The influence of sexual structure and dual incubation delays on spreading dynamics and optimal control of a gonorrhea model

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Abstract

An SIRS (Susceptible-Infectious-Recovered-Susceptible) model with sexual structure and dual incubation delays was proposed to characterize the effects of homosexual and heterosexual behaviors on the transmission dynamics and optimal control of gonorrhea. First, the nonnegativity and boundedness of solutions were obtained, and the basic reproduction number R₀ was calculated as well. Second, local and global asymptotical stability of the disease-free equilibrium was established if R₀ < 1. If R₀ > 1 the disease was uniformly persistent, and there existed at least one endemic equilibrium. An optimal control strategy was derived based on sensitivity analysis and practical intervention policy. Finally, the theoretical findings were illustrated through numerical simulations, revealing that targeted management of male infected individuals can markedly diminish gonorrhea prevalence, while disregarding the incubation periods tended to substantially overestimate the epidemic scale.

Keywords

gonorrhea / incubation delay / sexual structure / uniform persistence / optimal control strategy

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References

[1] R. C. Brunham, F. A. Plummer, A general model of sexually transmitted disease epidemiology and its implications for control, Med. Clin. N. Am., 74 (1990), 1339-1352. https://doi.org/10.1016/S0025-7125(16)30484-9

[2] World Health Organization, Implementing the global health sector strategies on HIV, viral hepatitis and sexually transmitted infections, 2022-2030: report on progress and gaps 2024, Geneva: WHO, 2024. Available from: https://www.who.int/publications/i/item/9789240094925.

[3] N. Agyepong, U. Govinden, A. Owusu-Ofori, S. Y. Essack, Multidrug-resistant gram-negative bacterial infections in a teaching hospital in Ghana, Antimicrob. Resist. Infect. Control, 7 (2018), 37. https://doi.org/10.1186/s13756-018-0324-2

[4] L. D. Maduna, M. M. Kock, B. M. J. W. van der Veer, O. Radebe, J. McIntyre, L. B. van Alphen, et al., Antimicrobial resistance of Neisseria gonorrhoeae isolates from high-risk men in Johannesburg, South Africa , Antimicrob. Agents Chemother., 64 (2020), e00906. https://doi.org/10.1128/aac.00906-20

[5] K. Jansen, G. Steffen, A. Potthoff, A. K. Schuppe, D. Beer, H. Jessen, et al., STI in times of PrEP: high prevalence of chlamydia, gonorrhea, and mycoplasma at different anatomic sites in men who have sex with men in Germany, BMC Infect. Dis., 20 (2020), 110. https://doi.org/10.1186/s12879-020-4831-4

[6] S. Kurz, A. Ressler, What to know about gonorrhea, JAMA , 330 (2023), 1397. https://doi.org/10.1001/jama.2023.15431

[7] M. A. Bishi, P. Kaur, M. Vyas, S. Sharma, Ameliorating gonorrhea: recent therapeutic adaptations and scope to improve its prevailing condition, Infect. Disord. Drug Targets, 24 (2024), e180124225807. https://doi.org/10.2174/0118715265258305231124105334

[8] M. Unemo, H. S. Seifert, E. W. III Hook, S. Hawkes, F. Ndowa, J. R. Dillon, Gonorrhoea, Nat. Rev. Dis. Primers, 5 (2019), 79. https://doi.org/10.1038/s41572-019-0128-6

[9] A. Nalado, A. Abdu, B. Adamu, M. H. Aliyu, F. A. Arogundade, A. A. Sanusi, et al., Prevalence of chronic kidney disease markers in Kumbotso rural northern Nigeria, Afr. J. Med. Med. Sci., 45 (2016), 61-65. Available from: https://pubmed.ncbi.nlm.nih.gov/28686828.

[10] L. Ben Said, A. Conrad, S. Souza, D. Alfaiate, F. Ader, A. S. Batalla, et al., Partner treatment strategies for chlamydia and gonorrhea: time for a reappraisal, Infect. Dis. Now, 55 (2025), 105038. https://doi.org/10.1016/j.idnow.2025.105038

[11] C. A. Carne, I. V. Weller, A. M. Johnson, C. Loveday, F. Pearce, A. Hawkins, et al., Prevalence of antibodies to human immunodeficiency virus, gonorrhoea rates, and changed sexual behaviour in homosexual men in London, Lancet, 1 (1987), 656-658. https://doi.org/10.1016/s0140-6736(87)90415-6

[12] N. P. Pai, J. R. Dillon, A lateral flow assay for Neisseria gonorrhoeae: a step forward for an inexpensive biomarker-based diagnosis of N. gonorrhoeae at the point of care? Lancet, 403 (2024), 594-595. https://doi.org/10.1016/S0140-6736(23)02569-2

[13] World Health Organization, WHO guidelines for the treatment of Neisseria gonorrhoeae, Geneva: WHO, 2016. Available from: https://www.who.int/publications/i/item/9789241549691.

[14] E. W. Landhuis, Multidrug-resistant “super gonorrhea” rallies multipronged effort, JAMA, 331 (2024), 1695-1697. https://doi.org/10.1001/jama.2023.15355

[15] G. A. Bolan, P. F. Sparling, J. N. Wasserheit, The emerging threat of untreatable gonococcal infection, N. Engl. J. Med., 366 (2012), 485-487. https://doi.org/10.1056/NEJMp1112456

[16] J. M. Hyman, E. A. Stanley, Using mathematical models to understand the AIDS epidemic, Math. Biosci., 90 (1988), 415-473. https://doi.org/10.1016/0025-5564(88)90078-8

[17] K. J. Foreman, N. Marquez, A. Dolgert, K. Fukutaki, N. Fullman, M. McGaughey, et al., Forecasting life expectancy, years of life lost, and all-cause and cause-specific mortality for 250 causes of death: reference and alternative scenarios for 2016-40 for 195 countries and territories, Lancet, 392 (2018), 2052-2090. https://doi.org/10.1016/S0140-6736(18)31694-5

[18] J. Zhang, W. Hao, Z. Jin, The dynamics of sexually transmitted diseases with men who have sex with men, J. Math. Biol., 84 (2021), 1. https://doi.org/10.1007/s00285-021-01694-z

[19] C. C. McCluskey, E. Roth, P. van den Driessche, Implication of Ariaal sexual mixing on gonorrhea, Am. J. Hum. Biol., 17 (2005), 293-301. https://doi.org/10.1002/ajhb.20123

[20] C. Castillo-Chavez, W. Huang, J. Li, The effects of females’ susceptibility on the coexistence of multiple pathogen strains of sexually transmitted diseases, J. Math. Biol., 35 (1997), 503-522. https://doi.org/10.1007/s002850050063

[21] C. Castillo-Chavez, W. Huang, J. Li, Competitive exclusion and coexistence of multiple strains in an SIS STD model, SIAM J. Appl. Math., 59 (1999), 1790-1811. https://doi.org/10.1137/S0036139997325862

[22] Y. H. Hsieh, C. H. Chen, Modelling the social dynamics of a sex industry: its implications for spread of HIV/AIDS, Bull. Math. Biol., 66 (2004), 143-166. https://doi.org/10.1016/j.bulm.2003.08.004

[23] Y. H. Hsieh, Y. S. Wang, Basic reproduction number for HIV model incorporating commercial sex and behavior change, Bull. Math. Biol., 68 (2006), 551-575. https://doi.org/10.1007/s11538-005-9050-z

[24] B. R. Morin, L. Medina-Rios, E. T. Camacho, C. Castillo-Chavez, Static behavioral effects on gonorrhea transmission dynamics in a MSM population, J. Theor. Biol., 267 (2010), 35-40. https://doi.org/10.1016/j.jtbi.2010.07.027

[25] F. Saldana, I. Barradas, The role of behavioral changes and prompt treatment in the control of STIs, Infect. Dis. Modell., 4 (2019), 1-10. https://doi.org/10.1016/j.idm.2018.12.001

[26] E. Bonyah, M. A. Khan, K. O. Okosun, J. F. Gómez-Aguilar, Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control, Math. Biosci., 309 (2019), 1-11. https://doi.org/10.1016/j.mbs.2018.12.015

[27] Y. A. Terefe, S. M. Kassa, M. D. Asfaw, C. Venter, The use of an imperfect vaccination and awareness campaign in the control of antibiotic resistant gonorrhoea infection: a mathematical modelling perspective, Appl. Math. Modell., 135 (2024), 149-172. https://doi.org/10.1016/j.apm.2024.06.042

[28] C. Kenyon, B. Herrmann, G. Hughes, H. J. C. de Vries, Management of asymptomatic sexually transmitted infections in Europe: towards a differentiated, evidence-based approach, Lancet Reg. Health Eur., 34 (2023), 100743. https://doi.org/10.1016/j.lanepe.2023.100743

[29] L. M. Gorgos, J. M. Marrazzo, Sexually transmitted infections among women who have sex with women, Clin. Infect. Dis., 53 (2011), S84-S91. https://doi.org/10.1093/cid/cir697

[30] J. Tran, C. K. Fairley, J. J. Ong, T. R. Phillips, E. T. Aung, E. P. F. Chow, The incubation for urethral gonorrhoea among men who have sex with men with and without oropharyngeal gonorrhoea, Epidemiol. Infect., 152 (2024), e104. https://doi.org/10.1017/S095026882400089X

[31] J. Hale, S. V. Lunel, An Introduction to Functional Differential Equations , Springer-Verlag, New York, 1993.

[32] H. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems , American Mathematical Society, Providence, RI, 1995. https://doi.org/10.1090/surv/041

[33] Z. Bai, X. Q. Zhao, Basic reproduction ratios for periodic and time-delayed compartmental models with impulses, J. Math. Biol., 80 (2020), 1095-1117. https://doi.org/10.1007/s00285-019-01452-2

[34] J. P. La Salle, The Stability of Dynamical Systems , Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976. https://doi.org/10.1137/1.9781611970432

[35] H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435. https://doi.org/10.1137/0524026

[36] X. Q. Zhao, Uniform persistence in processes with application to nonautonomous competitive models, J. Math. Anal. Appl., 258 (2001), 87-101. https://doi.org/10.1006/jmaa.2000.7361

[37] L. Göllmann, D. Kern, H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optim. Control. Appl. Methods, 30 (2009), 341-365. https://doi.org/10.1002/oca.843

[38] M. Martcheva, An Introduction to Mathematical Epidemiology , Springer, New York, 2015.

[39] R. Rishel, W. Fleming, Deterministic and Stochastic Optimal Control , New York, 1975.

[40] J. K. K. Asamoah, G. Q. Sun, Fractional Caputo and sensitivity heat map for a gonorrhea transmission model in a sex structured population, Chaos, Solitons Fractals, 175 (2023), 114026. https://doi.org/10.1016/j.chaos.2023.114026

[41] S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol., 254 (2008), 178-196. https://doi.org/10.1016/j.jtbi.2008.04.011