Analytical–Numerical Solution of a Temperature-Dependent Volterra Integro-Differential Model for Avian Influenza Transmission with Saturated Incidence and Environmental Feedback Via an Enhanced Adomian Decomposition Method

Ejes; Valentine; Liberty Ebiwareme

doi:10.56201/ijasmt.vol.12.no8.2026pg106.126

📄 Download Paper

Analytical–Numerical Solution of a Temperature-Dependent Volterra Integro-Differential Model for Avian Influenza Transmission with Saturated Incidence and Environmental Feedback Via an Enhanced Adomian Decomposition Method

Ejes, Valentine, Liberty Ebiwareme

Abstract

This study presents an analytical–numerical investigation of a temperature-dependent Volterra integro-differential model for the transmission dynamics of avian influenza incorporating saturated incidence and environmental feedback. The model captures both direct transmission among birds and indirect transmission through environmental viral load, while accounting for memory effects via an exponential kernel. The transmission process is modeled using a nonlinear saturated incidence function to reflect realistic contact limitations, and the environmental viral decay is assumed to depend explicitly on temperature. An enhanced Adomian Decomposition Method is employed to derive approximate analytical solutions of the nonlinear system. The method effectively handles the coupled integro-differential structure and nonlinear interaction terms. To validate the accuracy and convergence of the analytical solution, numerical simulations are performed using the fourth-order Runge–Kutta (RK4) method. A close agreement between the EADM and RK4 solutions is observed, confirming the reliability and efficiency of the proposed approach. Furthermore, the basic reproduction number (Ro) is derived using the next-generation matrix method, revealing contributions from both direct and environmental transmission pathways. Stability analysis shows that the disease-free equilibrium is locally asymptotically stable when ( Ro< 1) and unstable otherwise, while an endemic equilibrium exists when ( Ro> 1). The influence of temperature on viral decay is shown to play a significant role in reducing infection persistence, highlighting its importance in disease control strategies. The results demonstrate that incorporating environmental memory, temperature effects, and nonlinear incidence provides a more realistic framework for understanding avian influenza dynamics, and the enhanced Adomian Decomposition Method offers an efficient analytical tool for solving complex epidemiological models.

Keywords

Avian influenza; Volterra integro-differential model; Temperature-dependent transmission; Saturated incidence; Environmental viral load; Memory kernel; Enhanced Adomian Decomposition Method ; Runge–Ku

References

Adomian, G. (1988). A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications, 135(2), 501–544. Brown, J. D., Goekjian, G., Poulson, R., Valeika, S., & Stallknecht, D. E. (2009). Avian influenza virus in water: infectivity is dependent on pH, salinity and temperature. Veterinary Microbiology, 136(1–2), 20–26. Diekmann, O., & Heesterbeek, J. A. P. (2000). Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. John Wiley & Sons. Gripenberg, G., Londen, S.-O., & Staffans, O. (1990). Volterra Integral and Functional Equations. Cambridge University Press. Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653. Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society A, 115(772), 700–721. Miller, R. K. (1971). Nonlinear Volterra Integral Equations. W. A. Benjamin, Inc. Wazwaz, A.-M. (2009). A First Course in Integral Equations. World Scientific Publishing. Webster, R. G., Bean, W. J., Gorman, O. T., Chambers, T. M., & Kawaoka, Y. (1992). Evolution and ecology of influenza A viruses. Microbiological Reviews, 56(1), 152–179. Van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180, 29–48. Allen, L. J. S. (2008). An Introduction to Stochastic Epidemic Models. In Mathematical Epidemiology (pp. 81–130). Springer. Liu, W. M., Hethcote, H. W., & Levin, S. A. (1987). Dynamical behavior of epidemiological models with nonlinear incidence rates. Journal of Mathematical Biology, 25(4), 359–380. Zhao, X.-Q. (2017). Nonlinear Integral Equations and Applications in Epidemiology. Springer. Cui, J., Li, X., & Li, Z. (2018). Modeling avian influenza with environmental transmission and delay. Applied Mathematical Modelling, 60, 74–88. Brauer, F., Castillo-Chavez, C., & Feng, Z. (2019). Mathematical Models in Epidemiology. Springer. Chattopadhyay, J., & Chatterjee, A. (2007). Saturated incidence rates in epidemiological models. Journal of Biological Systems, 15(3), 353–374.

Analytical–Numerical Solution of a Temperature-Dependent Volterra Integro-Differential Model for Avian Influenza Transmission with Saturated Incidence and Environmental Feedback Via an Enhanced Adomian Decomposition Method

Abstract

Keywords

References

More Articles from INTERNATIONAL JOURNAL OF APPLIED SCIENCES AND MATHEMATICAL THEORY