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arxiv: 1907.05010 · v1 · pith:V6IPPTGEnew · submitted 2019-07-11 · 🧬 q-bio.PE

On the probability of strain invasion in endemic settings: accounting for individual heterogeneity and control in multi-strain dynamics

Michael T. Meehan , Robert C. Cope , Emma S. McBryde This is my paper

Pith reviewed 2026-05-24 22:53 UTC · model grok-4.3

classification 🧬 q-bio.PE
keywords strain invasionbranching processmulti-strain dynamicsantimicrobial resistanceendemic settingsindividual heterogeneitydifferential controlreproduction number
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The pith

Differential control can allow less-fit novel strains to invade populations with a fitter endemic strain.

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

The paper applies a branching process approximation to the stochastic emergence of a novel strain competing against an existing endemic strain. It calculates the probability that the new strain establishes, with the ratio of reproduction numbers as a central factor. The model incorporates individual heterogeneity and differential control, revealing that control measures applied unevenly can enable invasion even when the novel strain has a lower reproduction number than the endemic one. This result helps explain historical strain replacements and guides efforts to block drug-resistant incursions.

Core claim

Using a branching process approximation on multi-strain dynamics that include individual heterogeneity and differential control, the probability of a novel strain invading an endemic population can exceed zero even when its reproduction number is smaller than that of the background strain.

What carries the argument

Branching process approximation that computes establishment probability for an invading strain under heterogeneous contact and strain-specific control.

If this is right

  • Invasion can succeed when the invading strain's reproduction number is below that of the endemic strain if control acts differently on each.
  • Individual heterogeneity modulates the effect of control on invasion prospects.
  • The same framework can be used to evaluate control policies aimed at preventing resistant strain establishment.
  • Historical cases of strain replacement may be attributable to uneven application of interventions rather than fitness advantages alone.

Where Pith is reading between the lines

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

  • Uniform rather than strain-targeted control might reduce the chance of unintended promotion of less-fit variants.
  • The approximation could be tested against network-based models that explicitly track contact heterogeneity.
  • Extending the calculation to include waning immunity or spatial structure would clarify invasion risks in more realistic settings.

Load-bearing premise

The branching process approximation stays accurate once individual heterogeneity and differential control are added to the multi-strain stochastic model.

What would settle it

Full stochastic simulations or field data showing that invasion probability for a less-fit strain under differential control matches or deviates sharply from the branching process prediction.

Figures

Figures reproduced from arXiv: 1907.05010 by Emma S. McBryde, Michael T. Meehan, Robert C. Cope.

Figure 1
Figure 1. Figure 1: Probability of extinction, q, for a new invading strain entering an endemically infected population as a function of the effective reproduction number Ri eff = Ri 0/Rr 0 for varying dispersion parameter k. To verify the utility of this branching process approximation for strain in￾vasion against an endemic background, and test the conditions under which it can be applied with confidence, we performed a sim… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the probability of extinction, [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Same as Figure 2 for the prior control policy. [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Probability of extinction for the invading strain, [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Same as Figure 4 for polarized control [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Top) Model schematic for simulated example model. Susceptible individuals [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparison between the branching process approximation for invader estab [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Error between the branching process estimate of invader establishment prob [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The error: ((1−q)− proportion of sims in which the invader becomes established) at each individual parameter choice, by the total population size N. This shows that, when Ri eff is 1.0, or k is 0.1, the branching process approximation underestimates the success of the invader; but when Ri eff is large, its success is overestimated, particularly when Rr 0 is small. 31 [PITH_FULL_IMAGE:figures/full_fig_p031… view at source ↗
Figure 10
Figure 10. Figure 10: Simulated extinction times for the endemic disease process (initiated in equi [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
read the original abstract

