• Previous Article
    BMO type space associated with Neumann operator and application to a class of parabolic equations
  • DCDS-B Home
  • This Issue
  • Next Article
    Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production
doi: 10.3934/dcdsb.2021029
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Benaim, Stochastic persistence, preprint, arXiv: 1806.08450. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Z. CaoX. LiuX. WenL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. GrayD. GreenhalghX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

N. HieuN. DuP. 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.  Google Scholar

[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.  Google Scholar

[18]

M. Jacobsen, Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes, Springer Science & Business Media, 2006.  Google Scholar

[19]

M. LiuX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006.  doi: 10.1142/p473.  Google Scholar
[24]

R. M. May and R. M. Anderson, Population biology of infectious diseases: Part Ⅱ, Nature, 280 (1979), 455-461.  doi: 10.1038/280455a0.  Google Scholar

[25]

A. MishraB. AmbrosioS. Gakkhar and M. Aziz-Alaoui, A network model for control of dengue epidemic using sterile insect technique., Math. Biosci. Eng., 15 (2018), 441-460.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

W. H. Organization, et al., Global Strategic Framework for Integrated Vector Management, Technical report, World Health Organization, 2004. Google Scholar

[30]

Y. Shen, Mathematical models of dengue fever and measures to control it, Ph.D thesis, Florida State University in Tallahassee, 2014. Google Scholar

[31]

C. SunW. YangJ. 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.  Google Scholar

[32]

H. YangH. 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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

M. Benaim, Stochastic persistence, preprint, arXiv: 1806.08450. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

Z. CaoX. LiuX. WenL. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. GrayD. GreenhalghX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

N. HieuN. DuP. 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.  Google Scholar

[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.  Google Scholar

[18]

M. Jacobsen, Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes, Springer Science & Business Media, 2006.  Google Scholar

[19]

M. LiuX. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial college press, 2006.  doi: 10.1142/p473.  Google Scholar
[24]

R. M. May and R. M. Anderson, Population biology of infectious diseases: Part Ⅱ, Nature, 280 (1979), 455-461.  doi: 10.1038/280455a0.  Google Scholar

[25]

A. MishraB. AmbrosioS. Gakkhar and M. Aziz-Alaoui, A network model for control of dengue epidemic using sterile insect technique., Math. Biosci. Eng., 15 (2018), 441-460.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

W. H. Organization, et al., Global Strategic Framework for Integrated Vector Management, Technical report, World Health Organization, 2004. Google Scholar

[30]

Y. Shen, Mathematical models of dengue fever and measures to control it, Ph.D thesis, Florida State University in Tallahassee, 2014. Google Scholar

[31]

C. SunW. YangJ. 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.  Google Scholar

[32]

H. YangH. 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.  Google Scholar

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
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)
[1]

Qingkai Kong, Zhipeng Qiu, Zi Sang, Yun Zou. Optimal control of a vector-host epidemics model. Mathematical Control & Related Fields, 2011, 1 (4) : 493-508. doi: 10.3934/mcrf.2011.1.493

[2]

Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060

[3]

Yanzhao Cao, Dawit Denu. Analysis of stochastic vector-host epidemic model with direct transmission. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2109-2127. doi: 10.3934/dcdsb.2016039

[4]

Yanxia Dang, Zhipeng Qiu, Xuezhi Li. Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (4) : 901-931. doi: 10.3934/mbe.2017048

[5]

Shangzhi Li, Shangjiang Guo. Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lévy jumps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 5101-5134. doi: 10.3934/dcdsb.2020335

[6]

Tzy-Wei Hwang, Yang Kuang. Host Extinction Dynamics in a Simple Parasite-Host Interaction Model. Mathematical Biosciences & Engineering, 2005, 2 (4) : 743-751. doi: 10.3934/mbe.2005.2.743

[7]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041

[8]

Kaifa Wang, Yang Kuang. Fluctuation and extinction dynamics in host-microparasite systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1537-1548. doi: 10.3934/cpaa.2011.10.1537

[9]

Sebastian J. Schreiber. On persistence and extinction for randomly perturbed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 457-463. doi: 10.3934/dcdsb.2007.7.457

[10]

Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124

[11]

Xia Wang, Yuming Chen. An age-structured vector-borne disease model with horizontal transmission in the host. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1099-1116. doi: 10.3934/mbe.2018049

[12]

Daniel Franco, Chris Guiver, Phoebe Smith, Stuart Townley. A switching feedback control approach for persistence of managed resources. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021109

[13]

Wen Jin, Horst R. Thieme. Persistence and extinction of diffusing populations with two sexes and short reproductive season. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3209-3218. doi: 10.3934/dcdsb.2014.19.3209

[14]

Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447

[15]

Suqi Ma. Low viral persistence of an immunological model. Mathematical Biosciences & Engineering, 2012, 9 (4) : 809-817. doi: 10.3934/mbe.2012.9.809

[16]

Naveen K. Vaidya, Feng-Bin Wang. Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021048

[17]

Miaoran Yao, Yongxin Zhang, Wendi Wang. Bifurcation analysis for an in-host Mycobacterium tuberculosis model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2299-2322. doi: 10.3934/dcdsb.2020324

[18]

Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565

[19]

S. R.-J. Jang. Allee effects in a discrete-time host-parasitoid model with stage structure in the host. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 145-159. doi: 10.3934/dcdsb.2007.8.145

[20]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]