Epidemic Dynamics with Nonlinear Incidence Considering Vaccination Effectiveness

Putri Zahra Kamalia, Dipo Aldila

Abstract


This paper presents a mathematical model that examines the effect of nonlinear incidence on disease transmission dynamics.  Furthermore, we also accommodate newborn and adult vaccination strategy as the prevention strategy to prevent rapid spread of the disease due to nonlinear incidence rate. Assuming a constant population  size,  the  system is  reduced  to  a  two-dimensions and  nondimensionalized using  the  average infectious period as the time scale.   Analytical results reveal the existence of both disease-free and endemic equilibria, with the possibility of backward bifurcation when the nonlinear incidence parameter exceeds a critical threshold.   This implies that disease persistence may still occur even when the basic reproduction number is less than one.  Numerical simulations using MATCONT conducted to visualize the occurrence of both forward and backward bifurcations phenomena.    Using COVID-19 parameter values,  a  global sensitivity analysis via Partial Rank Correlation Coefficient - Latin Hypercube Sampling method indicates that newborn vaccination has a stronger impact on reducing the basic reproduction number. These findings provide important insights for designing effective vaccination strategies and understanding the complex dynamics arising from nonlinear transmission and imperfect immunization.

Keywords


Nonlinear incidence; Vaccination; Basic reproduction number; Global sensitivity analysis

Full Text:

PDF

References


W. O. Kermack and A. G. McKendrick, “A contribution to the mathematical theory of epidemics,” Proceedings of the Royal Society of London. Series A, vol. 115, no. 772, pp. 700–721, 1927. DOI:10.1098/rspa.1927.0118

H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Review, vol. 42, no. 4, pp. 599–653, 2000. DOI:10.1137/S0036144500371907

F. Brauer, “Compartmental models in epidemiology,” in Mathematical epidemiology, Berlin: Springer, 2008. DOI:10.1007/978-3-540-78911-6_2

D. Aldila et al., “A mathematical study on the spread of COVID-19 considering social distancing and rapid assessment: The case of Jakarta, Indonesia,” Chaos, Solitons & Fractals, vol. 139, p. 110042, 2020. DOI:10.1016/j.chaos.2020.110042

M. Chinazzi et al., “The effect of travel restrictions on the spread of the 2019 novel coronavirus (covid-19) outbreak,” Science, vol. 368, no. 6489, pp. 395–400, 2020. DOI:10.1126/science.aba9757

X. Wei et al., “Assessing the effectiveness of the intervention measures of COVID-19 in China based on dynamical method,” Infectious Disease Modelling, vol. 8, no. 1, pp. 159–171, 2023. DOI:10.1016/j.idm.2022.12.007

F. P. Polack et al., “Safety and efficacy of the bnt162b2 mrna covid-19 vaccine,” New England Journal of Medicine, vol. 383, no. 27, pp. 2603–2615, 2020. DOI:10.1056/NEJMoa2034577

CDC, “Measles (rubeola): For healthcare professionals,” https://www.cdc.gov/measles/hcp/index.html, 2022, Accessed on 18 July 2025.

B. Trunz, P. Fine, and C. Dye, “Effect of bcg vaccination on childhood tuberculous meningitis and miliary tuberculosis worldwide: a meta-analysis and assessment of cost-effectiveness,” The Lancet, vol. 367, no. 9517, pp. 1173–1180, 2006. DOI:10.1016/S0140-6736(06)68507-3

D. vAldila, C. A. Puspadani, and R. Rusin, “Mathematical Analysis of the Impact of Community Ignorance on the Population Dynamics of Dengue,” Frontiers in Applied Mathematics and Statistics, vol. 9, p. 1094971, 2023.

V. Capasso and G. Serio, “A generalization of the kermack-mckendrick deterministic epidemic model,” Mathematical Biosciences, vol. 42, no. 1–2, pp. 43–61, 1978. DOI:10.1016/0025-5564(78)90006-8

S. Funk, M. Salathé, and V. A. Jansen, “Modelling the influence of human behaviour on the spread of infectious diseases: a review,” Journal of the Royal Society Interface, vol. 7, no. 50, pp. 1247–1256, 2010. DOI:10.1098/rsif.2010.0142

M. A. Safi and A. B. Gumel, “Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine,” Computers & Mathematics with Applications, vol. 61, no. 10, pp. 3044–3070, 2011. DOI:10.1016/j.camwa.2011.03.095

B. D. Handari et al., “An optimal control model to understand the potential impact of the new vaccine and transmission-blocking drugs for malaria: A case study in papua and west papua, indonesia,” Vaccines, vol. 10, no. 8, p. 1174, 2022. DOI:10.3390/vaccines10081174

D. Aldila et al., “A tuberculosis epidemic model as a proxy for the assessment of the novel m72/as01e vaccine,” Communications in Nonlinear Science and Numerical Simulation, vol. 120, p. 107162, 2023. DOI:10.1016/j.cnsns.2023.107162

H. A. Fatahillah and D. Aldila, “Forward and backward bifurcation analysis from an imperfect vaccine efficacy model with saturated treatment and saturated infection,” Jambura Journal of Biomathematics (JJBM), vol. 5, no. 2, pp. 132–143, 2024.

DOI:10.37905/jjbm.v5i2.28810

D. Aldila, C. A. Puspadani, and R. Rusin, “Mathematical analysis of the impact of community ignorance on the population dynamics of dengue,” Frontiers in Applied Mathematics and Statistics, vol. 9, p. 1094971, 2023. DOI:/10.3389/fams.2023.1094971

O. Diekmann, J. A. P. Heesterbeek, and M. G. Roberts, “The construction of next-generation matrices for compartmental epidemic models,” Journal of the Royal Society Interface, vol. 7, no. 47, pp. 873–885, 2009. DOI:10.1098/rsif.2009.0386

D. Aldila et al., “An analytical transmission model for evaluating pneumonia vaccination and control strategies,” Healthcare Analytics, vol. 7, p. 100394, 2025. DOI:10.1016/j.health.2025.100394

D. Aldila et al., “Evaluating vaccination and quarantine for measles intervention strategy in jakarta, indonesia through mathematical modeling,” Partial Differential Equations in Applied Mathematics, vol. 14, p. 101191, 2025. DOI:10.1016/j.padiff.2025.101191

D. Aldila et al., “Reassessment of public awareness and prevention strategies for hiv and covid-19 co-infections through epidemic modeling,” PLOS ONE, vol. 20, no. 7, pp. 1–37, 2025. DOI:10.1371/journal.pone.0328488

D. Aldila, “Change in stability direction induced by temporal interventions: a case study of a tuberculosis transmission model with relapse and reinfection,” Frontiers in Applied Mathematics and Statistics, vol. 11, 2025. DOI:10.3389/fams.2025.1541981

D. Aldila, “Optimal control for dengue eradication program under the media awareness effect,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 24, no. 1, pp. 95–122, 2023. DOI:10.1515/ijnsns-2020-0142

D. Aldila et al., “Optimal control of pneumonia transmission model with seasonal factor: Learning from jakarta incidence data,” Heliyon, vol. 9, no. 7, p. e18096, 2023. DOI:10.1016/j.heliyon.2023.e18096

D. Aldila et al., “Forward Bifurcation with Hysteresis Phenomena from Atherosclerosis Mathematical Model,” Communication in Biomathematical Sciences, vol. 4, no. 2, pp. 125–137, 2021. DOI:10.5614/cbms.2021.4.2.4

O. Diekmann, J. A. P. Heesterbeek, and M. G. Roberts, “The construction of next-generation matrices for compartmental epidemic models,” Journal of the Royal Society Interface, vol. 7, no. 47, pp. 873–885, 2010. DOI:10.1098/rsif.2009.0386

C. Castillo-Chavez and B. Song, “Dynamical models of tuberculosis and their applications,” Mathematical Biosciences and Engineering, vol. 1, no. 2, pp. 361–404, 2004. DOI:10.3934/mbe.2004.1.361

C. K. Mahadhika and D. Aldila, “A deterministic transmission model for analytics-driven optimization of covid-19 post- pandemic vaccination and quarantine strategies,” Mathematical Biosciences and Engineering, vol. 21, no. 4, pp. 4956–4988, 2024. DOI:10.3934/mbe.2024219

S. Marino et al., “A methodology for performing global uncertainty and sensitivity analysis in systems biology,” Journal of Theoretical Biology, vol. 254, no. 1, pp. 178–196, 2008. DOI:10.1016/j.jtbi.2008.04.011

Z. Chazuka et al., “Strategic approaches to mitigating Hookworm infection: An optimal control and cost-effectiveness analysis,” Results in Control and Optimization, vol. 17, p. 100477, 2024. DOI:10.1016/j.rico.2024.100477

A. H. Hassan et al., “An Optimal Control Problem for a Monkeypox Transmission Model with Activity-Driven Contact Patterns,” Brazilian Journal of Physics, vol. 55, no. 5, p. 213, 2025. DOI:10.1007/s13538-025-01844-4

C. W. Chukwu et al., “Exploring the epidemiological impact of Pneumonia–Listeriosis co-infection in the human population: a modeling and optimal control study,” International Journal of Dynamics and Control, vol. 13, p. 197, 2025. DOI:10.1007/s40435-025-01714-6

C. M. Kribs-Zaleta and J. X. Velasco-Hernández, “A simple vaccination model with multiple endemic states,” Mathematical Biosciences, vol. 164, no. 2, pp. 183–201, 2000. DOI:10.1016/S0025-5564(00)00003-1

D. Aldila et al., “Backward bifurcation and periodic dynamics in a tuberculosis model with integrated control strategies,” Mathematical Biosciences and Engineering, vol. 22, no. 10, pp. 2720–2760, 2025. DOI:10.3934/mbe.2025100

W. Wang and S. Ruan, “Bifurcations in an epidemic model with constant removal rate of the infectives,” Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 775–793, 2004. DOI:10.1016/j.jmaa.2003.11.043




DOI: https://doi.org/10.37905/jjbm.v6i3.33815

Copyright (c) 2025 Putri Zahra Kamalia, Dipo Aldila

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Jambura Journal of Biomathematics (JJBM) has been indexed by:

EDITORIAL OFFICE OF JAMBURA JOURNAL OF BIOMATHEMATICS

 Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango 96554, Gorontalo, Indonesia
 Email: [email protected]
 Jambura Journal of Biomathematics (JJBM) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Powered by Public Knowledge Project OJS.