pith. sign in

arxiv: 2505.22134 · v2 · submitted 2025-05-28 · 🧬 q-bio.PE

Infection dynamics for fluctuating infection or removal rates regarding the number of infected and susceptible individuals

Seong Jun Park , M.Y. Choi This is my paper

Pith reviewed 2026-05-19 14:12 UTC · model grok-4.3

classification 🧬 q-bio.PE
keywords infection dynamicsnonlinear ratesSIR modelanalytic expressionstochastic dynamicsepidemiologyinfected population
0
0 comments X

The pith

An analytic expression gives the number of infected individuals at each moment for arbitrary nonlinear infection and removal rates.

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

The paper derives a closed-form analytic expression for the time-dependent number of infected people in a stochastic model where both the infection rate and the removal rate are general nonlinear functions of the current susceptible and infected population sizes. This matters because classic SIR models restrict rates to linear forms, yet real epidemics often involve nonlinear dependencies, and analytic results for the stochastic case have been scarce. The derivation tracks how the infected count changes as new cases appear, supplying an explicit formula that replaces simulation for this broader class of rate functions. If the expression holds, it supplies a direct quantitative route to predicting infection trajectories under more realistic rate assumptions.

Core claim

This work presents an analytic expression for the number of infected individuals when infection and removal rates are arbitrary nonlinear functions of the susceptible and infected populations. In particular, the expression accounts for the variation in the number of infected individuals at each moment as cases emerge in the stochastic dynamics.

What carries the argument

Analytic closed-form expression for the infected population size obtained from the stochastic dynamics with general nonlinear transition rates.

If this is right

  • The infected count at any time follows directly from the chosen nonlinear rate functions without requiring stochastic simulation.
  • The approach covers fluctuating rates that depend on both susceptible and infected numbers simultaneously.
  • It extends earlier deterministic nonlinear models to the stochastic setting.
  • The expression supplies a quantitative foundation for analyzing case emergence under general rate assumptions.

Where Pith is reading between the lines

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

  • The same derivation strategy might be tested on real outbreak data to check whether the analytic form improves short-term forecasts over linear-rate models.
  • Similar closed-form techniques could be explored for multi-strain or spatially structured extensions of the same nonlinear-rate framework.
  • The method may link to analytic treatments of nonlinear birth-death processes in other population models.

Load-bearing premise

That a closed-form analytic solution can be derived for the stochastic process when infection and removal rates are arbitrary nonlinear functions of the current population sizes.

What would settle it

A direct numerical simulation of the stochastic process for a specific nonlinear rate pair that produces a trajectory differing from the claimed analytic expression.

Figures

Figures reproduced from arXiv: 2505.22134 by M.Y. Choi, Seong Jun Park.

Figure 1
Figure 1. Figure 1: Infected (I), removed (R), and susceptible (S) populations with nonlinear infection and removal rates without the arrival of infected cases. Circles and lines plot data from simulations and theoretical results from Eqs. (4) to (6), respectively, for rates (a) 1.2 0.6 , 0.07 ns  = ns and 1.3 0.2 n  = n ; (b) 0.8 , 0.2( ) ns  =+ns and 1.2 0.1 n  = n . It is of interest to compare Eq. (3) with the number … view at source ↗
Figure 2
Figure 2. Figure 2: Infected (I), removed (R), and susceptible (S) populations resulting from Eqs. (4) to (6) (solid lines) and from the SIR model described by Eq. (7) (broken lines). Rates are given by (a) ( n s n , , 0.1 ,0.12 ) = ( ns n) in Eqs. (4) - (6) and (, 0.1,0.12 ) = ( ) in Eq. (7); (b) ( n s n , , 0.05 ,0.2 ) = ( ns n) in Eqs. (4) - (6) and (, 0.05,0.2 ) = ( ) in Eq. (7). 3. Infection dynamics with the arr… view at source ↗
Figure 3
Figure 3. Figure 3: Infected (I), removed (R), and susceptible (S) populations with nonlinear infection and removal rates with the arrival of infected cases. The number of infected individuals introduced from outside is 10, and introduced infected cases occur every 1.3-time units. Circles and lines plot data from simulations and [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
read the original abstract

In general, the rates of infection and removal (whether through recovery or death) are nonlinear functions of the number of infected and susceptible individuals. One of the simplest models for the spread of infectious diseases is the SIR model, which categorizes individuals as susceptible, infectious, recovered or deceased. In this model, the infection rate, governing the transition from susceptible to infected individuals, is given by a linear function of both susceptible and infected populations. Similarly, the removal rate, representing the transition from infected to removed individuals, is a linear function of the number of infected individuals. While nonlinear infection and removal rates have been extensively studied in deterministic epidemiological models, analytic results for stochastic dynamics with general nonlinear rates remain limited. This work presents an analytic expression for the number of infected individuals considering nonlinear infection and removal rates. In particular, we examine how the number of infected individuals varies as cases emerge and obtain the expression accounting for the number of infected individuals at each moment. This work paves the way for new quantitative approaches to understanding infection dynamics.

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 / 2 minor

Summary. The manuscript claims to derive an analytic expression for the time-dependent number of infected individuals I(t) in a stochastic SIR-type model in which the infection rate is an arbitrary nonlinear function of both S and I and the removal rate is an arbitrary nonlinear function of I.

Significance. If a genuine closed-form solution to the master equation exists for unrestricted nonlinear rates, the result would be significant: it would supply an exact, non-numerical description of stochastic infection trajectories that is currently unavailable for generic nonlinear epidemiological models and could support new quantitative predictions without simulation.

major comments (2)
  1. [Abstract] Abstract, paragraph 3: the central claim that an analytic expression has been obtained for arbitrary nonlinear rates is not accompanied by any derivation, explicit functional form, or verification; for generic β(S,I) and γ(I) the Kolmogorov forward equation for P(S,I,t) has no known closed-form solution, so the manuscript must either restrict the functional forms, reduce to the deterministic limit, or present an unevaluated integral/series whose validity is demonstrated.
  2. [Derivation] The manuscript does not show how the stochastic master equation is solved without introducing an unstated ansatz that effectively linearizes the rates or converts the problem to moment closure; this omission is load-bearing for the headline claim of a general analytic result in the stochastic setting.
minor comments (2)
  1. Define the stochastic process (continuous-time Markov chain on (S,I)) and the precise form of the transition rates explicitly before stating the analytic expression.
  2. Clarify whether the reported expression is for the expectation E[I(t)], the most probable value, or the full probability distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important issues of clarity and completeness in presenting the derivation, which we address point by point below. We will revise the manuscript to improve transparency while preserving the core contribution of an analytic description for the infected population under nonlinear rates.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 3: the central claim that an analytic expression has been obtained for arbitrary nonlinear rates is not accompanied by any derivation, explicit functional form, or verification; for generic β(S,I) and γ(I) the Kolmogorov forward equation for P(S,I,t) has no known closed-form solution, so the manuscript must either restrict the functional forms, reduce to the deterministic limit, or present an unevaluated integral/series whose validity is demonstrated.

    Authors: We agree that the abstract and main text require additional detail to substantiate the claim. The analytic expression we derive is an implicit integral form for the time-dependent expected number of infected individuals, obtained by integrating the nonlinear rate equations directly; this holds exactly in the deterministic (mean-field) limit and for the expectation in the stochastic setting without further approximation. We will revise the abstract to state this scope explicitly, add the explicit integral expression and its derivation in the main text, and include numerical verification for representative nonlinear β(S,I) and γ(I) to demonstrate validity. This addresses the concern by presenting the unevaluated integral representation rather than asserting a fully closed elementary form for arbitrary rates. revision: yes

  2. Referee: [Derivation] The manuscript does not show how the stochastic master equation is solved without introducing an unstated ansatz that effectively linearizes the rates or converts the problem to moment closure; this omission is load-bearing for the headline claim of a general analytic result in the stochastic setting.

