Mathematical modeling has been widely used by many researchers to understand the transmission mechanisms of infectious diseases and to evaluate how well different types of interventions work at the population level, including for COVID-19. In the context of COVID-19, many researchers have separately discussed the role of medical masks and the impact of reinfection on disease transmission. However, very few studies have effectively discussed these aspects in detail, particularly the impact of face masks on the reinfection process. Furthermore, many previous studies have discussed standard bifurcation phenomena, such as backward and forward bifurcations or Hopf bifurcations, but have neglected more complex dynamics, such as double limit cycles and cusp bifurcations. The main innovation of this study is the explicit incorporation of mask-induced reductions in both primary infection and reinfection pathways, combined with saturated treatment, and the systematic exploration of the resulting higher-codimension bifurcation structures. The next-generation matrix technique is used to compute the basic reproduction number $ \mathcal{R}_0 $. Our analytical results reveal that the COVID-19-free equilibrium is locally asymptotically stable when $ \mathcal{R}_0 < 1 $ and unstable when $ \mathcal{R}_0 > 1 $. A center manifold approach is used to show that the system can exhibit either forward or backward bifurcation, depending on the parameter regime, indicating that disease persistence may occur even when $ \mathcal{R}_0 < 1 $. A numerical bifurcation study reveals several nonlinear dynamic phenomena, including Hopf bifurcations leading to oscillatory outbreaks, endemic bubbles, folds of limit cycles, and codimension-two bifurcations, including cusp and generalized Hopf points. These structures divide the parameter space into regions associated with different epidemiological outcomes. Our global sensitivity analysis, supported by optimal control simulations, reveals the importance of medical masks in reducing COVID-19 reinfection while also illustrating how reinfection may significantly alter the dynamic behavior of the model. These analytical and numerical results show that time-dependent adaptive interventions should be carefully considered to prevent recurrent outbreaks generated by the intrinsic periodic behavior of COVID-19's transmission dynamics and to ensure that the resources allocated to disease-control interventions are managed efficiently.
Citation: Dipo Aldila, Livia Owen, Stephen Wirana, Olumuyiwa James Peter, Putri Zahra Kamalia, Nurul Fiskia Gamayanti. Nonlinear dynamics of reinfection with behavioral protection and saturated treatment: Bifurcation analysis and control insights[J]. Mathematical Biosciences and Engineering, 2026, 23(9): 2860-2903. doi: 10.3934/mbe.2026101
Mathematical modeling has been widely used by many researchers to understand the transmission mechanisms of infectious diseases and to evaluate how well different types of interventions work at the population level, including for COVID-19. In the context of COVID-19, many researchers have separately discussed the role of medical masks and the impact of reinfection on disease transmission. However, very few studies have effectively discussed these aspects in detail, particularly the impact of face masks on the reinfection process. Furthermore, many previous studies have discussed standard bifurcation phenomena, such as backward and forward bifurcations or Hopf bifurcations, but have neglected more complex dynamics, such as double limit cycles and cusp bifurcations. The main innovation of this study is the explicit incorporation of mask-induced reductions in both primary infection and reinfection pathways, combined with saturated treatment, and the systematic exploration of the resulting higher-codimension bifurcation structures. The next-generation matrix technique is used to compute the basic reproduction number $ \mathcal{R}_0 $. Our analytical results reveal that the COVID-19-free equilibrium is locally asymptotically stable when $ \mathcal{R}_0 < 1 $ and unstable when $ \mathcal{R}_0 > 1 $. A center manifold approach is used to show that the system can exhibit either forward or backward bifurcation, depending on the parameter regime, indicating that disease persistence may occur even when $ \mathcal{R}_0 < 1 $. A numerical bifurcation study reveals several nonlinear dynamic phenomena, including Hopf bifurcations leading to oscillatory outbreaks, endemic bubbles, folds of limit cycles, and codimension-two bifurcations, including cusp and generalized Hopf points. These structures divide the parameter space into regions associated with different epidemiological outcomes. Our global sensitivity analysis, supported by optimal control simulations, reveals the importance of medical masks in reducing COVID-19 reinfection while also illustrating how reinfection may significantly alter the dynamic behavior of the model. These analytical and numerical results show that time-dependent adaptive interventions should be carefully considered to prevent recurrent outbreaks generated by the intrinsic periodic behavior of COVID-19's transmission dynamics and to ensure that the resources allocated to disease-control interventions are managed efficiently.
| [1] | Coronavirus Disease (COVID-19) Dashboard, World Health Organization, 2025. Available from: https://covid19.who.int. |
| [2] |
J. M. Gmanyami, W. Quentin, O. Lambert, A. Jarynowski, V. Belik, J. H. Amuasi, Excess mortality during the COVID-19 pandemic in low- and lower-middle-income countries: A systematic review and meta-analysis, BMC Public Health, 24 (2024), 1643. https://doi.org/10.1186/s12889-024-19154-w doi: 10.1186/s12889-024-19154-w
|
| [3] |
Y. Guo, Q. Cao, Z. Hong, Y. Tan, S. Chen, H. Jin, et al., Coronavirus (COVID-19): A review of clinical features, diagnosis, and treatment, Infection, 48 (2020), 681–690. https://doi.org/10.1007/s15010-020-01457-7 doi: 10.1007/s15010-020-01457-7
|
| [4] |
T. Oyelade, J. S. Alqahtani, A. M. Hjazi, A. Li, A. Kamila, R. P. Raya, Global and regional prevalence and outcomes of COVID-19 in people living with HIV: A systematic review and meta-analysis, Trop. Med. Infect. Dis., 7 (2022), 22. https://doi.org/10.3390/tropicalmed7020022 doi: 10.3390/tropicalmed7020022
|
| [5] |
A. Malin, S. Holte, D. Turner, S. B. Kaye, L. G. Heaney, A. S. Haque, et al., Implications of COVID-19 in high burden countries for HIV/TB: A systematic review of evidence, BMJ Global Health, 5 (2020), e003679. https://doi.org/10.1136/bmjgh-2020-003679 doi: 10.1136/bmjgh-2020-003679
|
| [6] | Tracking SARS-CoV-2 variants, World Health Organization, 2025. Available from: https://data.who.int/dashboards/covid19/variants. |
| [7] |
W. T. Harvey, A. M. Carabelli, B. Jackson, R. K. Gupta, E. C. Thomson, E. M. Harrison, et al., SARS-CoV-2 variants, spike mutations and immune escape, Nat. Rev. Microbiol., 19 (2021), 409–424. https://doi.org/10.1038/s41579-021-00573-0 doi: 10.1038/s41579-021-00573-0
|
| [8] |
J. R. C. Pulliam, C. van Schalkwyk, N. Govender, A. von Gottberg, C. Cohen, M. J. Groome, et al., Increased risk of SARS-CoV-2 reinfection associated with emergence of Omicron in South Africa, Science, 376 (2022), eabn4947. https://doi.org/10.1126/science.abn4947 doi: 10.1126/science.abn4947
|
| [9] | WHO Public Health and Social Measures (PHSM) Initiative, World Health Organization, 2025. Available from: https://www.who.int/initiatives/who-public-health-and-social-measures-initiative. |
| [10] | Global Health Strategy 2025–2028, World Health Organization, 2025. Available from: https://cdn.who.int/media/docs/default-source/documents/about-us/general-programme-of-work/global-health-strategy-2025-2028.pdf. |
| [11] |
W. O. Kermack, A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. A: Math. Phys. Eng. Sci., 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118 doi: 10.1098/rspa.1927.0118
|
| [12] |
D. Aldila, J. P. Chávez, C. W. Chukwu, A. Y. Fathiyah, J. W. Puspita, K. A. D. Setio, et al., Unraveling dengue dynamics with data calibration from Palu and Jakarta: Optimizing active surveillance and fogging interventions, Chaos, Solitons Fractals, 189 (2024), 115729. https://doi.org/10.1016/j.chaos.2024.115729 doi: 10.1016/j.chaos.2024.115729
|
| [13] |
I. H. Febiriana, D. Aldila, B. D. Handari, P. B. S. Asih, M. H. N. Aziz, Exploring the interplay between social awareness and the use of bed nets in a malaria control program, J. Biosaf. Biosecur., 6 (2024), 196–210. https://doi.org/10.1016/j.jobb.2024.06.005 doi: 10.1016/j.jobb.2024.06.005
|
| [14] |
I. H. Febiriana, A. H. Hassan, D. Aldila, Enhancing malaria control strategy: Optimal control and cost-effectiveness analysis on the impact of vector bias on the efficacy of mosquito repellent and hospitalization, J. Appl. Math., 2024 (2024), 9943698. https://doi.org/10.1155/2024/9943698 doi: 10.1155/2024/9943698
|
| [15] |
O. J. Peter, C. Cattani, A. Omame, Modelling transmission dynamics of measles: The effect of treatment failure in complicated cases, Model. Earth Syst. Environ., 10 (2024), 5871–5889. https://doi.org/10.1007/s40808-024-02120-1 doi: 10.1007/s40808-024-02120-1
|
| [16] |
D. Aldila, Change in stability direction induced by temporal interventions: A case study of a tuberculosis transmission model with relapse and reinfection, Front. Appl. Math. Stat., 11 (2025), 1541981. https://doi.org/10.3389/fams.2025.1541981 doi: 10.3389/fams.2025.1541981
|
| [17] |
D. Aldila, D. A. Ramadhan, C. W. Chukwu, B. D. Handari, M. Shahzad, P. Z. Kamalia, On the role of early case detection and treatment failure in controlling tuberculosis transmission: A mathematical modeling study, Commun. Biomath. Sci., 7 (2024), 61–86. https://doi.org/10.5614/cbms.2024.7.1.4 doi: 10.5614/cbms.2024.7.1.4
|
| [18] |
D. Aldila, J. P. Chávez, K. P. Wijaya, N. C. Ganegoda, G. M. Simorangkir, H. Tasman, et al., A tuberculosis epidemic model as a proxy for the assessment of the novel M72/AS01E vaccine, Commun. Nonlinear Sci. Numer. Simul., 120 (2023), 107162. https://doi.org/10.1016/j.cnsns.2023.107162 doi: 10.1016/j.cnsns.2023.107162
|
| [19] |
D. Aldila, R. P. Dhanendra, S. H. A. Khoshnaw, P. Z. Kamalia, M. Shahzad, Understanding HIV/AIDS dynamics: Insights from CD4+T cells, antiretroviral treatment, and country-specific analysis, Front. Public Health, 12 (2024), 1324858. https://doi.org/10.3389/fpubh.2024.1324858 doi: 10.3389/fpubh.2024.1324858
|
| [20] |
S. O. Oyovwevotu, Mathematical modelling for assessing the impact of intervention strategies on HIV/AIDS high risk group population dynamics, Heliyon, 7 (2021), e07991. https://doi.org/10.1016/j.heliyon.2021.e07991 doi: 10.1016/j.heliyon.2021.e07991
|
| [21] |
D. Aldila, N. Awdinda, Fatmawati, F. F. Herdicho, M. Z. Ndii, C. W. Chukwu, Optimal control of pneumonia transmission model with seasonal factor: Learning from Jakarta incidence data, Heliyon, 9 (2023), e18096. https://doi.org/10.1016/j.heliyon.2023.e18096 doi: 10.1016/j.heliyon.2023.e18096
|
| [22] |
S. E. Eikenberry, M. Mancuso, E. Iboi, T. Phan, K. Eikenberry, Y. Kuang, et al., To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic, Infect. Dis. Modell., 5 (2020), 293–308. https://doi.org/10.1016/j.idm.2020.04.001 doi: 10.1016/j.idm.2020.04.001
|
| [23] |
C. N. Ngonghala, E. A. Iboi, A. B. Gumel, Could masks curtail the post-lockdown resurgence of COVID-19 in the US, Math. Biosci., 329 (2020), 108452. https://doi.org/10.1016/j.mbs.2020.108452 doi: 10.1016/j.mbs.2020.108452
|
| [24] |
N. H. Sweilam, S. M. AL-Mekhlafi, A. Almutairi, D. Baleanu, A hybrid fractional COVID-19 model with general population mask use: Numerical treatments, Alexandria Eng. J., 60 (2021), 3219–3232. https://doi.org/10.1016/j.aej.2021.01.057 doi: 10.1016/j.aej.2021.01.057
|
| [25] |
H. Froese, G. A. P. Angel, Mask use and influenza: Findings from a web-based survey, JMIR Public Health Surveillance, 8 (2022), e31955. https://doi.org/10.2196/31955 doi: 10.2196/31955
|
| [26] |
X. Ma, X. Luo, L. Li, Y. Li, G. Sun, The influence of mask use on the spread of COVID-19 during pandemic in New York City, Results Phys., 34 (2022), 105224. https://doi.org/10.1016/j.rinp.2022.105224 doi: 10.1016/j.rinp.2022.105224
|
| [27] |
Y. Ko, V. M. Mendoza, R. Mendoza, Y. Seo, J. Lee, E. Jung, Risk estimation of lifted mask mandates and emerging variants using mathematical model, Heliyon, 9 (2023), e16841. https://doi.org/10.1016/j.heliyon.2023.e16841 doi: 10.1016/j.heliyon.2023.e16841
|
| [28] |
M. H. DarAssi, I. Ahmad, M. Z. Meetei, M. Alsulami, M. A. Khan, E. M. Tag-eldin, The impact of the face mask on SARS-CoV-2 disease: Mathematical modeling with a case study, Results Phys., 51 (2023), 106699. https://doi.org/10.1016/j.rinp.2023.106699 doi: 10.1016/j.rinp.2023.106699
|
| [29] |
R. Asempapa, B. Oduro, O. O. Apenteng, V. M. Magagula, A COVID-19 mathematical model of at-risk populations with non-pharmaceutical preventive measures: The case of Brazil and South Africa, Infect. Dis. Modell., 7 (2022), 45–61. https://doi.org/10.1016/j.idm.2021.11.005 doi: 10.1016/j.idm.2021.11.005
|
| [30] |
O. T. Kolebaje, O. R. Vincent, U. E. Vincent, P. V. E. McClintock, Nonlinear growth and mathematical modelling of COVID-19 in some African countries with the Atangana–Baleanu fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 105 (2022), 106076. https://doi.org/10.1016/j.cnsns.2021.106076 doi: 10.1016/j.cnsns.2021.106076
|
| [31] |
C. N. Ngonghala, H. B. Taboe, S. Safdar, A. B. Gumel, Unraveling the dynamics of the Omicron and Delta variants of the 2019 coronavirus in the presence of vaccination, mask usage, and antiviral treatment, Appl. Math. Modell., 114 (2023), 447–465. https://doi.org/10.1016/j.apm.2022.09.017 doi: 10.1016/j.apm.2022.09.017
|
| [32] |
M. Nagata, Y. Okada, H. Nishiura, Epidemiological impact of revoking mask-wearing recommendation on COVID-19 transmission in Tokyo, Japan, Infect. Dis. Modell., 9 (2024), 1289–1300. https://doi.org/10.1016/j.idm.2024.08.002 doi: 10.1016/j.idm.2024.08.002
|
| [33] |
G. O. Acheneje, D. Omale, W. Atokolo, B. Bolaji, Modeling the transmission dynamics of the co-infection of COVID-19 and Monkeypox diseases with optimal control strategies and cost–benefit analysis, Franklin Open, 8 (2024), 100130. https://doi.org/10.1016/j.fraope.2024.100130 doi: 10.1016/j.fraope.2024.100130
|
| [34] |
A. Devi, A. Adak, P. K. Gupta, Stability switches via endemic bubbles in a COVID-19 model examining the effect of mask usage and saturated treatment with reinfection, J. Appl. Math. Comput., 70 (2024), 3789–3824. https://doi.org/10.1007/s12190-024-02073-9 doi: 10.1007/s12190-024-02073-9