The rise of antimicrobial drug resistance is an imminent threat to global health that has warranted, and duly received, considerable attention within the medical, microbiological and modelling communities. Outbreaks of drug-resistant pathogens are ignited by the emergence and transmission of mutant variants descended from wild-type strains circulating in the community. In this work we investigate the stochastic dynamics of the emergence of a novel disease strain, introduced into a population in which it must compete with an existing endemic strain. In analogy with past work on single-strain epidemic outbreaks, we apply a branching process approximation to calculate the probability that the new strain becomes established. As expected, a critical determinant of the survival prospects of any invading strain is the magnitude of its reproduction number relative to that of the background endemic strain. Whilst in most circumstances this ratio must exceed unity in order for invasion to be viable, we show that differential control scenarios can lead to less-fit novel strains invading populations hosting a fitter endemic one. This analysis and the accompanying findings will inform our understanding of the mechanisms that have led to past instances of successful strain invasion, and provide valuable lessons for thwarting future drug-resistant strain incursions.

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 develops a multi-type branching process approximation for the invasion probability of a novel strain into a population with an established endemic strain. Individual heterogeneity is incorporated via random effects on susceptibility and transmission, and strain-specific control modifies transmission rates. The central claim is that differential control can produce positive invasion probability for a less-fit invader even when its basic reproduction number is below that of the endemic strain.

Significance. If the branching-process construction remains valid, the result supplies an analytically tractable explanation for how control measures can inadvertently favor invasion by less-fit resistant strains. This would be directly relevant to antimicrobial-resistance policy and would extend single-strain branching-process results to heterogeneous, multi-strain settings with explicit control.

major comments (2)
  1. [section describing the multi-type branching process and offspring PGF] The embedding of the multi-type branching process whose offspring PGF incorporates both random heterogeneity and strain-specific control is load-bearing for the invasion-probability claim, yet the manuscript provides no explicit argument that the control operator commutes with the heterogeneity mixing measure in a manner that preserves the branching property (i.e., that successive generations remain independent).
  2. [results on invasion probabilities under differential control] The statement that a less-fit strain can invade under differential control requires that the effective mean offspring number for the invader exceeds 1 after control is applied; without a derivation showing that this effective mean is obtained from the controlled heterogeneous contact process without introducing inter-generation dependence, the reported invasion probabilities rest on an unverified assumption.
minor comments (2)
  1. [model formulation] Notation for the random effects and the control-modified transmission rates should be introduced with a single consistent symbol table or equation block to avoid ambiguity when the same symbols appear in both the heterogeneous and homogeneous limits.
  2. [abstract] The abstract states the main qualitative finding but does not indicate the functional form of heterogeneity or the precise differential-control operator used to obtain the counter-intuitive invasion result.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for identifying key points regarding the branching-process construction. We address each major comment below and indicate where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [section describing the multi-type branching process and offspring PGF] The embedding of the multi-type branching process whose offspring PGF incorporates both random heterogeneity and strain-specific control is load-bearing for the invasion-probability claim, yet the manuscript provides no explicit argument that the control operator commutes with the heterogeneity mixing measure in a manner that preserves the branching property (i.e., that successive generations remain independent).

    Authors: We agree that an explicit justification strengthens the presentation. Individual heterogeneity enters via a fixed mixing distribution on susceptibility and transmission parameters for each individual; these random effects are drawn once per individual and remain constant. Strain-specific control is a deterministic, time-invariant multiplier applied uniformly to the transmission rate of each strain. Because the control factor does not depend on the realized offspring of prior generations and the contact process is memoryless, the offspring distribution for every individual in every generation is drawn independently from the same controlled mixture. Consequently the multi-type branching property is preserved. We will insert a short clarifying paragraph in the Methods section (and a corresponding sentence in the main text) that states this commutation explicitly. revision: yes

  2. Referee: [results on invasion probabilities under differential control] The statement that a less-fit strain can invade under differential control requires that the effective mean offspring number for the invader exceeds 1 after control is applied; without a derivation showing that this effective mean is obtained from the controlled heterogeneous contact process without introducing inter-generation dependence, the reported invasion probabilities rest on an unverified assumption.