    Authors: We acknowledge the omission of explicit steps and will expand the derivation section substantially. Starting from the master equation for the joint probability, we take the expectation of the infected count to obtain a closed ODE for ⟨I(t)⟩ that incorporates the arbitrary nonlinear β(S,I) and γ(I) directly via the definitions of the transition rates; no linearization or additional moment closure is applied beyond computing the first moment. The resulting expression for I(t) is then obtained by quadrature. The revised manuscript will walk through this reduction from the master equation to the integral form step by step, with intermediate equations shown, to eliminate any appearance of an unstated ansatz. revision: yes

Circularity Check

0 steps flagged

No circularity: analytic claim stands on derivation from master equation without reduction to inputs

full rationale

The provided abstract and context describe a derivation of an analytic expression for I(t) from the stochastic master equation with arbitrary nonlinear β(S,I) and γ(I). No equations, self-citations, or steps are shown that define the result in terms of itself, rename a fit as a prediction, or import uniqueness via author overlap. The claim is therefore treated as self-contained pending external verification of the solution method.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the central claim rests on the existence of an unspecified analytic derivation for nonlinear-rate SIR dynamics.

pith-pipeline@v0.9.0 · 5711 in / 1164 out tokens · 54352 ms · 2026-05-19T14:12:25.938039+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Viral Infections of Humans: Epidemiology and Control

    Evans, A.S. Viral Infections of Humans: Epidemiology and Control. (Springer US, 2013)

    work page 2013

  2. [2]

    & Skalka, A.M

    Flint, J., Racaniello, V .R., Rall, G.F ., Hatziioannou, T. & Skalka, A.M. Principles of Virology, Volume 1: Molecular Biology. (Wiley, 2020)

    work page 2020

  3. [3]

    & Sironi, M

    Forni, D., Cagliani, R., Clerici, M. & Sironi, M. Disease-causing human viruses: novelty and legacy. Trends in Microbiology 30, 1232-1242 (2022)

    work page 2022

  4. [4]

    Fields Virology, 6th Edition

    Liu, L. Fields Virology, 6th Edition. Clinical Infectious Diseases 59, 613-613 (2014)

    work page 2014

  5. [5]

    Carroll, D. et al. The Global Virome Project. Science 359, 872-874 (2018)

    work page 2018

  6. [6]

    G., & Walker, G

    Kermack, W.O., McKendrick, A. G., & Walker, G. T. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 115, 700-721 (1927)

    work page 1927

  7. [7]

    The Mathematics of Infectious Diseases

    Hethcote, H.W. The Mathematics of Infectious Diseases. SIAM Review 42, 599-653 (2000)

    work page 2000

  8. [8]

    & Mak, M.K

    Harko, T., Lobo, F .S.N. & Mak, M.K. Exact analytical solutions of the Susceptible- Infected-Recovered (SIR) epidemic model and of the SIR model with equal death and birth rates. Appl. Math. Comput. 236, 184-194 (2014)

    work page 2014

  9. [9]

    & Schlickeiser, R

    Kröger, M. & Schlickeiser, R. Analytical solution of the SIR-model for the temporal evolution of epidemics. Part A: time-independent reproduction factor. Journal of Physics A: Mathematical and Theoretical 53, 505601 (2020)

    work page 2020

  10. [10]

    in Applied Mathematical Ecology

    Hethcote, H.W. in Applied Mathematical Ecology. (eds. S.A. Levin, T.G. Hallam & L.J. Gross) 119-144 (Springer Berlin Heidelberg, Berlin, Heidelberg; 1989)

    work page 1989

  11. [11]

    Computational Epidemiology: Data-Driven Modeling of COVID-19

    Kuhl, E. Computational Epidemiology: Data-Driven Modeling of COVID-19. (Springer International Publishing, 2021)

    work page 2021

  12. [12]

    Contributions to the Mathematical Theory of Epidemics

    Kermack, W.O.a.M., A.G. Contributions to the Mathematical Theory of Epidemics. III.—Further Studies of the Problem of Endemicity. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 141, 94-122 (1933)

    work page 1933

  13. [13]

    & Castillo-Chavez, C

    Brauer, F . & Castillo-Chavez, C. Mathematical Models in Population Biology and Epidemiology. (Springer New York, 2013)

    work page 2013

  14. [14]

    & Wang, W

    Cai, Y ., Kang, Y ., Banerjee, M. & Wang, W. A stochastic SIRS epidemic model with infectious force under intervention strategies. Journal of Differential Equations 259, 7463-7502 (2015)

    work page 2015

  15. [15]

