Analysis of a time-delayed vector-host epidemic model with nonlinear incidences

Volume 38, Issue 4, pp 502--520 https://dx.doi.org/10.22436/jmcs.038.04.06

Publication Date: February 11, 2025 Submission Date: August 06, 2024 Revision Date: November 11, 2024 Accteptance Date: January 08, 2025

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 515 Downloads
• 1744 Views

Authors

S. Jothika - Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, 603203, Tamil Nadu, India. M. Radhakrishnan - Department of Mathematics, College of Engineering and Technology, SRM Institute of Science and Technology, Kattankulathur, 603203, Tamil Nadu, India. - Directorate of Learning and Development, SRM Institute of Science and Technology, Kattankulathur, 603203, Tamil Nadu, India.

Abstract

In this study, we investigate a time-delayed vector-host epidemic model with nonlinear incidence rates to gain a deeper understanding of the dynamics of vector-borne diseases, particularly those transmitted by vectors like mosquitoes. The model incorporates a constant human recruitment rate, exponential natural death, and an asymptotically constant vector population. We rigorously analyze the stability of both the disease-free and endemic equilibria, introducing a threshold parameter, the basic reproduction number \(R_0\), to determine the long-term behavior of the epidemic. For \(R_0<1\), the disease-free equilibrium is globally asymptotically stable, indicating disease eradication. When \(R_0>1\), the endemic equilibrium emerges, and its stability is assessed through local stability analysis. The impact of time delay on the system dynamics is examined, revealing conditions under which a Hopf bifurcation occurs, leading to sustained periodic oscillations. This phenomenon highlights the critical role of time delay in influencing the epidemiology and control of vector-borne diseases. Our findings provide valuable insights into the complex interplay between time delays and nonlinear transmission dynamics, offering implications for effective disease management and control strategies.

Share and Cite

Keywords

• Time-delayed nonlinear incidence epidemic model
• vector-borne diseases
• basic reproduction number
• endemic equilibrium
• disease-free equilibrium
• Hopf bifurcation

MSC

•  37N25
•  34C23
•  34D23

References

• [1] R. M. Anderson, The persistence of direct life cycle infectious diseases within populations of hosts, Amer. Math. Soc., Providence, RI, 12 (1979), 1–67
  • View Article
  • Google Scholar
  • MATH


• [2] R. M. Anderson, R. M. May, Infectious diseases of humans: dynamics and control, Oxford University Press, (1991)
  • View Article
  • Google Scholar


• [3] J. L. Aron, R. M. May, The population dynamics of malaria, in The population dynamics of infectious diseases: theory and applications, Springer, Boston, MA, (1982), 139–179
  • View Article
  • Google Scholar


• [4] C. Bowman, A. B. Gumel, P. Van den Driessche, J. Wu, H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107–1133
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] L.-M. Cai, X.-Z. Li, Global analysis of a vector-host epidemic model with nonlinear incidences, Appl. Math. Comput., 217 (2010), 3531–3541
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43–61
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] Y. Chen, S. Zou, J. Yang, Global analysis of an SIR epidemic model with infection age and saturated incidence, Nonlinear Anal. Real World Appl., 30 (2016), 16–31
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [8] L. Esteva, H. M. Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), 132–147
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] Z. Feng, J. X. Velasco-Hernández, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523–544
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] G. Huang, Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125–139
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] G. Huang, Y. Takeuchi, W. Ma, D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192–1207
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] A. Kumar, Nilam, Mathematical analysis of a delayed epidemic model with nonlinear incidence and treatment rates, J. Eng. Math., 115 (2019), 1–20
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] M. Y. Li, J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125 (1995), 155–164
  • View Article
  • MathSciNet
  • Google Scholar


• [14] M. Li, X. Liu, An SIR epidemic model with time delay and general nonlinear incidence rate, Abstr. Appl. Anal., 2014 (2014), 7 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] W.-M. Liu, H. W. Hethcote, S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359–380
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] T. Ma, X. Meng, Global analysis and Hopf-bifurcation in a cross-diffusion prey-predator system with fear effect and predator cannibalism, Math. Biosci. Eng., 19 (2022), 6040–6071
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] M. J. Mackinnon, Drug resistance models for malaria, Acta Trop., 94 (2005), 207–217
  • View Article
  • Google Scholar


• [18] M. Manivel, A. Venkatesh, K. Arunkumar, M. P. Raj, Shyamsunder, A mathematical model of the dynamics of the transmission of monkeypox disease using fractional differential equations, Adv. Theory Simul., 7 (2024),
  • View Article
  • Google Scholar


• [19] C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. Real World Appl., 11 (2010), 3106–3109
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] X. Meng, L. Chen, H. Cheng, Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl. Math. Comput., 186 (2007), 516–529
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [21] X. Meng, F. Li, S. Gao, Global analysis and numerical simulations of a novel stochastic eco-epidemiological model with time delay, Appl. Math. Comput., 339 (2018), 701–726
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] P. R. Murugadoss, V. Ambalarajan, V. Sivakumar, P. B. Dhandapani, D. Baleanu, Analysis of dengue transmission dynamic model by stability and Hopf bifurcation with two-time delays, Front. Biosci., 28 (2023), 13 pages
  • View Article
  • Google Scholar


• [23] R. Naresh, A. Tripathi, J. M. Tchuenche, D. Sharma, Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate, Comput. Math. Appl., 58 (2009), 348–359
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Comput. Math. Appl., 56 (2008), 3118–3129
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] R. Ross, The prevention of malaria, John Murray, London (1911)
  • View Article
  • Google Scholar


• [26] P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] A. Venkatesh, M. P. Raj, B. Baranidharan, Analyzing dynamics and stability of single delay differential equations for the dengue epidemic model, Results Control Optim., 15 (2024), 17 pages
  • View Article
  • Google Scholar


• [28] A. Venkatesh, M. P. Raj, B. Baranidharan, M. K. I. Rahmani, K. T. Tasneem, M. Khan, J. Giri, Analyzing steady-state equilibria and bifurcations in a time-delayed SIR epidemic model featuring Crowley-Martin incidence and Holling Type II treatment rates, Heliyon, 10 (2024), 15 pages
  • View Article
  • Google Scholar


• [29] A. Venkatesh, M. A. Rao, M. P. Raj, K. A. Kumar, D. K. K. Vamsi, Mathematical modelling of COVID-19 dynamics using SVEAIQHR model, Comput. Math. Biophys., 12 (2024), 23 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] R. Xu, Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. Real World Appl., 10 (2009), 3175–3189
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] R. Xu, Z. Ma, Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos Solitons Fractals, 41 (2009), 2319–2325
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] X.-P. Yan, W.-T. Li, Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Appl. Math. Comput., 177 (2006), 427–445
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH