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Muhammad Gul
Muhammad Usman
Nawab Ali
Muhammad Ilyas
Sayyida Aiza Bukhari
Muhammad Haidar Zaman
Muhammad Abubakar


Biological model, Stability theory, Routh-Hurwitz Criteria, Endemic equilibrium


The purpose of this research is to examine a biological model of vector-borne disease. The paper’s research demonstrates that its dynamics are solely dependent on the basic reproduction number R0. Our investigation is consisted on stability theory and numerical simulations. The Routh-Hurwitz Criteria and the Lyapunov approach are used to determine the local and global asymptotic stability of the disease-free equilibrium. In this paper, we use the work of McCluskey and van den Driessche to show that endemic equilibrium is stable locally. We also use the geometric approach method developed by Li and Muldowney to show that endemic equilibrium is stable at the global level, where the disease stays latent if it already exists. If R0 ≤1, the disease-free equilibrium is globally asymptotically stable, and the disease will vanish, and a unique endemic equilibrium exists if R0 >1.


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1 Gubler, D.J., The global threat of emergent/re-emergent vector-borne diseases. Vector biology, ecology and control, 2010: p. 39-62.
2 Savić, S., et al., Emerging vector-borne diseases–incidence through vectors. Frontiers in public health, 2014. 2: p. 267.
3 Harvell, C.D., et al., Climate warming and disease risks for terrestrial and marine biota. Science, 2002. 296(5576): p. 2158-2162.
4 Chala, B. and F. Hamde, Emerging and Re-emerging Vector-Borne Infectious Diseases and the Challenges for Control: A Review. Front Public Health, 2021. 9: p. 715759.
5 Gryseels, B., et al., Human schistosomiasis. The lancet, 2006. 368(9541): p. 1106-1118.
6 Chala, B. and W. Torben, An epidemiological trend of urogenital schistosomiasis in Ethiopia. Frontiers in public Health, 2018. 6: p. 60.
7 Morens, D.M., G.K. Folkers, and A.S. Fauci, Emerging infections: a perpetual challenge. The Lancet infectious diseases, 2008. 8(11): p. 710-719.
8 LaSalle JP. The Stability of Dynamical Systems, SIAM, Philadelphia, Pa, USA 1976.
9 Buonomo B, Vargas-De-Leon C. Stability and bifurcation analysis of a vector-bias model of malaria transmission, Mathematical Biosciences 2013; 242(1): 59-67.
10 Muldowney JS. Compound matrices and ordinary differential equations, The Rocky Mountain Journal of Mathematics 1990; 20(4): 857-872.
11 van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 2002; 180(1-2): 29-48.
12 Fonda A. Uniformly persistent semidynamical systems, Proceedings of the American Mathematical Society 1988; 104(1): 111-116.
13 Tumwiine J, Mugisha JYT, Luboobi LS. A hostvector model for malaria with infective immigrants, Journal of Mathematical Analysis and Applications 2010; 361(1): 139-149.
14 Li MY, Muldowney JS. Global stability for the SEIR model in epidemiology, Mathematical Biosciences 1995; 125(2): 155-164.

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