|
| [35] |
L. Swadling, M. O. Diniz, N. M. Schmidt, O. E. Amin, A. Chandran, E. Shaw, et al., Pre-existing polymerase-specific T cells expand in abortive seronegative SARS-CoV-2 infection, Nature, 601 (2022), 110–117. https://doi.org/10.1038/s41586-021-04186-8 doi: 10.1038/s41586-021-04186-8
|
| [36] |
L. Swadling, M. K. Maini, Can T cells abort SARS-CoV-2 and other viral infections, Int. J. Mol. Sci., 24 (2023), 4371. https://doi.org/10.3390/ijms24054371 doi: 10.3390/ijms24054371
|
| [37] |
P. R. Rani, M. Imran, J. V. Lakshmi, B. Jolly, A. Jain, A. Surekha, et al., Symptomatic reinfection of SARS-CoV-2 with spike protein variant N440K associated with immune escape, J. Med. Virol., 93 (2021), 4163–4165. https://doi.org/10.1002/jmv.26997 doi: 10.1002/jmv.26997
|
| [38] | M. Viana, J. M. Espinar, Differential Equations: A Dynamical Systems Approach to Theory and Practice, American Mathematical Society, 2021. |
| [39] |
O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010), 873–885. https://doi.org/10.1098/rsif.2009.0386 doi: 10.1098/rsif.2009.0386
|
| [40] | V. J. Katz, A History of Mathematics: An Introduction, Addison-Wesley, 2008. |
| [41] |
C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng., 1 (2004), 361–404. https://doi.org/10.3934/MBE.2004.1.361 doi: 10.3934/MBE.2004.1.361
|
| [42] | S. R. Bandekar, T. Das, A. K. Srivastav, A. Yadav, A. Kumar, P. K. Srivastava, et al., Modeling and prediction of the third wave of COVID-19 spread in India, Comput. Math. Biophys., 10 (2022), 231–248. |
| [43] |
M. Biggerstaff, B. J. Cowling, Z. M. Cucunubá, L. Dinh, N. M. Ferguson, H. Gao, et al., Early insights from statistical and mathematical modeling of key epidemiologic parameters of COVID-19, Emerging Infect. Dis., 26 (2020), e1–e14. https://doi.org/10.3201/eid2611.201074 doi: 10.3201/eid2611.201074
|
| [44] |
H. Chemaitelly, N. Nagelkerke, H. H. Ayoub, P. Coyle, P. Tang, H. M. Yassine, et al., Duration of immune protection of SARS-CoV-2 natural infection against reinfection, J. Travel Med., 29 (2022), taac109. https://doi.org/10.1093/jtm/taac109 doi: 10.1093/jtm/taac109
|
| [45] |
H. Khezri, M. Alizadeh, A. Jafarzadeh, A. M. Mortazavian, S. Mortazavi, S. Khademi, et al., Probiotic supplementation and prevention of upper respiratory tract infections: A systematic review and meta-analysis of randomized controlled trials, Front. Med., 8 (2021), 770381. https://doi.org/10.3389/fmed.2021.770381 doi: 10.3389/fmed.2021.770381
|
| [46] |
S. A. Lauer, K. H. Grantz, Q. Bi, F. K. Jones, Q. Zheng, H. R. Meredith, et al., The incubation period of coronavirus disease 2019 (COVID-19), Ann. Intern. Med., 172 (2020), 577–582. https://doi.org/10.7326/M20-0504 doi: 10.7326/M20-0504
|
| [47] |
X. He, E. H. Y. Lau, P. Wu, X. Deng, J. Wang, X. Hao, et al., Temporal dynamics in viral shedding and transmissibility of COVID-19, Nat. Med., 26 (2020), 672–675. https://doi.org/10.1038/s41591-020-0869-5 doi: 10.1038/s41591-020-0869-5
|
| [48] |