    & May, R.M

    Anderson, R.M. & May, R.M. (Oxford University Press, 1991)

    work page 1991

  16. [16]

    & Britton, T

    Diekmann, O., Heesterbeek, H. & Britton, T. (Princeton University Press, 2012)

    work page 2012

  17. [17]

    & Altman, N

    Bjørnstad, O.N., Shea, K., Krzywinski, M. & Altman, N. The SEIRS model for infectious disease dynamics. Nature Methods 17, 557-558 (2020)

    work page 2020

  18. [18]

    & Torres, A

    Gao, S., Teng, Z., Nieto, J.J. & Torres, A. Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay. BioMed Research International 2007, 064870 (2007)

    work page 2007

  19. [19]

    Allen, L.J.S. et al. Mathematical Epidemiology. (Springer Berlin Heidelberg, 2008)

    work page 2008

  20. [20]

    & Rohani, P

    Keeling, M.J. & Rohani, P . Modeling Infectious Diseases in Humans and Animals. (Princeton University Press, 2008)

    work page 2008

  21. [21]

    & Gorban, A.N

    Kastalskiy, I.A., Pankratova, E.V ., Mirkes, E.M., Kazantsev, V .B. & Gorban, A.N. Social stress drives the multi-wave dynamics of COVID-19 outbreaks. Scientific Reports 11, 22497 (2021)

    work page 2021

  22. [22]

    & Shen, B.-W

    Paxson, W. & Shen, B.-W. A KdV–SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves. International Journal of Bifurcation and Chaos 32, 2250199 (2022)

    work page 2022

  23. [23]

    & Devika, K.B

    Rohith, G. & Devika, K.B. Dynamics and control of COVID-19 pandemic with nonlinear incidence rates. Nonlinear Dynamics 101, 2013-2026 (2020)

    work page 2013

  24. [24]

    & Mena, D

    Calvo-Monge, J., Sanchez, F ., Calvo, J.G. & Mena, D. A nonlinear relapse model with disaggregated contact rates: Analysis of a forward-backward bifurcation. Infectious Disease Modelling 8, 769-782 (2023)

    work page 2023

  25. [25]

    & Choi, M

    Park, S.J. & Choi, M. Y . Counting statistics based on the analytic solutions of the differential-difference equation for birth-death processes. Chaos, Solitons & Fractals 173, 113679 (2023)

    work page 2023

  26. [26]

    & Zimmerman, D.L

    Hogg, R.V ., Tanis, E.A. & Zimmerman, D.L. Probability and Statistical Inference. (Pearson, 2013)

    work page 2013

  27. [27]

    Introduction to Probability Models

    Ross, S.M. Introduction to Probability Models. (Academic Press, 2019)

    work page 2019

  28. [28]

    & Choi, M

    Park, S.J. & Choi, M. Y . Product molecule numbers and reaction rate fluctuations in elementary reactions. AIP Advances 12, 065308 (2022)

    work page 2022

  29. [29]

    A Guide to First-Passage Processes

    Redner, S. A Guide to First-Passage Processes. (Cambridge University Press, 2001)

    work page 2001

  30. [30]

    & Choi, M

    Park, S.J. & Choi, M. Y . Analytic expression of the probability density function for the first-passage time in birth-death processes. Chaos, Solitons & Fractals 186, 115307 (2024)

    work page 2024

  31. [31]

    & Gail, R

    Kleinrock, L. & Gail, R. Queueing Systems: Problems and Solutions. (Wiley, 1996)

    work page 1996

  32. [32]

    & Harris, C.M

    Gross, D., Shortle, J.F ., Thompson, J.M. & Harris, C.M. Fundamentals of Queueing Theory. (Wiley, 2011)

    work page 2011

  33. [33]

    Principles of Random Walk

    Spitzer, F . Principles of Random Walk. (Springer New York, 2013)

    work page 2013

  34. [34]

    Song, S. et al. Transport dynamics of complex fluids. Proceedings of the National Academy of Sciences 116, 12733-12742 (2019)

    work page 2019