The Impact of Imperfect Vaccination on Infectious Disease Transmission in an Age-Structured Population
Authors: Hathout Fatima Zohra , Touaoula Tarik Mohammed 
Published: January 2025

In this paper, we consider the influence of imperfect vaccination on the spread of infectious diseases in an age-structured population. The benefits of vaccination, even if not perfect, generally outweigh the risks of severe diseases. In a mathematical system, we consider the compartment of susceptible s; vaccinated v and infected i individuals with an age structure. The proposed model is globally analyzed by introducing total trajectories and employing a suitable Lyapunov functional. To illustrate our theoretical findings, we include numerical simulations at the end of the paper.
AMS subject classification: 35Q92, 37N25, 92D30.
REFERENCES
[1] Adimy, M., Chekroun, A., & Ferreira, C. P. (2020). Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase. Mathematical Biosciences and Engineering, 17(2), 1329-1354.‏ Search in Google Scholar. https://dx.doi.org/10.3934/mbe.2020067
[2] Benchaira, S., Mancer, S., & Necir, A. (2024). A Log-Probability-Weighted-Moments type estimator for the extreme value index in a truncation scheme. International Journal of Applied Mathematics and Simulation, 1(2).‏ Search in Google Scholar. https://doi.org/10.69717/ijams.v1.i2.99
[3] Soufiane, B., & Touaoula, T. M. (2016). Global analysis of an infection age model with a class of nonlinear incidence rates. Journal of Mathematical Analysis and Applications, 434(2), 1211-1239.‏ Search in Google Scholar. https://doi.org/10.1016/j.jmaa.2015.09.066
[4] Boudjema, I., & Touaoula, T. M. (2018). Global stability of an infection and vaccination age-structured model with general nonlinear incidence. J. Nonlinear Funct. Anal, 2018(33), 1-21.‏ Search in Google Scholar. https://doi.org/10.23952/jnfa.2018.33
[5] Bubar, K. M., Reinholt, K., Kissler, S. M., Lipsitch, M., Cobey, S., Grad, Y. H., & Larremore, D. B. (2021). Model-informed COVID-19 vaccine prioritization strategies by age and serostatus. Science, 371(6352), 916-921.‏ Search in Google Scholar. https://doi.org/10.1126/science.abe6959
[6] Buckner, J. H., Chowell, G., & Springborn, M. R. (2020). Optimal dynamic prioritization of scarce COVID-19 vaccines. Medrxiv, 2020-09.‏ Search in Google Scholar. https://doi.org/10.1073/pnas.2025786118
[7] Cai, L., Martcheva, M., & Li, X. Z. (2013). Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete and Continuous Dynamical Systems Series B, 18, 2239-2265.‏ Search in Google Scholar. View in Academia
[8] Cai, L. M., Modnak, C., & Wang, J. (2017). An age-structured model for cholera control with vaccination. Applied Mathematics and Computation, 299, 127-140.‏ Search in Google Scholar. https://doi.org/10.1016/j.amc.2016.11.013
[9] Castillo-Chavez, C., Hethcote, H. W., Andreasen, V., Levin, S. A., & Liu, W. M. (1989). Epidemiological models with age structure, proportionate mixing, and cross-immunity. Journal of mathematical biology, 27, 233-258.‏ Search in Google Scholar. https://doi.org/10.1007/BF00275810
[10] Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases: model building, analysis and interpretation (Vol. 5). John Wiley & Sons.‏ Search in Google Scholar. View
[11] Halloran, M. E., Haber, M., & Longini Jr, I. M. (1992). Interpretation and estimation of vaccine efficacy under heterogeneity. American Journal of Epidemiology, 136(3), 328-343.‏ Search in Google Scholar. https://doi.org/10.1093/oxfordjournals.aje.a116498
[12] Feng, Z., Feng, Y., & Glasser, J. W. (2020). Influence of demographically-realistic mortality schedules on vaccination strategies in age-structured models. Theoretical population biology, 132, 24-32.‏ Search in Google Scholar. https://doi.org/10.1016/j.tpb.2020.01.005
[13] Galazka, A. M., Robertson, S. E., & Oblapenko, G. P. (1995). Resurgence of diphtheria. European journal of epidemiology, 11, 95-105.‏ Search in Google Scholar. https://doi.org/10.1007/BF01719954
[14] Grenfell, B. T., & Anderson, R. M. (1989). Pertussis in England and Wales: an investigation of transmission dynamics and control by mass vaccination. Proceedings of the Royal Society of London. B. Biological Sciences, 236(1284), 213-252.‏ Search in Google Scholar. https://doi.org/10.1098/rspb.1989.0022
[15] Hathout, F. Z., Touaoula, T. M., & Djilali, S. (2022). Mathematical analysis of a triple age dependent epidemiological model with including a protection strategy. Discrete & Continuous Dynamical Systems-Series B, 27(12).‏     Search in Google Scholar. https://doi.org/10.3934/dcdsb.2022048
[16] Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM review, 42(4), 599-653.‏ Search in Google Scholar. https://doi.org/10.1137/S0036144500371907
 [17] Janaszek, W., Gay, N. J., & Gut, W. (2003). Measles vaccine efficacy during an epidemic in 1998 in the highly vaccinated population of Poland. Vaccine, 21(5-6), 473-478.‏ Search in Google Scholar. https://doi.org/10.1016/S0264-410X(02)00482-6
[18] Kanaan, M. N., & Farrington, C. P. (2002). Estimation of waning vaccine efficacy. Journal of the American Statistical Association, 97(458), 389-397.‏ Search in Google Scholar. https://doi.org/10.1198/016214502760046943
[19] Magal, P., & Thieme, H. R. (2004). Eventual compactness for semiflows generated by nonlinear age-structured models. Commun. Pure Appl. Anal, 3(4), 695-727.‏ Search in Google Scholar. View
[20] Magal, P., & Zhao, X. Q. (2005). Global attractors and steady states for uniformly persistent dynamical systems. SIAM journal on mathematical analysis, 37(1), 251-275.‏ Search in Google Scholar. https://doi.org/10.1137/S0036141003439173
[21] Magpantay, F. M., Riolo, M. A., De Celles, M. D., King, A. A., & Rohani, P. (2014). Epidemiological consequences of imperfect vaccines for immunizing infections. SIAM Journal on Applied Mathematics, 74(6), 1810-1830.‏ Search in Google Scholar. https://doi.org/10.1137/140956695
[22] McLean, A. R., & Blower, S. M. (1993). Imperfect vaccines and herd immunity to HIV. Proceedings of the Royal Society of London. Series B: Biological Sciences, 253(1336), 9-13.‏ Search in Google Scholar. https://doi.org/10.1098/rspb.1993.0075
[23] Mossong, J., Nokes, D. J., Edmunds, W. J., Cox, M. J., Ratnam, S., & Muller, C. P. (1999). Modeling the impact of subclinical measles transmission in vaccinated populations with waning immunity. American journal of epidemiology, 150(11), 1238-1249.‏ Search in Google Scholar. https://doi.org/10.1093/oxfordjournals.aje.a009951
[24] Scherer, A., & McLean, A. (2002). Mathematical models of vaccination. British medical bulletin, 62(1), 187-199.‏ Search in Google Scholar. https://doi.org/10.1093/bmb/62.1.187
[25] Shim, E., & Galvani, A. P. (2012). Distinguishing vaccine efficacy and effectiveness. Vaccine, 30(47), 6700-6705.‏ Search in Google Scholar. https://doi.org/10.1016/j.vaccine.2012.08.045
[26] Smith, H. L., & Thieme, H. R. (2011). Dynamical systems and population persistence (Vol. 118). American Mathematical Soc..‏ Search in Google Scholar. View a book
[27] Smith, H., & Zhao, X. Q. (2001). Robust persistence for semidynamical systems. Nonlinear Analysis, Theory, Methods and Applications, 47(9), 6169-6179.‏ Search in Google Scholar. https://doi.org/10.1016/S0362-546X(01)00678-2
[28] Webb, G. F. (1985). Theory of nonlinear age-dependent population dynamics. CRC Press.‏ Search in Google Scholar. View a book
 
Communicated Editor: Boumediène Abdellaoui
Manuscript received Mar 09, 2024; revised Jan 05, 2025; accepted Jan 19, 2025; published Jan 23, 2025.