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December  2021, 26(12): 6463-6481. doi: 10.3934/dcdsb.2021029

## Dynamics of a vector-host model under switching environments

 1 Department of Mathematics, The University of Alabama, Tuscaloosa, Alabama 35487-0350, USA 2 Faculty of Basic Sciences, Ho Chi Minh University of Transport, 2 Vo Oanh, Ho Chi Minh, Vietnam

** Corresponding author: Tran D. Tuong

Received  January 2020 Revised  November 2020 Published  December 2021 Early access  February 2021

Fund Project: This author is supported in part by NSF grant DMS-1853467

In this paper, the stochastic vector-host model has been proposed and analysed using nice properties of piecewise deterministic Markov processes (PDMPs). A threshold for the stochastic model is derived whose sign determines whether the disease will eventually disappear or persist. We show mathematically the existence of scenarios where switching plays a significant role in surprisingly reversing the long-term properties of deterministic systems.

Citation: Harrison Watts, Arti Mishra, Dang H. Nguyen, Tran D. Tuong. Dynamics of a vector-host model under switching environments. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6463-6481. doi: 10.3934/dcdsb.2021029
##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0. [2] Y. Asai, X. Han and P. E. Kloeden, Dynamics of Zika virus epidemic in random environment, in Mathematics Applied to Engineering, Modelling, and Social Issues, Springer, 2019,665–684. [3] K. Bao, L. Rong and Q. Zhang, Analysis of a stochastic sirs model with interval parameters, Discrete & Continuous Dynamical Systems-B, 24 (2019), 4827. doi: 10.3934/dcdsb.2019033. [4] M. Benaim, Stochastic persistence, preprint, arXiv: 1806.08450. [5] M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Annales de l'IHP Probabilités et Statistiques, 51 (2015), 1040–1075. doi: 10.1214/14-AIHP619. [6] M. Benaïm, E. Strickler, et al., Random switching between vector fields having a common zero, The Annals of Applied Probability, 29 (2019), 326-375. doi: 10.1214/18-AAP1418. [7] D. Bichara, Effects of migration on vector-borne diseases with forward and backward stage progression, preprint, arXiv: 1810.06777. doi: 10.3934/dcdsb.2019140. [8] Z. Cao, X. Liu, X. Wen, L. Liu and L. Zu, A regime-switching sir epidemic model with a ratio-dependent incidence rate and degenerate diffusion, Scientific Reports, 9 (2019), 1-7.  doi: 10.1186/s13662-017-1355-3. [9] M. H. Davis, Piecewise-deterministic markov processes: A general class of non-diffusion stochastic models, Journal of the Royal Statistical Society: Series B (Methodological), 46 (1984), 353-376.  doi: 10.1111/j.2517-6161.1984.tb01308.x. [10] N. H. Du and D. H. Nguyen, Dynamics of kolmogorov systems of competitive type under the telegraph noise, Journal of Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023. [11] Y. Dumont and F. Chiroleu, Vector control for the Chikungunya disease, Mathematical Biosciences & Engineering, 7 (2010), 313. doi: 10.3934/mbe.2010.7.313. [12] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The sis epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications, 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [13] Q. He and G. Yin, Large deviations for multi-scale markovian switching systems with a small diffusion, Asymptotic Analysis, 87 (2014), 123-145.  doi: 10.3233/ASY-131198. [14] A. Hening, D. H. Nguyen, et al., Coexistence and extinction for stochastic kolmogorov systems, Annals of Applied Probability, 28 (2018), 1893–1942. doi: 10.1214/17-AAP1347. [15] H. W. Hethcote and P. Van den Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.  doi: 10.1007/BF00160539. [16] N. Hieu, N. Du, P. Auger and D. H. Nguyen, Dynamical behavior of a stochastic sirs epidemic model, Mathematical Modelling of Natural Phenomena, 10 (2015), 56-73.  doi: 10.1051/mmnp/201510205. [17] J. Hui and L. Chen, Impulsive vaccination of sir epidemic models with nonlinear incidence rates, Discrete & Continuous Dynamical Systems-B, 4 (2004), 595. doi: 10.3934/dcdsb.2004.4.595. [18] M. Jacobsen, Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes, Springer Science & Business Media, 2006. [19] M. Liu, X. He and J. Yu, Dynamics of a stochastic regime-switching predator–prey model with harvesting and distributed delays, Nonlinear Analysis: Hybrid Systems, 28 (2018), 87-104.  doi: 10.1016/j.nahs.2017.10.004. [20] Q. Lu, Stability of sirs system with random perturbations, Physica A: Statistical Mechanics and Its Applications, 388 (2009), 3677-3686.  doi: 10.1016/j.physa.2009.05.036. [21] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and Applications, 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [22] P. M. Luz, C. J. Struchiner and A. P. Galvani, Modeling transmission dynamics and control of vector-borne neglected tropical diseases, PLoS Negl. Trop. Dis., 4 (2010), e761. doi: 10.1371/journal.pntd.0000761. [23] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006.  doi: 10.1142/p473. [24] R. M. May and R. M. Anderson, Population biology of infectious diseases: Part Ⅱ, Nature, 280 (1979), 455-461.  doi: 10.1038/280455a0. [25] A. Mishra, B. Ambrosio, S. Gakkhar and M. Aziz-Alaoui, A network model for control of dengue epidemic using sterile insect technique., Math. Biosci. Eng., 15 (2018), 441-460. [26] A. Mishra and S. Gakkhar, The effects of awareness and vector control on two strains dengue dynamics, Applied Mathematics and Computation, 246 (2014), 159-167.  doi: 10.1016/j.amc.2014.07.115. [27] A. Mishra and S. Gakkhar, Non-linear dynamics of two-patch model incorporating secondary dengue infection, International Journal of Applied and Computational Mathematics, 4 (2018), 19. doi: 10.1007/s40819-017-0460-z. [28] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka–Volterra models, Journal of Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [29] W. H. Organization, et al., Global Strategic Framework for Integrated Vector Management, Technical report, World Health Organization, 2004. [30] Y. Shen, Mathematical models of dengue fever and measures to control it, Ph.D thesis, Florida State University in Tallahassee, 2014. [31] C. Sun, W. Yang, J. Arino and K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Mathematical Biosciences, 230 (2011), 87-95.  doi: 10.1016/j.mbs.2011.01.005. [32] H. Yang, H. Wei and X. Li, Global stability of an epidemic model for vector-borne disease, Journal of Systems Science and Complexity, 23 (2010), 279-292.  doi: 10.1007/s11424-010-8436-7.

show all references

##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0. [2] Y. Asai, X. Han and P. E. Kloeden, Dynamics of Zika virus epidemic in random environment, in Mathematics Applied to Engineering, Modelling, and Social Issues, Springer, 2019,665–684. [3] K. Bao, L. Rong and Q. Zhang, Analysis of a stochastic sirs model with interval parameters, Discrete & Continuous Dynamical Systems-B, 24 (2019), 4827. doi: 10.3934/dcdsb.2019033. [4] M. Benaim, Stochastic persistence, preprint, arXiv: 1806.08450. [5] M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Qualitative properties of certain piecewise deterministic Markov processes, Annales de l'IHP Probabilités et Statistiques, 51 (2015), 1040–1075. doi: 10.1214/14-AIHP619. [6] M. Benaïm, E. Strickler, et al., Random switching between vector fields having a common zero, The Annals of Applied Probability, 29 (2019), 326-375. doi: 10.1214/18-AAP1418. [7] D. Bichara, Effects of migration on vector-borne diseases with forward and backward stage progression, preprint, arXiv: 1810.06777. doi: 10.3934/dcdsb.2019140. [8] Z. Cao, X. Liu, X. Wen, L. Liu and L. Zu, A regime-switching sir epidemic model with a ratio-dependent incidence rate and degenerate diffusion, Scientific Reports, 9 (2019), 1-7.  doi: 10.1186/s13662-017-1355-3. [9] M. H. Davis, Piecewise-deterministic markov processes: A general class of non-diffusion stochastic models, Journal of the Royal Statistical Society: Series B (Methodological), 46 (1984), 353-376.  doi: 10.1111/j.2517-6161.1984.tb01308.x. [10] N. H. Du and D. H. Nguyen, Dynamics of kolmogorov systems of competitive type under the telegraph noise, Journal of Differential Equations, 250 (2011), 386-409.  doi: 10.1016/j.jde.2010.08.023. [11] Y. Dumont and F. Chiroleu, Vector control for the Chikungunya disease, Mathematical Biosciences & Engineering, 7 (2010), 313. doi: 10.3934/mbe.2010.7.313. [12] A. Gray, D. Greenhalgh, X. Mao and J. Pan, The sis epidemic model with Markovian switching, Journal of Mathematical Analysis and Applications, 394 (2012), 496-516.  doi: 10.1016/j.jmaa.2012.05.029. [13] Q. He and G. Yin, Large deviations for multi-scale markovian switching systems with a small diffusion, Asymptotic Analysis, 87 (2014), 123-145.  doi: 10.3233/ASY-131198. [14] A. Hening, D. H. Nguyen, et al., Coexistence and extinction for stochastic kolmogorov systems, Annals of Applied Probability, 28 (2018), 1893–1942. doi: 10.1214/17-AAP1347. [15] H. W. Hethcote and P. Van den Driessche, Some epidemiological models with nonlinear incidence, Journal of Mathematical Biology, 29 (1991), 271-287.  doi: 10.1007/BF00160539. [16] N. Hieu, N. Du, P. Auger and D. H. Nguyen, Dynamical behavior of a stochastic sirs epidemic model, Mathematical Modelling of Natural Phenomena, 10 (2015), 56-73.  doi: 10.1051/mmnp/201510205. [17] J. Hui and L. Chen, Impulsive vaccination of sir epidemic models with nonlinear incidence rates, Discrete & Continuous Dynamical Systems-B, 4 (2004), 595. doi: 10.3934/dcdsb.2004.4.595. [18] M. Jacobsen, Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes, Springer Science & Business Media, 2006. [19] M. Liu, X. He and J. Yu, Dynamics of a stochastic regime-switching predator–prey model with harvesting and distributed delays, Nonlinear Analysis: Hybrid Systems, 28 (2018), 87-104.  doi: 10.1016/j.nahs.2017.10.004. [20] Q. Lu, Stability of sirs system with random perturbations, Physica A: Statistical Mechanics and Its Applications, 388 (2009), 3677-3686.  doi: 10.1016/j.physa.2009.05.036. [21] Q. Luo and X. Mao, Stochastic population dynamics under regime switching, Journal of Mathematical Analysis and Applications, 334 (2007), 69-84.  doi: 10.1016/j.jmaa.2006.12.032. [22] P. M. Luz, C. J. Struchiner and A. P. Galvani, Modeling transmission dynamics and control of vector-borne neglected tropical diseases, PLoS Negl. Trop. Dis., 4 (2010), e761. doi: 10.1371/journal.pntd.0000761. [23] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006.  doi: 10.1142/p473. [24] R. M. May and R. M. Anderson, Population biology of infectious diseases: Part Ⅱ, Nature, 280 (1979), 455-461.  doi: 10.1038/280455a0. [25] A. Mishra, B. Ambrosio, S. Gakkhar and M. Aziz-Alaoui, A network model for control of dengue epidemic using sterile insect technique., Math. Biosci. Eng., 15 (2018), 441-460. [26] A. Mishra and S. Gakkhar, The effects of awareness and vector control on two strains dengue dynamics, Applied Mathematics and Computation, 246 (2014), 159-167.  doi: 10.1016/j.amc.2014.07.115. [27] A. Mishra and S. Gakkhar, Non-linear dynamics of two-patch model incorporating secondary dengue infection, International Journal of Applied and Computational Mathematics, 4 (2018), 19. doi: 10.1007/s40819-017-0460-z. [28] D. H. Nguyen and G. Yin, Coexistence and exclusion of stochastic competitive Lotka–Volterra models, Journal of Differential Equations, 262 (2017), 1192-1225.  doi: 10.1016/j.jde.2016.10.005. [29] W. H. Organization, et al., Global Strategic Framework for Integrated Vector Management, Technical report, World Health Organization, 2004. [30] Y. Shen, Mathematical models of dengue fever and measures to control it, Ph.D thesis, Florida State University in Tallahassee, 2014. [31] C. Sun, W. Yang, J. Arino and K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Mathematical Biosciences, 230 (2011), 87-95.  doi: 10.1016/j.mbs.2011.01.005. [32] H. Yang, H. Wei and X. Li, Global stability of an epidemic model for vector-borne disease, Journal of Systems Science and Complexity, 23 (2010), 279-292.  doi: 10.1007/s11424-010-8436-7.
Sample paths of $I_H(t)$ (Example 4.1). In the deterministic systems (LEFT) there is persistence in state 1 and extinction in state 2. In the switched system (RIGHT), the infection persists
Sample paths of $I_H(t)$ (Example 4.2). In both deterministic systems (LEFT), $I_H(t)$ converges exponentially fast to 0. Switching makes the disease persist (RIGHT)
Joint density of $(S_H(t),I_H(t),\xi_t)$ in state 1 (LEFT) and state 2 (RIGHT), according to the invariant measure (Example 4.2)
Sample paths of $I_H(t)$ (Example 4.3). In both deterministic systems, $I_H(t)$ converges to a positive equilibrium (LEFT). Switching allows for extinction (RIGHT)
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