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Analysis of stochastic vector-host epidemic model with direct transmission
1. | Department of Mathematics & Statistics, Auburn University, Auburn, AL 36849 |
2. | Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, United States |
References:
[1] |
M. Aguiar, N. Stollenwerk and B. W. Kooi, Modeling Infectious Diseases Dynamics: Dengue Fever, a Case Study,Epidemiology Insights,, ISBN: 978-953-51-0565-7, (): 978.
|
[2] |
M. Andraud, N. Hens, C. Marais and P. Beutels, Dynamic epidemiological models for dengue transmission: A systematic review of structural approaches, PLoS One, 7 (2012), e49085.
doi: 10.1371/journal.pone.0049085. |
[3] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, A Wiley-Interscience Publication, 1971. |
[4] |
J. R. Beddington and R. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.
doi: 10.1126/science.197.4302.463. |
[5] |
K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of west nile virus, Bull. Math. Biol., 72 (2010), 1006-1028.
doi: 10.1007/s11538-009-9480-0. |
[6] |
K. W. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 587-611.
doi: 10.3934/dcdsb.2009.11.587. |
[7] |
Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic dynamics of an sirs epidemic model with ratio-dependent incidence rate, Abstract and Applied Analysis, 2013 (2013), Article ID 172631, 11pp. |
[8] |
L. Cai and Xuezhi Li, Analysis of a simple vector-host epidemic model with direct transmission, Discrete Dynamics in Nature and Society, (2010), Article ID 679613, 12pp.
doi: 10.1155/2010/679613. |
[9] |
N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53.
doi: 10.1016/j.jmaa.2006.01.055. |
[10] |
L. C. Evans, An Introduction to Stochastic Differential Equations, University of California, Berkeley, CA, 2013.
doi: 10.1090/mbk/082. |
[11] |
A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation sis epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.
doi: 10.1137/10081856X. |
[12] |
D. O, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, Math. Biol., 35 (1990), 503-522. |
[13] |
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.
doi: 10.1137/S0036144500378302. |
[14] | |
[15] |
, , ().
|
[16] |
, , ().
|
[17] |
, , ().
|
[18] |
M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission, Applied Mathematical Modelling, 36 (2012), 5214-5228.
doi: 10.1016/j.apm.2011.11.087. |
[19] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[20] |
X. Ling, Modeling and Analysis of Vector-borne Diseases on Complex Networks, PhD Thesis, Kansas State University, 2013. |
[21] |
S. Mandal, R. R. Sarkar and S. Sinha, Mathematical Models Of Malaria - A Review, Malar J., 10, 2011, 202. |
[22] |
X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, second edition, January 13, 2008.
doi: 10.1533/9780857099402. |
[23] |
M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61. Springer, New York, 2015.
doi: 10.1007/978-1-4899-7612-3. |
[24] |
F. E. Mckenzie, Why Model Malaria?, Parasitology Today, 16 (2000), 511-516.
doi: 10.1016/S0169-4758(00)01789-0. |
[25] |
G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32 (2000), 747-763.
doi: 10.1016/S0895-7177(00)00169-2. |
[26] |
K. Okosunl and O. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate, Appl. Comput. Math., 12 (2013), 20-32. |
[27] |
Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Computers and Mathematics with Applications, 56 (2008), 3118-3129.
doi: 10.1016/j.camwa.2008.09.002. |
[28] |
K. Rafail, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012.
doi: 10.1007/978-3-642-23280-0. |
[29] |
R. Rosenberg and C. B. Beard, Vector-borne infections, CDCEID journal, 17 (2011), 2pp.
doi: 10.3201/eid1705.110310. |
[30] |
M. Samsuzzoha, M. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915.
doi: 10.1016/j.apm.2012.03.029. |
[31] |
Smallpox: Disease, Prevention, and Intervention., The CDC and the World Health Organization,, History and Epidemiology of Global Smallpox Eradication From the training course, (): 16.
|
[32] |
M. O. Souza, Multiscale analysis for a vector-borne epidemic model, J. Math. Biol., 68 (2014), 1269-1293.
doi: 10.1007/s00285-013-0666-6. |
[33] |
D. R. Stirzaker, A perturbation method for the stochastic recurrent epidemic, EpidemicIMA J Appl Math., 15 (1975), 135-160.
doi: 10.1093/imamat/15.2.135-a. |
[34] |
N. Stollenwerk, M. Aguiar, S. Ballesteros, J. Boto, B. Kooi and L. Mateus, Dynamic Noise, Chaos and Parameter Estimation in Population Biology, Interface Focus, 2012.
