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Mathematical Modeling of Toxoplasmosis in Cats with Two Time Delays under Environmental Effects

Author

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  • Sharmin Sultana

    (Department of Mathematics, New Mexico Tech, Leroy Place, Socorro, NM 87801, USA)

  • Gilberto González-Parra

    (Department of Mathematics, New Mexico Tech, Leroy Place, Socorro, NM 87801, USA)

  • Abraham J. Arenas

    (Departamento de Matemáticas y Estadística, Universidad de Córdoba, Montería 230002, Córdoba, Colombia)

Abstract

In this paper, we construct a more realistic mathematical model to study toxoplasmosis dynamics. The model considers two discrete time delays. The first delay is related to the latent phase, which is the time lag between when a susceptible cat has effective contact with an oocyst and when it begins to produce oocysts. The second discrete time delay is the time that elapses from when the oocysts become present in the environment to when they are able to infect. The main aim in this paper is to find the conditions under which the toxoplasmosis can disappear from the cat population and to study whether the time delays can affect the qualitative properties of the model. Thus, we investigate the impact of the combination of two discrete time delays on the toxoplasmosis dynamics. Using dynamical systems theory, we are able to find the basic reproduction number R 0 d that determines the global long-term dynamics of the toxoplasmosis. We prove that, if R 0 d < 1 , the toxoplasmosis will be eradicated and that the toxoplasmosis-free equilibrium is globally stable. We design a Lyapunov function in order to prove the global stability of the toxoplasmosis-free equilibrium. We also prove that, if the threshold parameter R 0 d is greater than one, then there is only one toxoplasmosis-endemic equilibrium point, but the stability of this point is not theoretically proven. However, we obtained partial theoretical results and performed numerical simulations that suggest that, if R 0 d > 1 , then the toxoplasmosis-endemic equilibrium point is globally stable. In addition, other numerical simulations were performed in order to help to support the theoretical stability results.

Suggested Citation

  • Sharmin Sultana & Gilberto González-Parra & Abraham J. Arenas, 2023. "Mathematical Modeling of Toxoplasmosis in Cats with Two Time Delays under Environmental Effects," Mathematics, MDPI, vol. 11(16), pages 1-20, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3463-:d:1214146
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    References listed on IDEAS

    as
    1. Shyam Sundar Santra & Omar Bazighifan & Hijaz Ahmad & Shao-Wen Yao, 2020. "Second-Order Differential Equation with Multiple Delays: Oscillation Theorems and Applications," Complexity, Hindawi, vol. 2020, pages 1-6, November.
    2. Zahra Hosseininejad & Mehdi Sharif & Shahabeddin Sarvi & Afsaneh Amouei & Seyed Abdollah Hosseini & Tooran Nayeri Chegeni & Davood Anvari & Reza Saberi & Shaban Gohardehi & Azadeh Mizani & Mitra Sadeg, 2018. "Toxoplasmosis seroprevalence in rheumatoid arthritis patients: A systematic review and meta-analysis," PLOS Neglected Tropical Diseases, Public Library of Science, vol. 12(6), pages 1-13, June.
    3. Sharmin Sultana & Gilberto González-Parra & Abraham J. Arenas, 2023. "A Generalized Mathematical Model of Toxoplasmosis with an Intermediate Host and the Definitive Cat Host," Mathematics, MDPI, vol. 11(7), pages 1-17, March.
    4. Turner, Matthew & Lenhart, Suzanne & Rosenthal, Benjamin & Zhao, Xiaopeng, 2013. "Modeling effective transmission pathways and control of the world’s most successful parasite," Theoretical Population Biology, Elsevier, vol. 86(C), pages 50-61.
    5. García, M.A. & Castro, M.A. & Martín, J.A. & Rodríguez, F., 2018. "Exact and nonstandard numerical schemes for linear delay differential models," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 337-345.
    6. Sykes, David & Rychtář, Jan, 2015. "A game-theoretic approach to valuating toxoplasmosis vaccination strategies," Theoretical Population Biology, Elsevier, vol. 105(C), pages 33-38.
    7. Gilberto González-Parra & Sharmin Sultana & Abraham J. Arenas, 2022. "Mathematical Modeling of Toxoplasmosis Considering a Time Delay in the Infectivity of Oocysts," Mathematics, MDPI, vol. 10(3), pages 1-20, January.
    8. Liu, Yifan & Cai, Jiazhi & Xu, Haowen & Shan, Minghe & Gao, Qingbin, 2023. "Stability and Hopf bifurcation of a love model with two delays," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 558-580.
    9. Abraham J. Arenas & Gilberto González-Parra & Jhon J. Naranjo & Myladis Cogollo & Nicolás De La Espriella, 2021. "Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay," Mathematics, MDPI, vol. 9(3), pages 1-21, January.
    10. María Ángeles Castro & Miguel Antonio García & José Antonio Martín & Francisco Rodríguez, 2019. "Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems," Mathematics, MDPI, vol. 7(11), pages 1-14, November.
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