The North Carolina Journal of Mathematics and Statistics

Analysis of a Vector-Borne Diseases Model with a Two-Lag Delay Differential Equation

Yusuf Qaddura, Nsoki Mavinga

Abstract


We are concerned with the stability analysis of equilibrium solutions for a two-lag delay differential equation which models the spread of vector-borne diseases, where the lags are incubation periods in humans and vectors. We show that there are some values of transmission and recovery rates for which the disease dies out and others for which the disease spreads into an endemic. The proofs of the main stability results are based on the linearization method and the analysis of roots of transcendental equations. We then simulate numerical solutions using MATLAB. We observe that the solution could possess chaotic and sometimes unbounded behaviors.


Keywords


delay differential equations; epidemiology; vector-borne diseases; linearization; transcendental equations; stability.

Full Text:

PDF

References


Agusto, F., Bewick, S., and Fagan, W. (2017). Mathematical model of zika virus with vertical transmission. Infectious Disease Modelling 2(2):244–267

Braddock, R. and van den Driessche, P. (1983). On a two lag differential delay equation. The ANZIAM Journal, 24(3):292–317.

Cai, L.-M., Li, X.-Z., Fang, B., and Ruan, S. (2017). Global properties of vector–host disease models with time delays. Journal of mathematical biology, 74(6):1397–1423.

Cooke, K. L. (1979). Stability analysis for a vector disease model. The Rocky Mountain Journal of Mathematics, 9(1):31–42.

Cooke, K. L. and van ven Driessche, P. (1996). Analysis of an seirs epidemic model with two delays. Journal of Mathematical Biology, 35(2):240–260.

Erneux, T. (2009). Applied delay differential equations, volume 3. Springer Science & Business Media.

Hayes, N. (1950). Roots of the transcendental equation associated with a certain difference-differential equation. Journal of the London Mathematical Society, 1(3):226–232.

Smith, H. L. (2011). An introduction to delay differential equations with applications to the life sciences, volume 57. Springer New York.

van den Driessche, P. (1994). Some epidemiological models with delays. Technical report. http://dspace.library.uvic.ca/handle/1828/1767

Xiao, Y. and Zou, X. (2013). Latencies in malaria infections and their impact on the disease dynamics. Math Biosci Eng, 10(2):463–481.