doi: 10.1098/rsfs.2011.0103. |
[35] |
J. E. Truscott and C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment, PNAS, 100 ( 2003), 9067-9072.
doi: 10.1073/pnas.1436273100. |
[36] |
Q. Wei, Z. Xiong and F. Wang, Dynamic of a Stochastic SIR Model Under Regime Switching, Journal of Information & Computational Science, 10 (2013), 2727-2734.
doi: 10.12733/jics20101856. |
[37] |
H. M. Wei, X. Z. Li and M. Martcheva, An epidemic model of a vector-borne disease with direct transmission and time delay, Journal of Mathematical Analysis and Applications, 342 (2008), 895-908.
doi: 10.1016/j.jmaa.2007.12.058. |
[38] |
M. J. Wonham and M. A. Lewis, A Comparative Analysis of Models for West Nile Virus, Mathematical Epidemiology Lecture Notes in Mathematics, 1945 (2008), 365-390.
doi: 10.1007/978-3-540-78911-6_14. |
[39] |
H. Yang, H. Wei and X. Li, Global stability of an epidemic model for vector-borne disease, J Syst Sci Complex Journal, 23 (2010), 279-292.
doi: 10.1007/s11424-010-8436-7. |
[40] |
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Control Optim, 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
[41] |
L. Zu, D. Jiang and D. O'Regan, Stochastic Permanence, Stationary Distribution and Extinction of a Single-Species Nonlinear Diffusion System withRandom Perturbation, Abstract and Applied Analysis, Article ID 320460, 2014.
doi: 10.1155/2014/320460. |
show all references
References:
[1] |
M. Aguiar, N. Stollenwerk and B. W. Kooi, Modeling Infectious Diseases Dynamics: Dengue Fever, a Case Study,Epidemiology Insights,, ISBN: 978-953-51-0565-7, (): 978.
|
[2] |
M. Andraud, N. Hens, C. Marais and P. Beutels, Dynamic epidemiological models for dengue transmission: A systematic review of structural approaches, PLoS One, 7 (2012), e49085.
doi: 10.1371/journal.pone.0049085. |
[3] |
L. Arnold, Stochastic Differential Equations: Theory and Applications, A Wiley-Interscience Publication, 1971. |
[4] |
J. R. Beddington and R. May, Harvesting natural populations in a randomly fluctuating environment, Science, 197 (1977), 463-465.
doi: 10.1126/science.197.4302.463. |
[5] |
K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of west nile virus, Bull. Math. Biol., 72 (2010), 1006-1028.
doi: 10.1007/s11538-009-9480-0. |
[6] |
K. W. Blayneh, Y. Cao and H.-D. Kwon, Optimal control of vector-borne diseases: Treatment and prevention, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 587-611.
doi: 10.3934/dcdsb.2009.11.587. |
[7] |
Y. Cai, X. Wang, W. Wang and M. Zhao, Stochastic dynamics of an sirs epidemic model with ratio-dependent incidence rate, Abstract and Applied Analysis, 2013 (2013), Article ID 172631, 11pp. |
[8] |
L. Cai and Xuezhi Li, Analysis of a simple vector-host epidemic model with direct transmission, Discrete Dynamics in Nature and Society, (2010), Article ID 679613, 12pp.
doi: 10.1155/2010/679613. |
[9] |
N. Dalal, D. Greenhalgh and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325 (2007), 36-53.
doi: 10.1016/j.jmaa.2006.01.055. |
[10] |
L. C. Evans, An Introduction to Stochastic Differential Equations, University of California, Berkeley, CA, 2013.
doi: 10.1090/mbk/082. |
[11] |
A. Gray, D. Greenhalgh, L. Hu, X. Mao and J. Pan, A stochastic differential equation sis epidemic model, SIAM J. Appl. Math., 71 (2011), 876-902.
doi: 10.1137/10081856X. |
[12] |
D. O, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcalR_0$ in models for infectious diseases, Math. Biol., 35 (1990), 503-522. |
[13] |
D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546.
doi: 10.1137/S0036144500378302. |
[14] | |
[15] |
, , ().
|
[16] |
, , ().
|
[17] |
, , ().
|
[18] |
M. Jovanovic and M. Krstic, Stochastically perturbed vector-borne disease models with direct transmission, Applied Mathematical Modelling, 36 (2012), 5214-5228.
