| Citation: |
Divine Wanduku. ESTIMATING WHITE NOISE INTENSITY REGIONS FOR COMPARABLE PROPERTIES OF A CLASS OF SEIRS STOCHASTIC AND DETERMINISTIC EPIDEMIC MODELS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1095-1037. doi: 10.11948/20190372
|
ESTIMATING WHITE NOISE INTENSITY REGIONS FOR COMPARABLE PROPERTIES OF A CLASS OF SEIRS STOCHASTIC AND DETERMINISTIC EPIDEMIC MODELS
-
Abstract
A comparative stochastic and deterministic study of a family of epidemic models for vector-borne diseases e.g. malaria and dengue fever etc. is presented. The family type is determined by a general nonlinear incidence rate of the disease. Two major sources of environmental white noises are considered: disease transmission and natural death rates. The impacts of each source of noise on the disease dynamics are examined. The basic reproduction numbers and other threshold values for the disease in the stochastic and deterministic settings are determined and compared to determine the impacts of the noises on the dynamics. The question about the extend that stability conditions for steady states in the noise-free disease dynamics, remain valid for the stochastic stability of the steady state is answered in this paper. Moreover, noise intensity regions are computed, within which all stability conditions for both systems are the same, and both systems behave similarly.
-
-
References
[1] E. J. Allen, L. J. S. Allen, A. Arciniega and P. Greenwood, Construction of equivalent stochastic differential equation models, Stochastic Analysis and Applications, 2008, 26, 274-297. doi: 10.1080/07362990701857129 [2] R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991. [3] E. Avila and V. Buonomo, Analysis of a mosquito-borne disease transmission model with vector stages and nonlinear forces of infection, Ricerche di Matematica., 2015, 64(2), 377-390. doi: 10.1007/s11587-015-0245-9 [4] Z. Bai and Y. Zhou, Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate, Nonlinear Analysis: Real World Applications, 2012, 13(3), 1060-1068. doi: 10.1016/j.nonrwa.2011.02.008 [5] E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delay influenced by stochastic perturbations, Mathematics and Computers in Simulation, 1998, 45, 269-277. doi: 10.1016/S0378-4754(97)00106-7 [6] M. T. Bretscher, Nicolas Maire, Ingrid Felger, Seth Owusu-Agyei and T. Smit, Asymptomatic Plasmodium falciparum infections may not be shortened by acquired immunity, Malaria Journal, 2015, 14, 294, DOI:10.1186/s12936-015-0813-1. [7] Y. Cai, J. jiao, Z. Gui, Y. liu et al., Environmental variability in a stochastic epidemic model, Applied mathematics and compuation, 2018, 329, 210-226. doi: 10.1016/j.amc.2018.02.009 [8] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, volume 97, 1993. [9] V. Capasso and G. A. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Mathematical Biosciences, 1978, 42(1-2), 43-61. doi: 10.1016/0025-5564(78)90006-8 [10] C. Chen and Y. Kang, Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence, Abstract and Applied Analysis, 2014, 2014(723825), 13 pages. [11] L. Chen and J. Chen, Nonlinear Biologiical Dynamical System, Science Press, Beijing, 1993. [12] N. Chitnis, J. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of Mathematical Biology, 2008, 70(5), 1272-96. doi: 10.1007/s11538-008-9299-0 [13] R. Cont and D. A. Fournie, Functional It$\hat{o}$ calculus and stochastic integral representation of martingales, Annals of Probability, 2013, 41, 109-133. [14] K. L. Cooke, Stability analysis for a vector disease model, Rocky Mountain Journal of Mathematics, 1979, 9(1), 31-42. [15] K. L. Cooke and P. van den Driessche, Analysis of an SEIRS epidemic model with two delays, Journal of Mathematical Biology, 1996, 35(2), 240-60. doi: 10.1007/s002850050051 [16] J. M. Crutcher, S. L. Hoffman and Malaria, Chapter 83-malaria, Medical Microbiology, 4th edition, Galveston (TX): University of Texas Medical Branch at Galveston, 1996. [17] D. L. Doolan, C. Dobano and J. K. Baird, Acquired Immunity to Malaria, clinical microbiology reviews, 2009, 22(1), 13-36. doi: 10.1128/CMR.00025-08 [18] N. Du and N. N. Nhu, Permanence and extinction of certain stochastic SIR models perturbed by a complex type of noises, Applied Mathematics Letters, 2017, 64, 223-230. doi: 10.1016/j.aml.2016.09.012 [19] N. Dieu, N. Du, D. Nguyen and G. Yin, Classification of asymptotic behavior in a stochastic SIR model, SIAM Journal of Applied Dynamical Systems, 2016, 15, 1062-1084. doi: 10.1137/15M1043315 [20] S. Gao, Z. Teng and D. Xie, The effects of pulse vaccination on SEIR model with two time delays, Applied Mathematics and Computation, 2008, 201(1-2), 282-292. doi: 10.1016/j.amc.2007.12.019 [21] H. Hai and M. Zhan, Dynamics of a delayed epidemic model with non-monotonic incidence rate, Communications in Nonlinear Science and Numerical Simulation, 2010, 15(2), 459-468. doi: 10.1016/j.cnsns.2009.04.018 [22] R. E. Howes, K. E. Battle, K. N. Mendis, et al., Global Epidemiology of Plasmodium vivax, The American Journal of Tropical Medicine and Hygiene, 2016, 95(6), 15-34, DOI:10.4269/ajtmh.16-0141. [23] L. Hviid, Naturally acquired immunity to Plasmodium falciparum malaria, Acta Tropica, 2005, 95(3), 270-5. doi: 10.1016/j.actatropica.2005.06.012 [24] M. Y. Hyun, Malaria transmission model for different levels of acquired immunity and temperature dependent parameters (vector), Revista. Saude Publica, 2000, 34(3), 223-231. doi: 10.1590/S0034-89102000000300003 [25] Z. Jianga, B. Wanbiao Mab and J. Wei, Global Hopf bifurcation and permanence of a delayed SEIRS epidemic model, Mathematics and Computers in Simulation, 2016, 122, 35-54. doi: 10.1016/j.matcom.2015.11.002 [26] A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Mathematical Biosciences and Engineering, 2004, 1(1), 57-60. doi: 10.3934/mbe.2004.1.57 [27] M. Krstic, The effect of stochastic perturbation on a nonlinear delay malaria epidemic model, Mathematics and Computers in Simulations, 2011, 82, 558-569. doi: 10.1016/j.matcom.2011.09.003 [28] Y. Kuang, Delay Differential Equations with Applications in population Dynamics, Academic Press, Boston, 1993. [29] Y. N. Kyrychko and K. B. Blyussb, Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear Analysis: Real World Applications, 2005, 6(30), 495-507. [30] G. S. Ladde, Cellular Systems-Ⅱ. Stability of Campartmental Systems, Mathematical Biosciences, 1976, 30, 1-21. doi: 10.1016/0025-5564(76)90013-4 [31] G. S. Ladde and V. Lakshmikantham, Random Differential Inequalities, Academic press, New York, 1980. [32] P. V. V. Le, P. Kumar and M. O. Ruiz, Stochastic lattice-based modelling of malaria dynamics, Malaria Journal, 2018, 17, 250. doi: 10.1186/s12936-018-2397-z [33] W. Liu, H. W. Hethcote and S. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, Journal of mathematical Biology, 1987, 25, 359-380. doi: 10.1007/BF00277162 [34] W. Liu, H. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, Journal of mathematical Biology, 1987, 25(4), 359-380. doi: 10.1007/BF00277162 [35] Q. Liu, D. Jiang, N. Shi, T. Hayat and A. Alsaedi, Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence, Communications in Nonlinear Science and Numerical Simulation, 2016, 40, 89-99. doi: 10.1016/j.cnsns.2016.04.023 [36] Q. Liu and Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Physica A, 2015, 428, 140-153. doi: 10.1016/j.physa.2015.01.075 [37] J. P. Mateusa and C. M. Silvab, Existence of periodic solutions of a periodic SEIRS model with general incidence, Nonlinear Analysis: Real World Applications, 2017, 34, 379-402. doi: 10.1016/j.nonrwa.2016.09.013 [38] J. P. Mateus and C. M. Silva, A non-autonomous SEIRS model with general incidence rate, Applied Mathematics and Computation, 2014, 247, 169-189. doi: 10.1016/j.amc.2014.08.078 [39] G. Macdonald, The analysis of infection rates in diseases in which superinfection occurs, Tropical diseases bulletine, 1961, 47, 907-915. [40] C. McCluskey, Global Stability of an SIR epidemic model with delay and general nonlinear incidence, Mathematical biosciences and engineering, 2010, 7(4), 837-850. doi: 10.3934/mbe.2010.7.837 [41] S. M. Moghadas and A. B. Gumel, Global Statbility of a two-stage epidemic model with generalized nonlinear incidence, Mathematics and computers in simulation, 2002, 60, 107-118. doi: 10.1016/S0378-4754(02)00002-2 [42] Y. Muroya, Y. Enatsu and Y. Nakata, Global stability of a delayed SIRS epidemic model with a non-monotonic incidence rate, Journal of Mathematical Analysis and Applications, 2011, 377(1), 1-14. doi: 10.1016/j.jmaa.2010.10.010 [43] C. N. Ngonghala, G. A. Ngwa and M. I. Teboh-Ewungkem, Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission, Mathematical Biosciences, 2012, 240(1), 45-62. doi: 10.1016/j.mbs.2012.06.003 [44] G. A. Ngwa and W. Shu, A mathematical model for endemic malaria with variable human and mosquito population, Mathematical and Computer Modelling, 2000, 32, 747-763. doi: 10.1016/S0895-7177(00)00169-2 [45] G. A. Ngwa, A. M. Niger and A. B. Gumel, Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector, Applied Mathematics and Computation, 2010, 74, 1351-1395. [46] D. Nguyen, N. Nguyen and G. Yin, Analysis of a spatially inhomogeneous stochastic partial differential equation epidemic model, Journal of Applied Probability, 2020, 57, 613-636. doi: 10.1017/jpr.2020.15 [47] L. Pang, S. Ruan, S. Liu, Z. Zhao and X. Zhang, Transmission dynamics and optimal control of measles epidemics, Applied Mathematics and Computation, 2015, 256, 131-147. doi: 10.1016/j.amc.2014.12.096 [48] S. Ruan, D. Xiao and J. C. Beier, On the delayed ross-macdonald model for malaria transmission, Bulletin of Mathematical Biology, 2008, 70(4), 1098-1114. doi: 10.1007/s11538-007-9292-z [49] R. Reiner Jr, M. Geary, et al., Seasonality of Plasmodium falciparum transmission: a systematic review, Malaria Journal, 2015, 14, 343, DOI:10.1186/s12936-015-0849-2. [50] R. Ross, The Prevention of Malaria, John Murray, London, 1911. [51] M. De la Sena, S. Alonso-Quesadaa and A. Ibeasb, On the stability of an SEIR epidemic model with distributed time-delay and a general class of feedback vaccination rules, Applied Mathematics and Computation, 2015, 270(1), 953-976. [52] B. G. Sampath Aruna Pradeep and W. Ma, Global Stability Analysis for Vector Transmission Disease Dynamic Model with Non-linear Incidence and Two Time Delays, Journal of Interdisciplinary Mathematics, 2015, 18(40), 395-415. [53] S. Syafruddin, M. Salmi and M. Noorani, Lyapunov function of SIR and SEIR model for transmission of dengue fever disease, International Journal of Simulation and Process Modelling, 2013, 8(2/3), 177-184, DOI: 10.1504/IJSPM.2013.057544. [54] Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Analysis, 2000, 42, 931-947. doi: 10.1016/S0362-546X(99)00138-8 [55] M. I. Teboh-Ewungkem and T. Yuster, A within-vector mathematical model of plasmodium falciparum and implications of incomplete fertilization on optimal gametocyte sex ratio, Journal of Theoretical biology, 2010, 264(2), 273-286, DOI:10.1016/j.jtbi.2009.12.017. [56] D. Wanduku, Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment, International Journal of Biomathematics, 2018, 11(6), 1850085(46 pages), DOI: 10.1142/S1793524518500857. [57] D. Wanduku, Modeling Highly Random Dynamical Infectious Systems. In: Dutta H., Peters J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham, 2020. Available from https://doi.org/10.1007/978-3-319-99918-0_17. [58] D. Wanduku, Analyzing the qualitative properties of white noise on a family of infectious disease models in a highly random environment, available at arXiv: 1808.09842[q-bio. PE]. [59] D. Wanduku, Complete Global Analysis of a Two-Scale Network SIRS Epidemic Dynamic Model with Distributed Delay and Random Perturbation, Applied Mathematics and Computation, 2017, 294, 49-76. doi: 10.1016/j.amc.2016.09.001 [60] D. Wanduku and G. S. Ladde, Global properties of a two-scale network stochastic delayed human epidemic dynamic model, Nonlinear Analysis: Real World Applications, 13(2012), 794-816. doi: 10.1016/j.nonrwa.2011.08.017 [61] D. Wanduku and G. S. Ladde, The global analysis of a stochastic two-scale Network Human Epidemic Dynamic Model With Varying Immunity Period, Journal of Applied Mathematics and Physics, 2017, 5, 1150-1173. doi: 10.4236/jamp.2017.55101 [62] D. Wanduku and G. S. Ladde, Global Stability of Two-Scale Network Human Epidemic Dynamic Model, Neural, Parallel, and Scientific Computations, 2011, 19, 65-90. [63] D. Wanduku and G. S. Ladde, Fundamental Properties of a Two-scale Network stochastic human epidemic Dynamic model, Neural, Parallel, and Scientific Computations, 2011, 19, 229-270. [64] D. Wanduku and G. S. Ladde, Global stability of a two-scale network SIR delayed epidemic dynamic model, Proceedings of Dynamic Systems and Applications, 2012, 6, 437-441. [65] D. Wanduku, Two-Scale Network Epidemic Dynamic Model for Vector Borne Diseases, Proceedings of Dynamic Systems and Applications, 2016, 6, 228-232. [66] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 2007, 208(2), 419-29. doi: 10.1016/j.mbs.2006.09.025 [67] R. Xue, Stochastic differential equations and applications, Horwood Publishing Ltd., 2nd ed., 2008. [68] Y. Xue and X. Duan, Dynamic Analysis Of An Sir Epidemic Model With Nonlinear Incidence Rate And Double Delays, International Journal Of Information And Systems Sciences, 2011, 7(1), 92-102. [69] Y. Zhou, W. Zhang, S. Yuan, H. Hu, Persistence And Extinction In Stochastic Sirs Models With General Nonlinear Incidence Rate, Electronic Journal of Differential Equations, 2014, 2014(42), 1-17. [70] L. Zhu, H. Hu, A stochastic SIR epidemic model with density dependent birth rate, Advances in Differential Equations, 2015, 2015(330), https://doi.org/10.1186/s13662-015-0669-2. [71] WHO, Dengue and severe dengue, 2020. Available from http://www.who.int/denguecontrol/human/en/. [72] CDC, Malaria, About malaria, Disease, 2020. Available from https://www.cdc.gov/malaria/about/disease.html. [73] CDC, dengue, About Dengue: What You Need to Know, 2020. Available from https://www.cdc.gov/dengue/about/index.html. -
-
Export File
Citation
Divine Wanduku. ESTIMATING WHITE NOISE INTENSITY REGIONS FOR COMPARABLE PROPERTIES OF A CLASS OF SEIRS STOCHASTIC AND DETERMINISTIC EPIDEMIC MODELS[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1095-1037. doi: 10.11948/20190372
DownLoad: