pith. sign in

arxiv: 1907.10679 · v2 · pith:Z25V5J42new · submitted 2019-07-24 · 🧬 q-bio.PE · math.DS· math.ST· stat.TH

Complete maximum likelihood estimation for SEIR epidemic models: theoretical development

Divine Wanduku , Chinmoy Rahul This is my paper

Pith reviewed 2026-05-24 16:13 UTC · model grok-4.3

classification 🧬 q-bio.PE math.DSmath.STstat.TH
keywords SEIR modelMarkov chainmaximum likelihood estimationEM algorithmepidemic modelingdiscrete timeinfectious disease dynamics
0
0 comments X

The pith

A class of discrete-time SEIR Markov chain models admits complete-data maximum likelihood estimation via the EM algorithm.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a family of SEIR Markov chain models for infectious diseases in a closed population where the numbers in each compartment evolve through random births, deaths, and state transitions at discrete time steps. Variants are distinguished by whether birth and death rates are zero or positive and by whether the incubation and infectious periods are fixed or drawn from distributions. Parameter estimation proceeds by writing the complete-data likelihood for the observed chain of states and maximizing it with the expectation-maximization algorithm. The resulting estimators are illustrated on simulated trajectories that recover the model dynamics. A sympathetic reader would care because the method supplies an explicit, approximation-free route from observed state counts to model parameters.

Core claim

The SEIR Markov chain models possess a complete-data likelihood that can be maximized directly by the expectation-maximization algorithm, yielding parameter estimates for the driving rates of births, deaths, and state transitions without further imputation or approximation steps.

What carries the argument

The complete-data likelihood of the discrete-time SEIR Markov chain, maximized via the expectation-maximization algorithm.

If this is right

  • Birth and death rates, together with transition probabilities, become identifiable from sequences of compartment counts.
  • The same estimation procedure applies uniformly to the four listed model subclasses (zero or positive birth-death, constant or random periods).
  • Numerical trajectories generated from the fitted models reproduce the statistical behavior of the original process.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same complete-data construction could be tried on related compartmental chains such as SIR or SEIRS.
  • If surveillance data supply the required state sequences, the estimators could be applied to real outbreaks without intermediate imputation.
  • The discrete-time formulation makes the method immediately compatible with regularly sampled public-health records.

Load-bearing premise

The complete-data likelihood for the SEIR Markov chain can be written explicitly and maximized by the EM algorithm without requiring extra approximations.

What would settle it

A set of simulated epidemic trajectories generated from known parameter values for which the EM procedure returns estimates that deviate systematically from the generating values.

Figures

Figures reproduced from arXiv: 1907.10679 by Chinmoy Rahul, Divine Wanduku.

Figure 1
Figure 1. Figure 1: Shows the states of the system: S, E, I, R, and the transition between states Ci , i = 1, 2, 3, 4, and also the births Bi , and deaths Di in the population. 2.2. Decomposition of the population over disease states and time In this section,we characterize the different disease subclasses namely: susceptible, exposed, infectious and recovered individuals over discrete time intervals of fixed length, for exam… view at source ↗
Figure 2
Figure 2. Figure 2: Shows three sample paths each for the states [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Shows the approximate distributions for the state [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Shows three sample paths each for the states [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Shows the approximate distributions for the state [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Shows the transition of the process {X(tk); k = 0, 1, 2, . . .} over time k = 0, 1, 2, . . . , T , and observed data Hˆ T = {xˆ(t0), xˆ(t1), xˆ(t2), . . . , xˆ(tT )}. The parameters Θ = (p, λ) are constant in the population at all times k = 0, 1, 2, . . . , T . We assume that we have data for the SEIR infectious disease such as Pneumonia or influenza over time units tk, k = 0, 1, 2, . . . , T denoted Hˆ T … view at source ↗
read the original abstract

We present a class of SEIR Markov chain models for infectious diseases observed over discrete time in a random human population living in a closed environment. The population changes over time through random births, deaths, and transitions between states of the population. The SEIR models consist of random dynamical equations for each state (S, E, I and R) involving driving events for the process. We characterize some special types of SEIR Markov chain models in the class including: (1) when birth and death are zero or non-zero, and (2) when the incubation and infectious periods are constant or random. A detailed parameter estimation applying the maximum likelihood estimation technique and expectation maximization algorithm are presented for this study. Numerical simulation results are given to validate the epidemic models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper presents a class of discrete-time SEIR Markov chain models for infectious disease spread in a closed population subject to random births, deaths, and state transitions. It characterizes special cases based on zero/non-zero birth-death rates and constant/random incubation/infectious periods. The central contribution is a claimed derivation of complete-data maximum likelihood estimation via the EM algorithm, with numerical simulations used for validation.

Significance. If the explicit complete-data likelihood, E-step, and M-step are correctly derived without hidden approximations or boundary issues, the work would supply a standard but useful rigorous estimation procedure for stochastic SEIR models that incorporate demographic processes. This could support more accurate inference in small or closed populations where continuous approximations are inappropriate. The simulation component provides a basic check but does not substitute for the missing analytic details.

major comments (2)
  1. [Abstract] Abstract (and parameter-estimation paragraph): the manuscript asserts that 'a detailed parameter estimation applying the maximum likelihood estimation technique and expectation maximization algorithm are presented' yet supplies no explicit expression for the complete-data likelihood, the observed-data likelihood, or the EM update equations. Without these, it is impossible to verify the central claim that EM yields the complete MLE or to check handling of boundary cases and convergence.
  2. [Abstract] The weakest assumption—that the models admit a complete-data likelihood maximizable by EM without further approximations—cannot be evaluated because the likelihood construction itself is not shown. This is load-bearing for the title and abstract claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We agree that the explicit derivations are necessary to support the central claims and will revise the manuscript to include them.

read point-by-point responses

  1. Referee: [Abstract] Abstract (and parameter-estimation paragraph): the manuscript asserts that 'a detailed parameter estimation applying the maximum likelihood estimation technique and expectation maximization algorithm are presented' yet supplies no explicit expression for the complete-data likelihood, the observed-data likelihood, or the EM update equations. Without these, it is impossible to verify the central claim that EM yields the complete MLE or to check handling of boundary cases and convergence.

    Authors: We agree that the explicit expressions for the complete-data likelihood, observed-data likelihood, and EM update equations must be provided to allow verification. The current manuscript states that these are presented but does not display the formulas in the abstract or estimation section. In the revision we will add the full construction of the complete-data likelihood from the Markov chain transition probabilities (including births, deaths, and SEIR state changes), the observed-data likelihood, the E-step expectation, and the closed-form M-step updates, together with a discussion of boundary cases and convergence. revision: yes

  2. Referee: [Abstract] The weakest assumption—that the models admit a complete-data likelihood maximizable by EM without further approximations—cannot be evaluated because the likelihood construction itself is not shown. This is load-bearing for the title and abstract claims.

    Authors: The referee is correct that the likelihood construction is load-bearing. The manuscript title and abstract claim a complete MLE via EM, yet the explicit likelihood is not displayed. We will expand the parameter-estimation section to derive the complete-data likelihood directly from the discrete-time Markov chain dynamics for the four model variants (zero/non-zero birth-death rates; constant/random incubation and infectious periods), showing that the EM algorithm applies without additional approximations when the complete data are available. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The provided abstract and context describe a standard construction of discrete-time SEIR Markov chain models with births/deaths, followed by application of MLE via the EM algorithm on the complete-data likelihood. This matches well-known statistical techniques for epidemic models and does not exhibit self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations. No equations or steps are quoted that reduce the central claim to its own inputs by construction. The derivation chain appears self-contained against external benchmarks for this model class.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; ledger entries cannot be populated from the provided text. The central claim rests on unstated assumptions about the form of the transition probabilities and the applicability of EM to the resulting incomplete-data likelihood.

pith-pipeline@v0.9.0 · 5659 in / 1099 out tokens · 18630 ms · 2026-05-24T16:13:57.752037+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    42(2000) 599

    H.W.Hethcote, the mathematics of infectious diseases, SIAM Rev. 42(2000) 599

    work page 2000

  2. [2]

    W. O. Kermack, A. G. McKendrick, A contribution to the mat hematical theory of epidemics part I, proc. Roy. Soc. Lond. A115 (1927) 700

    work page 1927

  3. [3]

    M Anderson, M

    R. M Anderson, M. Robert, Infectious diseases of humans: dynamics and control, Oxford university press, 1992

    work page 1992

  4. [4]

    Bernoulli, Essai d’une nouvelle analyse de la mortali te causee par la petite ve- role, et des avantage de l’Inoculation pour la prevenir, Mem

    D. Bernoulli, Essai d’une nouvelle analyse de la mortali te causee par la petite ve- role, et des avantage de l’Inoculation pour la prevenir, Mem . phys. Acade. Roy. Sci. 6(1760) 1. 50

    work page

  5. [5]

    Diekmann, J

    O. Diekmann, J. A. P . Heesterbeek, J. A. J. Metz, On the defi nition and the computa- tion of the basic reproduction ratio Ro in models for infecti ous diseases in heteroge- neous populations, J. Math. Biol. 28(1990):365-382

    work page 1990

  6. [6]

