Theoretical analysis of a measles model with nonlinear incidence functions. (English) Zbl 1505.92191
Summary: Measles is a highly contagious respiratory disease of global public health concern. A deterministic mathematical model for the transmission dynamics of measles in a population with Crowley-Martin incidence function to account for the inhibitory effect due to susceptible and infected individuals and vaccination is formulated and analyzed using standard dynamical systems methods. The basic reproduction number is computed. By constructing a suitable Lyapunov function, the disease-free equilibrium is shown to be globally asymptotically stable. Using the center manifold theory, the model exhibits a forward bifurcation, which implies that the endemic equilibrium is also globally asymptotically stable. To determine the optimal choice of intervention measures to mitigate the spread of the disease, an optimal control problem is formulated (by introducing a set of three time-dependent control variables representing the first and second vaccine doses, and the palliative treatment) and analyzed using Pontryagin’s maximum principle. To account for the scarcity of measles vaccines during a major outbreak or other causes such as the COVID-19 pandemic, a Holling type-II incidence function is introduced at the model simulation stage. The control strategies have a positive population level impact on the evolution of the disease dynamics. Graphical results reveal that when the mass-action incidence function is used, the number of individuals who received first and second vaccine dose is smaller compared to the numbers when the Crowley-Martin incidence-type function is used. Inhibitory effect of susceptibles tends to have the same effect on the population level as the Crowley-Martin incidence function, while the control profiles when inhibitory effect of the infectives is considered have similar effect as when the mass-action incidence is used, or when there is limitation in the availability of measles vaccines. Missing out the second measles vaccine dose has a negative impact on the initial disease prevalence.
MSC:
| 92D30 | Epidemiology |
| 92C60 | Medical epidemiology |
| 34A34 | Nonlinear ordinary differential equations and systems |
| 34D20 | Stability of solutions to ordinary differential equations |
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
References:
| [1] | Wang, Z.; Moreno, Y.; Stefano Boccaletti, S.; Perc, M., Vaccination and epidemics in networked populations - An introduction, Chaos Solitons Fractals, 103, 177-183 (2017) |
| [2] | Deng, X.; He, H.; Zhou, Y.; Xie, S.; Fang, Y.; Zeng, Y., Economic burden and associated factors of measles patients in Zhejiang Province, China, Hum Vaccin Immunother, 15, 11, 2571-2577 (2019) |
| [3] | De Broucker, G.; Ssebagereka, A.; Apolot, R. R.; Aloysius, M.; Kiracho, E. E.; Patenaude, B., The economic burden of measles in children under five in Uganda, Vaccine, 6, 100077 (2020) |
| [4] | Arsal, S. A.; Aldila, D.; Handari, B. D., Short review of mathematical model of measles, (AIP conference proceedings, vol. 2264 (2020)), Article 020003 pp. |
| [5] | Worldwide measles deaths climb 50 |
| [6] | Mulholl, K.; Kretsinger, K.; Wondwossen, L.; Crowcroft, N., Action needed now to prevent further increases in measles and measles deaths in the coming years, Lancet, 396, 10265, 1782-1784 (2020) |
