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Continuous and discrete compartmental models for infectious disease

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Pith's one-line read A curve-fitting comparison of cellular automaton and ODE versions of SI, SIR, and SEIR models finds power-law versus exponential early growth and proposes a hyperbolic tangent fit for both.

arxiv 2504.13953 v1 pith:T2Y5QWTB submitted 2025-04-16 q-bio.PE physics.bio-ph

Gustavo A. Sousa , Diogo L. M. Souza , Enrique C. Gabrick , Patrício D. C. dos Reis , Lucas E. Bentivoglio , Antonio M. Batista , José D. Szezech Jr This is my paper · ORCID
classification q-bio.PEphysics.bio-ph
keywords modelsmathematicaldiseasefunctiongrowthdiseasesexponentialhyperbolic
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The pith

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The reading

Epidemiologists often describe an outbreak with compartmental models that divide people into susceptible, infected, and recovered groups. This paper compares two ways of running those models: a cellular automaton on a grid, where each individual is a cell that infects nearby neighbors with some probability, and a set of differential equations, where the whole population is treated as one well-mixed fluid. The authors simulate SI, SIR, and SEIR versions with both approaches and look at how the number of infected people grows at the beginning of the outbreak. In the grid model, the early increase looks like a power law (a line on a log-log plot), while in the differential-equation version it looks exponential (a line on a semi-log plot), at least for two of the three models. They then fit a hyperbolic tangent function to every early outbreak curve and obtain high R-squared values, concluding that this sigmoid shape can represent both growth styles.

The main issue is that the comparison is descriptive curve fitting, not a derivation. The parameters in the fit are chosen after the data are generated, and no alternative functions are tested. The paper also sets the infection probability equal in both representations, but in the grid model this probability acts per infected neighbor per time step while in the equations it acts as a per-capita contact rate; equating the numbers does not make the models dynamically equivalent. For the SEIR differential-equation model the exponential fit is actually poor (R-squared 0.75), which contradicts the abstract's claim that ODE growth is well represented by an exponential. The hyperbolic tangent result is essentially a tautology: any smooth S-shaped curve can be fitted by a bounded sigmoid with enough parameters.

Extended reading notes

Core claim

The abstract states: 'the CA yields a power-law growth, while the ODE growth rate is well-represented by an exponential function' and 'a substantial contribution of our work is using a hyperbolic tangent to fit the initial growth of infected individuals for all the considered models.' If correct, the paper would show that a single sigmoid function describes epidemic onset in both discrete spatial and continuous mean-field frameworks.

Load-bearing premise

The assumption that using numerically equal parameter values in both representations makes the models comparable: in the CA, beta acts as a per-neighbor infection probability per time step, while in the ODE it is a per-capita contact rate per unit time (Tables 1-3 and the text 'once selected the parameters values, they need to be equal for both approaches'). If the rates are not truly equivalent, the observed difference between power-law and exponential growth could reflect a mismatch in transmission speed rather than an intrinsic property of discrete versus continuous representation.

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Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The paper's central claims rest on arbitrary parameter choices, in-sample curve fits, and an unexamined equivalence between CA and ODE rate parameters. No new entities are introduced.

free parameters (8)
  • Beta (SI, SIR, SEIR) = 0.1, 0.2, 0.25
    Chosen arbitrarily; set equal for CA and ODE to allow comparison, but the meaning differs between per-neighbor probability in CA and per-capita contact rate in ODE.
  • Gamma (SIR, SEIR) = 0.1
    Recovery probability/rate, chosen arbitrarily.
  • Omega (SEIR) = 0.2
    Incubation rate, chosen arbitrarily.
  • Initial infected/exposed fraction = 0.01 (100 cells in CA; 0.01 in ODE)
    Selected to match between representations, but the spatial arrangement in CA is random.
  • Fitting window cutoff = I = 0.15 N
    All fits are limited to the early phase up to 15 percent of the population; the cutoff is chosen by hand.
  • Power-law K and B for CA fits = SI: K=0.007, B=1.09; SIR: K=0.017, B=0.87; SEIR: K=0.0069, B=1.21
    Fitted to the simulated CA i(t) curves in Sections 3, 4, and 5.
  • Exponential K and B for ODE fits = SI: K=0.01076, B=0.09262; SIR: K=0.0175, B=0.0597; SEIR: K=0.00077, B=0.116
    Fitted to the simulated ODE i(t) curves in Sections 3, 4, and 5.
  • Tanh parameters alpha, xi, phi = CA SI: alpha=0.115, xi=0.108, phi=-1.27; ODE SI: alpha=0.500, xi=0.050, phi=-2.298; CA SIR: alpha=0.090, xi=0.150…
    Three parameters fitted to each early i(t) curve; these are the basis for the claimed hyperbolic tangent result.
assumptions (5)
  • standard math Standard SIR and SEIR ODE compartmental equations and their analytic solutions
    Used in Eqs. (1)-(4), (8)-(10), and (11)-(14).
  • domain assumption Von Neumann neighborhood and periodic boundary conditions in CA
    Defines the local interaction structure of the cellular automaton.
  • domain assumption Homogeneous mixing assumption in ODE models
    Standard mean-field assumption in compartmental ODEs.
  • ad hoc to paper Equivalence of numerically equal beta, gamma, and omega across CA and ODE
    The paper sets parameters equal across representations but does not justify that the per-neighbor probability in CA matches the per-capita rate in ODE; the comparison of growth rates relies on this.
  • standard math Series expansion and exponential representation of the hyperbolic tangent
    Used in the Conclusions to argue tanh captures both power-law and exponential growth.

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Pith. "Pith review of Continuous and discrete compartmental models for infectious disease." pith.science (2026). https://pith.science/paper/T2Y5QWTB

@misc{pith2026250413953,
  author       = {Pith},
  title        = {Pith review of: Continuous and discrete compartmental models for infectious disease},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T2Y5QWTB}},
  note         = {Machine review of arXiv:2504.13953}
}
read the original abstract

The study of infectious disease propagation is essential for understanding and controlling epidemics. One of the most useful tools for gaining insights into the spread of infectious diseases is mathematical modelling. In terms of mathematical epidemiology, the main models are based on compartments, such as SI, SIR, and SEIR. These models offer mathematical frameworks for representing the proliferation dynamics of various diseases, for instance flu and smallpox. In this work, we explore these models using two distinct mathematical approaches, Cellular Automata (CA) and ODEs. They are able to reproduce the spread dynamics of diseases with their own individuality. CA models incorporate the local interaction among individuals with discrete time and space, while ODEs provide a continuous and simplified view of a disease propagation in large and homogeneous populations. By comparing these two approaches, we find that the shape of the curves of all models is similar for both representations. Although, the growth rates differ between CA and ODE. One of our results is to show that the CA yields a power-law growth, while the ODE growth rate is well-represented by an exponential function. Furthermore, a substantial contribution of our work is using a hyperbolic tangent to fit the initial growth of infected individuals for all the considered models. Our results display a strong correlation between simulated data and adjusted function. We mainly address this successful result by the fact that the hyperbolic function captures both growing: the power-law (when considered the first terms of infinite sums) and combinations of exponential (when the hyperbolic function is written via exponential). Therefore, our work shows that when modelling a disease the choice of mathematical representation is crucial, in particular to model the onset of an epidemic.

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