    V . Islam, Stochastic models for epidemics: current issu es and developments, in: cel- ebrating statistics: papers in honor of Sir David Cox on his 8 0th birthday, Oxford University press, Oxford, 2005

    work page 2005

  7. [7]

    Andersson, T

    H. Andersson, T. Britton, stochastic epidemic models an d their statistical analysis. In lectures notes in statistics: vol. 151, New Y ork, Springer, 2000

    work page 2000

  8. [8]

    Allen, an introduction to stochastic epidemic models , In: Brauer F., van den Driess- che P ., Wu J

    L. Allen, an introduction to stochastic epidemic models , In: Brauer F., van den Driess- che P ., Wu J. (eds) Mathematical Epidemiology. Lecture Note s in Mathematics, vol

    work page

  9. [9]

    Springer, Berlin, Heidelberg, 81-130

    work page

  10. [10]

    Keeling, J.V

    M.j. Keeling, J.V . Ross, on methods for studying stochas tic disease dynamics, J. R. Soc. Interface 5(2008), 171-181

    work page 2008

  11. [11]

    N. T. J. Bailey, the mathematical theory of infectious d iseases, New york, Griffin & Co., 1975

    work page 1975

  12. [12]

    Tuckwell, R,J

    H.C. Tuckwell, R,J. Williams: Some properties of a simp le stochastic epideic mode of SIR type. Math. Biosci. 208(2007), 76-97

    work page 2007

  13. [13]

    J. V . Ross, P . K. Pollett, On parameter estimation in population models, Theor. popul. biol. 70(2006), 498-510

    work page 2006

  14. [14]

    Gamerman, Markov chain Monte carlo: stochastic simu lation for beyesiaon infer- ence, london, UK: Chapman& Hall, 1997

    D. Gamerman, Markov chain Monte carlo: stochastic simu lation for beyesiaon infer- ence, london, UK: Chapman& Hall, 1997

    work page 1997

  15. [15]

    Tsutsui, N

    T. Tsutsui, N. Minamib, M. Koiwai, et. al., a stochastic -modeling evaluation of the foot-and-mouth-disease survey conducted after the outbre ak in Miyazaki, japan in 2000, Prev. vet. med. 61 (2003) 45

    work page 2000

  16. [16]

    Canto, C

    B. Canto, C. Coll, E. Sanchez, estimation of parameters in a structured SIR model, advances in difference equations, 2017(2017):33

    work page 2017

  17. [17]

    Raftery, S.J

    L Alkema, A.E. Raftery, S.J. Clark, probabilitic proje ctions of HIV prevalence using Bayesian melding, the annals of applied statistics, 2007, 2 29-248

    work page 2007

  18. [18]

    Riley, C

    S. Riley, C. Fraser, CA. Donnelly, et. al., transmissio n dynamics of thr etiologi- cal agent of SARS in Hong kong: impact of public health interv entions, Science, 300(5627): 1961-1966. 51

    work page 1961

  19. [19]

    Choi, GA

    B. Choi, GA. Rempala, inference for discretly observed stochastic kinetic networks with applications to epidemic modelling, biostatistics, 1 3 (2012): 153-165

    work page 2012

  20. [20]

    Y ang, A

    W. Y ang, A. Karspeck, J. Shaman, comparison of filtering methods for the modeling and retrospective forecasting of influenza epidemics, PLOS computational biology, 2014, 10

    work page 2014

  21. [21]

    Zimmer, R

    C. Zimmer, R. Y aesoubi, T. Chen, a likelhood approach fo r real time calibration of stochastic compartmental epidemic models, PLOS,computat ional biology,13(2017 ), e1005257,

    work page 2017

  22. [22]

    G. Chowell, fitting dynamic models to epidemic outbreak s with quantitified uncer- tainty: aprimer for parameter uncertainty, identifiabilit y, and forcasts, , infectious disease modeling, 2(2017), 379-398

    work page 2017

  23. [23]

    Fierro, A class of stochastic epidemic models and its deterministic counterpart, Jorual of the korean statistical society, 39(2010), 397-40 7

    R. Fierro, A class of stochastic epidemic models and its deterministic counterpart, Jorual of the korean statistical society, 39(2010), 397-40 7

    work page 2010

  24. [24]

    Wanduku, Statistical inferences and diffussion app roximation for SEIR Markov- chain models with birth and death processes, to appear (2019 )

    D. Wanduku, Statistical inferences and diffussion app roximation for SEIR Markov- chain models with birth and death processes, to appear (2019 )

    work page 2019

  25. [25]