doi: 10.1016/j.apm.2011.11.087. |
[19] |
P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag Berlin Heidelberg, 1992.
doi: 10.1007/978-3-662-12616-5. |
[20] |
X. Ling, Modeling and Analysis of Vector-borne Diseases on Complex Networks, PhD Thesis, Kansas State University, 2013. |
[21] |
S. Mandal, R. R. Sarkar and S. Sinha, Mathematical Models Of Malaria - A Review, Malar J., 10, 2011, 202. |
[22] |
X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, second edition, January 13, 2008.
doi: 10.1533/9780857099402. |
[23] |
M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics, 61. Springer, New York, 2015.
doi: 10.1007/978-1-4899-7612-3. |
[24] |
F. E. Mckenzie, Why Model Malaria?, Parasitology Today, 16 (2000), 511-516.
doi: 10.1016/S0169-4758(00)01789-0. |
[25] |
G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32 (2000), 747-763.
doi: 10.1016/S0895-7177(00)00169-2. |
[26] |
K. Okosunl and O. Makinde, Optimal control analysis of malaria in the presence of non-linear incidence rate, Appl. Comput. Math., 12 (2013), 20-32. |
[27] |
Z. Qiu, Dynamical behavior of a vector-host epidemic model with demographic structure, Computers and Mathematics with Applications, 56 (2008), 3118-3129.
doi: 10.1016/j.camwa.2008.09.002. |
[28] |
K. Rafail, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012.
doi: 10.1007/978-3-642-23280-0. |
[29] |
R. Rosenberg and C. B. Beard, Vector-borne infections, CDCEID journal, 17 (2011), 2pp.
doi: 10.3201/eid1705.110310. |
[30] |
M. Samsuzzoha, M. Singh and D. Lucy, Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza, Applied Mathematical Modelling, 37 (2013), 903-915.
doi: 10.1016/j.apm.2012.03.029. |
[31] |
Smallpox: Disease, Prevention, and Intervention., The CDC and the World Health Organization,, History and Epidemiology of Global Smallpox Eradication From the training course, (): 16.
|
[32] |
M. O. Souza, Multiscale analysis for a vector-borne epidemic model, J. Math. Biol., 68 (2014), 1269-1293.
doi: 10.1007/s00285-013-0666-6. |
[33] |
D. R. Stirzaker, A perturbation method for the stochastic recurrent epidemic, EpidemicIMA J Appl Math., 15 (1975), 135-160.
doi: 10.1093/imamat/15.2.135-a. |
[34] |
N. Stollenwerk, M. Aguiar, S. Ballesteros, J. Boto, B. Kooi and L. Mateus, Dynamic Noise, Chaos and Parameter Estimation in Population Biology, Interface Focus, 2012.
doi: 10.1098/rsfs.2011.0103. |
[35] |
J. E. Truscott and C. A. Gilligan, Response of a deterministic epidemiological system to a stochastically varying environment, PNAS, 100 ( 2003), 9067-9072.
doi: 10.1073/pnas.1436273100. |
[36] |
Q. Wei, Z. Xiong and F. Wang, Dynamic of a Stochastic SIR Model Under Regime Switching, Journal of Information & Computational Science, 10 (2013), 2727-2734.
doi: 10.12733/jics20101856. |
[37] |
H. M. Wei, X. Z. Li and M. Martcheva, An epidemic model of a vector-borne disease with direct transmission and time delay, Journal of Mathematical Analysis and Applications, 342 (2008), 895-908.
doi: 10.1016/j.jmaa.2007.12.058. |
[38] |
M. J. Wonham and M. A. Lewis, A Comparative Analysis of Models for West Nile Virus, Mathematical Epidemiology Lecture Notes in Mathematics, 1945 (2008), 365-390.
doi: 10.1007/978-3-540-78911-6_14. |
[39] |
H. Yang, H. Wei and X. Li, Global stability of an epidemic model for vector-borne disease, J Syst Sci Complex Journal, 23 (2010), 279-292.
doi: 10.1007/s11424-010-8436-7. |
[40] |
C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, Control Optim, 46 (2007), 1155-1179.
doi: 10.1137/060649343. |
[41] |
L. Zu, D. Jiang and D. O'Regan, Stochastic Permanence, Stationary Distribution and Extinction of a Single-Species Nonlinear Diffusion System withRandom Perturbation, Abstract and Applied Analysis, Article ID 320460, 2014.
doi: 10.1155/2014/320460. |
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