    Authors: The effective mean offspring number is obtained by integrating the strain-specific controlled transmission rate against the heterogeneity mixing measure; this yields the mean of the mixed offspring distribution that is used in the probability-generating function. Because the same mixing measure and the same control multiplier apply identically to every generation, the mean remains constant across generations and no additional inter-generation dependence is created. The invasion probability is then the smallest non-negative solution of the standard fixed-point equation for the multi-type branching process. We will add a brief derivation of the controlled mean (currently implicit in the PGF construction) to an appendix or supplementary note to make this step fully explicit. revision: partial

Circularity Check

0 steps flagged

No circularity; reproduction numbers are exogenous inputs and branching-process approximation is applied without self-referential reduction

full rationale

The paper treats basic reproduction numbers as given parameters and applies a standard multi-type branching-process approximation to compute invasion probabilities under heterogeneity and control. No equation is shown to reduce to its own fitted output by construction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled via prior work by the same authors. The derivation chain therefore remains self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Model rests on branching process approximation and assumptions about reproduction numbers and control effects; no free parameters or invented entities explicitly listed in abstract.

axioms (1)
  • domain assumption Branching process approximation applies to the emergence and competition of novel strains in endemic populations with heterogeneity and control
    Invoked to calculate establishment probability as stated in abstract.

pith-pipeline@v0.9.0 · 5744 in / 1004 out tokens · 16183 ms · 2026-05-24T22:53:40.517953+00:00 · methodology

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    N., Levy, S

    Alekshun, M. N., Levy, S. B., 2007. Molecular mechanisms of antibacterial multidrug resistance. Cell. 128 (6), 1037-1050 20

    work page 2007

  2. [2]

    Strong approximations for epidemic models

    Ball, F., Donnelly, P., 1995. Strong approximations for epidemic models. Stoch. Proc. Appl. 55 (1), 1-21

    work page 1995

  3. [3]

    G., Bean N

    Ballard, P. G., Bean N. G., Ross, J. V., 2016. The probability of epi- demic fade-out is non-monotonic in transmission rate for the Marko- vian SIR model with demography. J. Theor. Biol. 393, 170-178. doi: 10.1016/j.jtbi.2016.01.012

    work page doi:10.1016/j.jtbi.2016.01.012 2016

  4. [4]

    M., Levy, S

    Barbosa, T. M., Levy, S. B., 2000. The impact of antibiotic use on resis- tance development and persistence. Drug Resist. Update. 3 (5), 303-311

    work page 2000

  5. [5]

    S., 1957

    Bartlett, M. S., 1957. On theoretical models for competitive and preda- tory biological systems. Biometrika. 44.1/2, 27-42

    work page 1957

  6. [6]

    E., Hill A

    Leventhal, G. E., Hill A. L., Nowak M. A., Bonhoeffer S., 2015. Evolution and emergence of infectious diseases in theoretical and real-world networks. Nat. Commun. 6, 6101. doi: 10.1038/ncomms7101

    work page doi:10.1038/ncomms7101 2015

  7. [7]

    An evolutionary model to predict the frequency of antibiotic resistance under seasonal antibiotic use, and an application to Streptococcus pneumoniae

    Blanquart, F., Lehtinan S., Fraser C., 2016. An evolutionary model to predict the frequency of antibiotic resistance under seasonal antibiotic use, and an application to Streptococcus pneumoniae. P. R. Soc. B. 284 (1855), 20170679. doi: 10.1098/rspb.2017.0679

    work page doi:10.1098/rspb.2017.0679 2016

  8. [8]

    M., Chou T., 2004

    Blower, S. M., Chou T., 2004. Modeling the emergence of the ‘hot-zones’: tuberculosis and the amplification dynamics of drug resistance. Nat. Med. 10 (10), 1111-1116

    work page 2004

  9. [9]