    Wanduku, The stochastic extinction and stability co nditions for nonlinear malaria epidemics, Mathematical Biosciences and Engineering, 16( 2019): 3771-3806

    D. Wanduku, The stochastic extinction and stability co nditions for nonlinear malaria epidemics, Mathematical Biosciences and Engineering, 16( 2019): 3771-3806

    work page 2019

  26. [26]

    Fierro, V

    R. Fierro, V . Leiva, N. balakrishnan, Statistical infe rence on stochastic epidemic model, Communications in statistics-simulations and comp utation, 44(2015): 2297- 2314

    work page 2015

  27. [27]

    H.Abbey, An examination of the Reed-Frost theory of epidemics , Hum. Biol. 24 (1952) 201

    work page 1952

  28. [28]

    Y aesoubi, T

    R. Y aesoubi, T. Cohen, Generalized Markov models of infectious disease spread: A novel framework for developing dynamic health policies , European Journal of opera- tional Research 215 (2011) 679-687

    work page 2011

  29. [29]

    D. Wanduku, Threshold conditions for a family of epidemic dynamic model s for malaria with distributed delays in a non-random environmen t, International Journal of Biomathematics, V ol.11, No.6 (2018) 1850085 (46 pages)

    work page 2018

  30. [30]

    M. Y . Li, J. R. Graef et al., Global dynamics of SEIR model with varying total popu- lation size, Mathematical Biosciences 160 (1999) 191-213. 52

    work page 1999

  31. [31]

    M. D. L. Sen, S. Alonso-Quesada et al., On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Applied Mathematics and Computation,270 (2015) 953-976

    work page 2015

  32. [32]

    Etbaigha, A

    F. Etbaigha, A. R. Willms et al., An Seir model of influenza A virus infection and reinfection within a farrow-to-finish swine farm , PLOS one 13(9)

    work page

  33. [33]

    D. Wanduku, Complete global analysis of a two-scale network SIRS epidem ic dy- namic model with distributed delay and random perturbation s, Applied Mathematics and Computation 294 (2017) 49-76

    work page 2017

  34. [34]

    Wanduku, Global properties of a two-scale network stochastic delaye d human epidemic dynamic model , Nonlinear Analysis: Real World Applications 13 (2012) 794-816

    D. Wanduku, Global properties of a two-scale network stochastic delaye d human epidemic dynamic model , Nonlinear Analysis: Real World Applications 13 (2012) 794-816

    work page 2012

  35. [35]

    O. J. Otieno, M. Joseph et al., Mathematical Model for Pneumonia Dynamics with Carriers, Int. Journal Of Math. Analysis, 7 (2013), no. 50, 2457-2473

    work page 2013

  36. [36]

    Statistical Inference, Second Edition

    Casella, Berger. Statistical Inference, Second Edition. Duxbury (2002)

    work page 2002

  37. [37]

    F oundations and Trends in Signal Processing V ol

    M.Gupta and Y .Chen, Theory and use of the EM algorithm. F oundations and Trends in Signal Processing V ol. 4 No.3 (2010)

    work page 2010

  38. [38]

    Bilmes, A gentle tutorial of the EM algorithm and it’s Application to parameter estimation for Gausian mixture and hidden Markov models

    J. Bilmes, A gentle tutorial of the EM algorithm and it’s Application to parameter estimation for Gausian mixture and hidden Markov models. In ternational Computer Science Institute, April, (1998)

    work page 1998

  39. [39]

    M.Greenwood, On the statistical measure of infectiousness, J. Hyg. Camb. 31 (1931) 336

    work page 1931

  40. [40]

    J. Gani, D. Jerwood, Markov chain methods in chain binomial epidemic models. Biometrics 27, (1971)

    work page 1971

  41. [41]

    Mochan, D

    E. Mochan, D. Swigon et al., A mathematical model of intrahost pneumococcal pneumonia infection dynamics in murine strains, Journal of Theoretical Biology, 353 (2014) 44-54

    work page 2014

  42. [42]

    A. M. Smith, J. A. McCullers et al., Mathematical model of a three-stage innate immune response to a pneumococcal lung infection, Journal o f Theoretical Biology, 276 (2011) 106-116. 53

    work page 2011

  43. [43]

    Teshome, O

    G. Teshome, O. Daniel et al., Co-dynamics of Pneumonia and Typhoid diseases with cost effective optimal control analysis, Applied Mathemat ics and Computation, 316 (2018) 438-459

    work page 2018

  44. [44]

    K. E. Lamb, D. Greenhalgh et al., A simple mathematical model for genetic effects in pneumococcal carriage and transmission, Journal of Comp utational and Applied Mathematics, 235 (2011) 1812-1818. 54

    work page 2011