| [7] | Aldila, D.; Asrianti, D., A deterministic model of measles with imperfect vaccination and quarantine intervention, J Phys Conf Ser, 1218, Article 012044 pp. (2019) |
| [8] | White, S. J.; Boldt, K. J.; Holditch, S. J.; Pol, G. A.; Jacobson, R. M., Measles, mumps, and rubella, Clin Obst Gyn, 55, 2, 550-559 (2012) |
| [9] | Sinha, D. N.; Klahn, N.; Pehlivan, S., Mathematical modeling of the 2019 measles outbreak on US population, Acta Sci Microb, 3, 4, 209-214 (2020) |
| [10] | Tilahun, G. T.; Demie, S.; Eyob, A., Stochastic model of measles transmission dynamics with double dose vaccination, Inf Dis Mod, 5, 478-494 (2020) |
| [11] | Garba, S. A.; Safi, M. A.; Usaini, S., Mathematical model for assessing the impact of vaccination and treatment on measles transmission dynamics, Math Methods Appl Sci, 40, 18, 6371-6388 (2017) · Zbl 1381.37106 |
| [12] | Sowole, S. O.; Ibrahim, A.; Sangare, D.; Lukman, A. O., Mathematical model for measles disease with control on the susceptible and exposed compartments, Open J Math Sci, 4, 1, 60-75 (2020) |
| [13] | Gold, J. E.; Baumgartl, W. H.; Okyay, R. A.; Licht, W. E.; Fidel Jr, P. L.; Noverr, M. C., Analysis of measles-mumps-rubella (MMR) titers of recovered COVID-19 patients, MBio, 11, 6, e02628-2620 (2020) |
| [14] | Arino, J.; McCluskey, C. C.; Driessche, P.van.den., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J Appl Math, 64, 1, 260-276 (2003) · Zbl 1034.92025 |
| [15] | The economic value of vaccination: Why prevention is wealth, J Mark Access Health Policy, 3 (2015) |
| [16] | Njeuhmeli, E.; Schnure, M.; Vazzano, A.; Gold, E.; Stegman, P.; Kripke, K., Using mathematical modeling to inform health policy: A case study from voluntary medical male circumcision scale-up in eastern and southern Africa and proposed framework for success, PLoS ONE, 14, 3, Article e0213605 pp. (2019) |
| [17] | Memon, Z.; Qureshi, S.; Memon, B. R., Mathematical analysis for a new nonlinear measles epidemiological system using real incidence data from Pakistan, Eur Phys J Plus, 135, 4, 378 (2020) |
| [18] | MacIntyre, C. R.; Gay, N. J.; Gidding, H. F.; Hull, B. P.; Gilbert, G. L.; McIntyre, P. B., A mathematical model to measure the impact of the measles control campaign on the potential for measles transmission in Australia, Int J Infect Dis, 6, 4, 277-282 (2002) |
| [19] | Bakhtiar, J. T., Control policy mix in measles transmission dynamics using vaccination, therapy, and treatment, Int J Math Math Sci, 2020 (2020), ID 1561569 · Zbl 1486.92236 |
| [20] | Thompson, K. M.; Badizadegan, N. D., Modeling the transmission of measles and rubella to support global management policy analyses and eradication investment cases, Risk Anal, 37, 6, 1109-1131 (2017) |
| [21] | Al-Darabsah, I., Threshold dynamics of a time-delayed epidemic model for continuous imperfect-vaccine with a generalized nonmonotone incidence rate, Nonlinear Dyn, 101, 1281-1300 (2020) · Zbl 1516.92100 |
| [22] | Stephen, E.; Kitengeso, R. E.; Kiria, G. T.; Felician, N.; Mwema, G. G.; Mafarasa, A. P., A mathematical model for control and elimination of the transmission dynamics of measles, Appl Comput Math, 4, 6, 396-408 (2015) |
| [23] | Upadhyay, R. K.; Pal, A. K.; Kumari, S.; Parimita, R., Dynamics of an \(S E I R\) epidemic model with nonlinear incidence and treatment rates, Nonlinear Dyn, 96, 2351-2368 (2019) · Zbl 1468.92060 |
| [24] | Balram, D.; Preeti, D.; Uma, S. D., Dynamics of an \(S I R\) model with nonlinear incidence and treatment rate, Appl Appl Math, 10, 2, 718-737 (2015) · Zbl 1331.34085 |
| [25] | Zhou, X.; Cui, J., Global stability of the viral dynamics with crowley-martin functional response, Bull Korean Math Soc, 48, 3, 555-574 (2011) · Zbl 1364.34076 |
| [26] | Rao, F.; Mandal, P. S.; Kang, Y., Complicated endemics of an \(S I R S\) model with a generalized incidence under preventive vaccination and treatment controls, Appl Math Model, 67, 38-61 (2019) · Zbl 1481.92159 |
| [27] | Goel, K.; Kumar, A.; Nilam, Nonlinear dynamics of a time-delayed epidemic model with two explicit aware classes, saturated incidences, and treatment, Nonlinear Dyn, 4, 1-23 (2020) · Zbl 1517.92030 |
| [28] | Yanju, X.; Weipeng, Z.; Guifeng, D.; Zhehua, L., Stability and Bogdanov-Takens Bifurcation of an \(S I S\) epidemic model with saturated treatment function, Math Probl Eng, 2015, Article 745732 pp. (2015) · Zbl 1394.92131 |
| [29] | Linhua, Z.; Meng, F., Dynamics of an \(S I R\) epidemic model with limited medical resources revisited, Nonlinear Anal Real World Appl, 13, 1, 312-324 (2012) · Zbl 1238.37041 |
| [30] | Elbasha, E. H.; Gumel, A. B., Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits, Bull Math Biol, 68, 3, 577-614 (2006) · Zbl 1334.91060 |
| [31] | Iboi, E. A.; Ngonghala, C. N.; Gumel, A. B., Will an imperfect vaccine curtail the COVID-19 pandemic in the U.S.?, Infect Dis Model, 5, 510-524 (2020) |
| [32] | Safi, M. A.; Gumel, A. B., Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine, Comp Math Appl, 61, 10, 3044-3070 (2011) · Zbl 1222.37102 |
| [33] | Teboh-Ewungkem, M. I.; Podder, C. N.; Gumel, A. B., Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics, Bull Math Biol, 72, 1, 63-93 (2010) · Zbl 1184.92033 |
| [34] | Magpantay, F. M.; Riolo, M. A.; De Celles, M. D.; King, A. A.; Rohani, P., Epidemiological consequences of imperfect vaccines for immunizing infections, SIAM J Appl Math, 74, 6, 1810-1830 (2014) · Zbl 1379.92063 |
| [35] | Gumel, A. B.; McCluskey, C. C.; Watmough, J., An \(S V E I R\) model for assessing potential impact of an imperfect anti-SARS vaccine, Math Biosci Eng, 3, 3, 485-512 (2006) · Zbl 1092.92039 |
| [36] | Gumel, A. B.; McCluskey, C. C.; van den Driessche, P., Mathematical study of a staged-progression HIV model with imperfect vaccine, Bull Math Biol, 68, 8, 2105-2128 (2006) · Zbl 1296.92124 |
| [37] | Cai, L.-M.; X-Zi, Li., Analysis of a \(S E I V\) epidemic model with a nonlinear incidence rate, Appl Math Model, 33, 7, 2919-2926 (2009) · Zbl 1205.34049 |
| [38] | Sahu, G. P.; Dhar, J., Analysis of an \(S V E I S\) epidemic model with partial temporary immunity and saturation incidence rate, Appl. Math. Mod, 36, 3, 908-923 (2012) · Zbl 1243.34068 |
| [39] | Nic-May, A. J.; Avila-Vales, E. J., Global dynamics of a two-strain flu model with a single vaccination and general incidence rate, Math Biosci Eng, 17, 6, 7862-7891 (2020) · Zbl 1471.92207 |
| [40] | Ghosh, J. K.; Ghosh, U.; Biswas, M. H.A.; Sarkar, S., Qualitative analysis and optimal control strategy of an \(S I R\) model with saturated incidence and treatment, Diff Equ Dyn Syst, 2019 (2019) |
| [41] | Rwezaura, H.; Mtisi, E.; Tchuenche, J. M., A mathematical analysis of influenza with treatment and vaccination, (Tchuenche, J. M.; Chiyaka, C., Infectious disease modelling research progress. Infectious disease modelling research progress, Series - public health in the 21st century, vol. 764 (2010), Nova Science Publishers: Nova Science Publishers NY, Inc), 31-84 |
| [42] | Malik, T. M.; Mohammed-Awel, J.; Gumel, A. B.; Elbasha, E. H., Mathematical assessment of the impact of cohort vaccination on pneumococcal carriage and serotype replacement, J Biol Dyn, 1-34 (2021) · Zbl 1484.92059 |
| [43] | Jaharuddin, Bakhtiar T., Control policy mix in measles transmission dynamics using vaccination, therapy, and treatment, Int J Math Math Sci, 2020, Article 1561569 pp. (2020) · Zbl 1486.92236 |
| [44] | Naik, P. A.; Zu, J.; Ghoreishi, M., Stability analysis and approximate solution of sir epidemic model with Crowley-Martin type functional response and Holling type-II treatment rate by using homotopy analysis method, J Appl Anal Comput, 10, 4, 1482-1515 (2020) · Zbl 1455.34039 |
| [45] | Zhou, X.; Cui, J., Analysis of stability and bifurcation for an \(S E I V\) epidemic model with vaccination and nonlinear incidence rate, Nonlinear Dyn, 63, 639-653 (2011) |
| [46] | van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math Biosci, 180, 1, 29-48 (2002) · Zbl 1015.92036 |
| [47] | Mtisi, E.; Rwezaura, H.; Tchuenche, J. M., A mathematical analysis of malaria and tuberculosis co-dynamics, Discrete Cont Dyn Syst B, 12, 4, 827-864 (2009) · Zbl 1181.92058 |
| [48] | Mukandavire, Z.; Gumel, A. B.; Garira, W.; Tchuenche, J. M., Mathematical analysis of a model for HIV-malaria co-infection, Math Biosci Eng, 6, 2, 333-362 (2009) · Zbl 1167.92020 |
| [49] | Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math Biosci Eng, 1, 2, 361-404 (2004) · Zbl 1060.92041 |
| [50] | Omame, A.; Rwezaura, H.; Diagne, M. L.; Inyama, S. C.; Tchuenche, J. M., COVID-19 and dengue co-infection in Brazil: optimal control and cost-effectiveness analysis, Eur Phys J Plus, 136, 10 (2021) |
| [51] | Tchoumi, S. Y.; Diagne, M. L.; Rwezaura, H.; Tchuenche, J. M., Malaria and COVID-19 co-dynamics: A mathematical model and optimal control, Appl Math Model, 9, 2021, 294-327 (2021) · Zbl 1481.92165 |
| [52] | Pontryagin, L.; Boltyanskii, V.; Gamkrelidze, R.; Mishchenko, E., The mathematical theory of optimal control process (1986) |
| [53] | Omame, A.; Nnanna, C. U.; Inyama, S. C., Optimal control and cost-effectiveness analysis of an HPV-chlamydia trachomatis co-infection model, Acta Biotheor (2021) |
| [54] | Amit, A. M.L.; Pepito, V. C.F.; Sumpaico-Tanchanco, L.; Dayrit, M. M., COVID-19 vaccine brand hesitancy and other challenges to vaccination in the Philippines, PLOS Glob Public Health, 2, 1, Article e0000165 pp. (2022) |
| [55] | Storey, D., COVID-19 vaccine hesitancy, Glob Health Sci Pract, 10, 1, Article e2200043 pp. (2022) |
| [56] | Khubchandani, J.; Sharma, S.; Price, J. H.; Wiblishauser, M. J.; Sharma, M.; Webb, F. J., COVID-19 vaccination hesitancy in the United States: A rapid national assessment, J Community Health, 46, 2, 270-277 (2021) |
| [57] | Fridman, A.; Gershon, R.; Gneezy, A., COVID-19 and vaccine hesitancy: A longitudinal study, PLoS ONE, 16, 4, Article e0250123 pp. (2021) |
| [58] | Rwezaura, H.; Tchoumi, S. Y.; J. M., Tchuenche, Impact of environmental transmission and contact rates on Covid-19 dynamics: A simulation study, Inform Med Unlocked, 27, Article 100807 pp. (2021) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