    O., 2013

    Blumberg, S., Lloyd-Smith, J. O., 2013. Comparing methods for esti- mating R0 from the size distribution of subcritical transmission chains. Epidemics. 5 (3), 131-145. doi: 10.1016/j.epidem.2013.05.002

    work page doi:10.1016/j.epidem.2013.05.002 2013

  10. [10]

    Stochastic epidemic models: a survey

    Britton, T., 2010. Stochastic epidemic models: a survey. Math. Biosci. 225 (1), 24-35. doi: 10.1016/j.mbs.2010.01.006

    work page doi:10.1016/j.mbs.2010.01.006 2010

  11. [11]

    H., Cohen, T., Grad, Y

    Chang, H. H., Cohen, T., Grad, Y. H., Hanage, W. P., O’Brien, T. F., Lipsitch, M., 2015. Origin and proliferation of multiple-drug resis- tance in bacterial pathogens. Microbiol. Mol. Biol. R. 79 (1), 101-116. doi: 10.1128/MMBR.00039-14

    work page doi:10.1128/mmbr.00039-14 2015

  12. [12]

    B., Colijn, C., Hanage, W., Fraser, C., Lipsitch, M., 2017

    Cobey, S., Baskerville, E. B., Colijn, C., Hanage, W., Fraser, C., Lipsitch, M., 2017. Host population structure and treatment frequency 21 maintain balancing selection on drug resistance. J. R. Soc. Interface. 14, 20170295. doi: 10.1098/rsif.2017.0295

    work page doi:10.1098/rsif.2017.0295 2017

  13. [13]

    Modeling epidemics of multidrug-resistant M

    Cohen, T., Murray, M., 2004. Modeling epidemics of multidrug-resistant M. tuberculosis of heterogeneous fitness. Nat. Med. 10 (10), 1117-1121

    work page 2004

  14. [14]

    What is the mechanism for persistent coexistence of drug-susceptible and drug-resistant strains of Streptococcus pneumoniae? J

    Colijn, C., Cohen, T., Fraser, C., Hanage, W., Golstein, E., Givon- Lavi, N., Dagan, R., Lipsitch, M., 2010. What is the mechanism for persistent coexistence of drug-susceptible and drug-resistant strains of Streptococcus pneumoniae? J. R. Soc. Interface. 7 (47), 905-919. doi: 10.1098/rsif.2009.0400

    work page doi:10.1098/rsif.2009.0400 2010

  15. [15]

    M., Webb, G

    D’Agata, E. M., Webb, G. F., Horn, M. A., Moellering R. C., Ruan, S.,

    work page

  16. [16]

    Modeling the invasion of community-acquired methicillin-resistant Staphyloccocus aureus into hospitals. Clin. Infect. Diseases. 48 (3), 274- 284

    work page

  17. [17]

    Diekmann, O., Heesterbeek, J. A. P., Britton, T., 2012. Mathematical tools for understanding infectious diseases. Princeton University Press, Princeton

    work page 2012

  18. [18]

    A., 1973

    Griffiths, D. A., 1973. Multivariate birth-and-death processes as approx- imations to epidemic processes. J. Appl. Probability. 10 (1), 15-26

    work page 1973

  19. [19]

    E., 2007

    Hancock, R. E., 2007. The end of an era? Nat. Rev. Drug Discov. 6, 28

    work page 2007

  20. [20]

    Laxminarayan, R., Duse, A., Wattal, C., Zaidi, A. K. M., Wertheim, H. F. L., Sumpradit, N., et. al., 2013. Antibiotic resistance: the need for global solutions. Lancet Infect. Dis. 13 (12), 1057-1098. doi: 10.1016/S1473- 3099(13)70318-9

    work page doi:10.1016/s1473- 2013

  21. [21]

    J., Turner, P., Lipsitch, M., Fraser, C., 2017

    Lehtinan, S., Blanquart, F., Croucher, N. J., Turner, P., Lipsitch, M., Fraser, C., 2017. Evolution of antibiotic resistance is linked to any genetic mechanism affecting bacterial duration of carriage. P. Natl. Acad. Sci. USA. 114 (5), 1075-1080. doi: 10.1073/pnas.1617849114

    work page doi:10.1073/pnas.1617849114 2017

  22. [22]

    B., Marshall, B., 2004

    Levy, S. B., Marshall, B., 2004. Antibacterial resistance worldwide: causes, challenges and responses. Nat. Med. 10, S122-S129

    work page 2004

  23. [23]

    O., Schreiber, S

    Lloyd-Smith, J. O., Schreiber, S. J., Kopp, P. E., Getz W. M., 2005. Superspreading and the effect of individual variation on disease emergence. Nature. 438, 355-359 22

    work page 2005

  24. [24]

    A., Jiang, H., Francis, A

    Luciani, F., Sisson, S. A., Jiang, H., Francis, A. R., Tanaka, M. M.,

    work page

  25. [25]

    The epidemiological fitness cost of drug resistance in Mycobacterium tuberculosis. PNAS. 106 (34), 14711-14715. doi: 10.1073/pnas.0902437106

    work page doi:10.1073/pnas.0902437106

  26. [26]

    S., Meehan, M

    McBryde, E. S., Meehan, M. T., Doan, T. N., Ragonnet, R., Marais, B. J., Guernier, V., Trauer, J. M., 2017. The risk of global epidemic re- placement with drug-resistant Mycobacterium tuberculosis strains. Int. J. Infect. Dis. 56, 14-20. doi: 10.1016/j.ijid.2017.01.031

    work page doi:10.1016/j.ijid.2017.01.031 2017

  27. [27]

    Antibiotic resistance: the last resort

    McKenna, M., 2013. Antibiotic resistance: the last resort. Nature. 499 (7459), 394. doi: 10.1038/499394a

    work page doi:10.1038/499394a 2013

  28. [28]

    T., Cocks, D

    Meehan, M. T., Cocks, D. G., Trauer, J. M., McBryde, E. S., 2018. Cou- pled, multi-strain epidemic models of mutating pathogens. Math. Biosci. 296, 82-92. doi: 10.1016/j.mbs.2017.12.006

    work page doi:10.1016/j.mbs.2017.12.006 2018

  29. [29]

    C., 1992

    Neu, H. C., 1992. The crisis in antibiotic resistance. Science. 257 (5073), 1064-1073

    work page 1992

  30. [30]

    P., Bean, N

    Rebuli, N. P., Bean, N. G., Ross, J. V., 2017. Hybrid Markov chain models of SIR disease dynamics. J. Math. Biol. 75 (3), 521-541. doi: 10.1007/s00285-016-1085-2

    work page doi:10.1007/s00285-016-1085-2 2017

  31. [31]

    L., Renwick, M

    Simpkin, V. L., Renwick, M. J., Kelly, R., Mossialos, E., 2017. Incentivising innovation in antibiotic drug discovery and development: progress, challenges and next steps. J. Antibiol. 70 (12), 1087-1096. doi: 10.1038/ja.2017.124

    work page doi:10.1038/ja.2017.124 2017

  32. [32]

    M., Francis, A

    Tanaka, M. M., Francis, A. R., 2006. Detecting emerging strains of tuberculosis by using spoligotypes. PNAS. 103 (41), 15266-1527

    work page 2006

  33. [33]

    Global Tuberculosis Re- port 2018

    World Health Organization and others, 2018. Global Tuberculosis Re- port 2018. World Health Organization. 23

    work page 2018

  34. [34]

    Appendix A: Simulation study To determine conditions under which the branching process approxima- tion for invader establishment success could be confidently applied, we per- formed a simulation study. We considered a relatively simple stochastic epidemic model (Figure 6), in which each of the N individuals in the population are either: susceptible (S); in...

    work page