Figure 1.
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
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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 |
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.
Mathematics Subject Classification:
34C12, 60H10, 92D25.
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:
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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. |
Figure 2.
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)
Figure 3.
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)
Figure 4.
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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