Open Science, Vol. 9, No. 9 | 7 September 2022
- Importation, Local Transmission, and Model Selection in Estimating the Transmissibility of COVID-19: The Outbreak in Shaanxi Province of China as a Case StudyTropical Medicine and Infectious Disease, Vol. 7, No. 9 | 3 September 2022
- The effects of local homogeneity assumptions in metapopulation models of infectious diseaseRoyal Society Open Science, Vol. 9, No. 7 | 13 July 2022
- Bayesian sequential data assimilation for COVID-19 forecastingEpidemics, Vol. 39 | 1 Jun 2022
- The importance of the generation interval in investigating dynamics and control of new SARS-CoV-2 variantsJournal of The Royal Society Interface, Vol. 19, No. 191 | 15 June 2022
- Face masking and COVID-19: potential effects of variolation on transmission dynamicsJournal of The Royal Society Interface, Vol. 19, No. 190 | 4 May 2022
- The role of time-varying viral shedding in modelling environmental surveillance for public health: revisiting the 2013 poliovirus outbreak in IsraelJournal of The Royal Society Interface, Vol. 19, No. 190 | 18 May 2022
- Fundamental limits on inferring epidemic resurgence in real time using effective reproduction numbersPLOS Computational Biology, Vol. 18, No. 4 | 11 April 2022
- Necessary and sufficient stability conditions for integral delay systemsInternational Journal of Robust and Nonlinear Control, Vol. 32, No. 6 | 23 November 2021
- Analytical solutions and parameter estimation of the SIR epidemic modelMathematical Analysis of Infectious Diseases | 1 Jan 2022
- Lyapunov Matrices for Integral Delay Systems with Piecewise Constant KernelIFAC-PapersOnLine, Vol. 55, No. 36 | 1 Jan 2022
- Stage duration distributions and intraspecific competition: a review of continuous stage-structured modelsMathematical Biosciences and Engineering, Vol. 19, No. 8 | 1 Jan 2022
- Model-based evaluation of school- and non-school-related measures to control the COVID-19 pandemicNature Communications, Vol. 12, No. 1 | 12 March 2021
- Understanding the effectiveness of government interventions against the resurgence of COVID-19 in EuropeNature Communications, Vol. 12, No. 1 | 5 October 2021
- On the use of aggregated human mobility data to estimate the reproduction numberScientific Reports, Vol. 11, No. 1 | 2 December 2021
- Spatio-temporal dynamics of a model for the effect of variable ages at reproductionNonlinearity, Vol. 34, No. 9 | 19 July 2021
- Computer Simulation and Epidemic Preventive Countermeasures Using SEIR Model and Big Data TechnologyJournal of Physics: Conference Series, Vol. 2033, No. 1 | 1 Sep 2021
- Cure and death play a role in understanding dynamics for COVID-19: Data-driven competing risk compartmental models, with and without vaccinationPLOS ONE, Vol. 16, No. 7 | 15 July 2021
- Building mean field ODE models using the generalized linear chain trick & Markov chain theoryJournal of Biological Dynamics, Vol. 15, No. sup1 | 13 April 2021
- A phenomenological estimate of the true scale of CoViD-19 from primary dataChaos, Solitons & Fractals, Vol. 146 | 1 May 2021
- Speed and strength of an epidemic interventionProceedings of the Royal Society B: Biological Sciences, Vol. 288, No. 1947 | 24 March 2021
- SEAHIR: A Specialized Compartmental Model for COVID-19International Journal of Environmental Research and Public Health, Vol. 18, No. 5 | 6 March 2021
- Forecasting hospital demand in metropolitan areas during the current COVID-19 pandemic and estimates of lockdown-induced 2nd wavesPLOS ONE, Vol. 16, No. 1 | 22 January 2021
- Forward-looking serial intervals correctly link epidemic growth to reproduction numbersProceedings of the National Academy of Sciences, Vol. 118, No. 2 | 23 December 2020
- Distributed Delay Differential Equation Representations of Cyclic Differential EquationsSIAM Journal on Applied Mathematics, Vol. 81, No. 4 | 24 August 2021
- Range of reproduction number estimates for COVID-19 spreadBiochemical and Biophysical Research Communications, Vol. 538 | 1 Jan 2021
- A generalized differential equation compartmental model of infectious disease transmissionInfectious Disease Modelling, Vol. 6 | 1 Jan 2021
- Bayesian analysis of Glucose dynamics during the Oral Glucose Tolerance Test (OGTT)Mathematical Biosciences and Engineering, Vol. 18, No. 4 | 1 Jan 2021
- Practical considerations for measuring the effective reproductive number, RtPLOS Computational Biology, Vol. 16, No. 12 | 10 December 2020
- Estimating epidemic coupling between populations from the time to invasionJournal of The Royal Society Interface, Vol. 17, No. 172 | 25 November 2020
- Spatial and temporal regularization to estimate COVID-19 reproduction number R(t): Promoting piecewise smoothness via convex optimizationPLOS ONE, Vol. 15, No. 8 | 20 August 2020
- A Primer on COVID‐19 Mathematical ModelsObesity, Vol. 28, No. 8 | 22 July 2020
- Hitchhiking, collapse, and contingency in phage infections of migrating bacterial populationsThe ISME Journal, Vol. 14, No. 8 | 1 May 2020
- Evaluation of COVID-19 epidemic outbreak caused by temporal contact-increase in South KoreaInternational Journal of Infectious Diseases, Vol. 96 | 1 Jul 2020
- Inferring generation-interval distributions from contact-tracing dataJournal of The Royal Society Interface, Vol. 17, No. 167 | 24 June 2020
- Building New Models: Rethinking and Revising ODE Model AssumptionsAn Introduction to Undergraduate Research in Computational and Mathematical Biology | 18 February 2020
- Spatially Adjusted Time-varying Reproductive Numbers: Understanding the Geographical Expansion of Urban Dengue OutbreaksScientific Reports, Vol. 9, No. 1 | 16 December 2019
- Preface to theme issue ‘Modelling infectious disease outbreaks in humans, animals and plants: epidemic forecasting and control’Philosophical Transactions of the Royal Society B: Biological Sciences, Vol. 374, No. 1776 | 20 May 2019
View Options
View options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].