# Age Structured Deterministic Model of Diphtheria Infection

• Ekere Sunday Udofia Department of Mathematics, Akwa Ibom State University, Ikot Akpaden, Akwa Ibom State, Nigeria
• Ubong Dominic Akpan Department of Mathematics, Akwa Ibom State University, Ikot Akpaden, Akwa Ibom State, Nigeria
• Joy Ijeoma Uwakwe Department of Mathematics, Alvan Ikoku University of Education, Owerri, Imo State, Nigeria
• Henry Sylvester Thomas Department of Mathematics, Akwa Ibom State University, Ikot Akpaden, Akwa Ibom State, Nigeria
Keywords: Lyapunov function, global stability, diphtheria infection, reproduction ratio, vaccination

### Abstract

Age-structured mathematical model of diphtheria infection has been formulated with specific epidemiological classes such as S1, susceptible infant at time t (0-1years), S2, susceptible school children population at time t, V, vaccination population at time t, E, exposed population at time t, I1, asymptomatic infection population at time t, I2, symptomatic infection population at time t, ID, detected infectious humans at time t (asymptomatic and symptomatic) population through testing, R, recovered population at time t. It was established through theorems and proofs that the model is epidemiologically meaningful, and that all its state variables are positive (non-negative) at time t>0 in the domain ℘, and that the domain ℘ is indeed bounded. Using the next generation matrix, the reproduction ratio Rb of the system was determined. Using dynamical system theory, it was established that the system is locally stable. A matrix-theoretic method was used in the construction of an appropriate Lyapunov function for the global stability analysis of the formulated model, and also established that the system is globally asymptotically stable if Rb≤1 and unstable otherwise.

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Published
2024-02-16
How to Cite
Udofia, E. S., Akpan, U. D., Uwakwe, J. I., & Thomas, H. S. (2024). Age Structured Deterministic Model of Diphtheria Infection. Earthline Journal of Mathematical Sciences, 14(3), 391-404. https://doi.org/10.34198/ejms.14324.391404
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