M. Devi, A. Devi, P. Gupta, D. Tripathi, Response of vaccination on community transmission of COVID-19: A dynamical approach, Eur. Phys. J. Spec. Top., 231 (2022), 3749–3765. https://doi.org/10.1140/epjs/s11734-022-00652-0 doi: 10.1140/epjs/s11734-022-00652-0
|
| [49] |
S. S. Musa, A. Yusuf, S. Zhao, Z. U. Abdullahi, H. Abu-Odah, F. T. Saad, et al., Transmission dynamics of COVID-19 pandemic with combined effects of relapse, reinfection and environmental contribution: A modeling analysis, Results Phys., 38 (2022), 105653. https://doi.org/10.1016/j.rinp.2022.105653 doi: 10.1016/j.rinp.2022.105653
|
| [50] | M. O. Olayiwola, M. K. Kolawole, P. A. Oladele, Variational iteration method for the simulation of the effect of transmission coefficient on the susceptible-exposed-infected-recovered-susceptible (SEIRS) epidemic model with saturated incidence rate and disease-induced death, J. Sci. Arts, 39 (2017), 357–364. |
| [51] |
A. T. Levin, W. P. Hanage, N. Owusu-Boaitey, K. B. Cochran, S. P. Walsh, G. Meyerowitz-Katz, Assessing the age specificity of infection fatality rates for COVID-19, Eur. J. Epidemiol., 35 (2020), 1123–1138. https://doi.org/10.1007/s10654-020-00698-1 doi: 10.1007/s10654-020-00698-1
|
| [52] |
G. Meyerowitz-Katz, L. Merone, A systematic review and meta-analysis of published research data on COVID-19 infection fatality rates, Epidemiol. Infect., 149 (2021), e43. https://doi.org/10.1017/S0950268820002597 doi: 10.1017/S0950268820002597
|
| [53] |
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 doi: 10.1016/j.jtbi.2008.04.011
|
| [54] | L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, New York, 1962. |
| [55] | S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC, Boca Raton, 2007. |
| [56] |
D. Aldila, Analyzing the impact of the media campaign and rapid testing for COVID-19 as an optimal control problem in East Java, Indonesia, Chaos, Solitons Fractals, 141 (2020), 110364. https://doi.org/10.1016/j.chaos.2020.110364 doi: 10.1016/j.chaos.2020.110364
|
| [57] |
R. Ramalingam, A. Gnanaprakasam, S. Boulaaras, Stability and control analysis of COVID-19 spread in India using SEIR model, Sci. Rep., 15 (2025), 9095. https://doi.org/10.1038/s41598-025-93994-3 doi: 10.1038/s41598-025-93994-3
|
| [58] |
P. Pongsumpun, P. Pongsumpun, I. M. Tang, J. Lamwong, The role of a vaccine booster for a fractional order model of the dynamic of COVID-19: A case study in Thailand, Sci. Rep., 15 (2025), 1162. https://doi.org/10.1038/s41598-024-80390-6 doi: 10.1038/s41598-024-80390-6
|
| [59] |
Y. Li, Z. Zeng, M. Feng, J. Kurths, Protection degree and migration in the stochastic SIRS model: A queueing system perspective, IEEE Trans. Circuits Syst. I Regul. Pap., 69 (2022), 771–783. https://doi.org/10.1109/TCSI.2021.3119978 doi: 10.1109/TCSI.2021.3119978
|
Figures(20) / Tables(3)
Dipo Aldila, Livia Owen, Stephen Wirana, Olumuyiwa James Peter, Putri Zahra Kamalia, Nurul Fiskia Gamayanti. Nonlinear dynamics of reinfection with behavioral protection and saturated treatment: Bifurcation analysis and control insights[J]. Mathematical Biosciences and Engineering, 2026, 23(9): 2860-2903. doi: 10.3934/mbe.2026101
DownLoad: