Dynamical behavior of a coupling SEIR epidemic model with transmission in body and vitro, incubation and environmental effects

Abulajiang Aili; Zhidong Teng; Long Zhang; Abulajiang Aili; Zhidong Teng; Long Zhang

doi:10.3934/mbe.2023023

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 1: 505-533. doi: 10.3934/mbe.2023023

Previous Article Next Article

Research article

Dynamical behavior of a coupling SEIR epidemic model with transmission in body and vitro, incubation and environmental effects

1.
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, Xinjiang, China
2.
College of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830017, Xinjiang, China

Received: 30 December 2021 Revised: 18 July 2022 Accepted: 29 August 2022 Published: 11 October 2022

In this paper, a coupling SEIR epidemic model is proposed to characterize the interaction of virus spread in the body of hosts and between hosts with environmentally-driven infection, humoral immunity and incubation of disease. The threshold criteria on the local (or global) stability of feasible equilibria with or without antibody response are established. The basic reproduction number $ R_{b0} $ is obtained for the SEIR model without an antibody response, by which we find that the disease-free equilibrium is locally asymptotically stable if $ R_{b0} < 1 $. Two endemic equilibria exist if $ R_{b0} < 1 $, in which one is locally asymptotically stable under some additional conditions but the other is unstable, which means there is backward bifurcation. In addition, the uniform persistence of this model is discussed. For the SEIR model with an antibody response, the basic reproduction number $ R_{0} $ is calculated, from which the disease-free equilibrium is globally asymptotically stable if $ R_0\leq1 $, and the unique endemic equilibrium is globally asymptotically stable if $ R_0 > 1 $. Antibody immunity in the host plays a great role in the control of disease transmission, especially when the diseases between the hosts are entirely extinct once antibody cells in the host reach a proper level. Finally, the main conclusions are illustrated by some special examples and numerical simulations.
- virus infection model,
- humoral immunity,
- antibody response,
- incubation,
- environmental effect,
- stability
Citation: Abulajiang Aili, Zhidong Teng, Long Zhang. Dynamical behavior of a coupling SEIR epidemic model with transmission in body and vitro, incubation and environmental effects[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 505-533. doi: 10.3934/mbe.2023023

Related Papers:

Abstract

In this paper, a coupling SEIR epidemic model is proposed to characterize the interaction of virus spread in the body of hosts and between hosts with environmentally-driven infection, humoral immunity and incubation of disease. The threshold criteria on the local (or global) stability of feasible equilibria with or without antibody response are established. The basic reproduction number $ R_{b0} $ is obtained for the SEIR model without an antibody response, by which we find that the disease-free equilibrium is locally asymptotically stable if $ R_{b0} < 1 $. Two endemic equilibria exist if $ R_{b0} < 1 $, in which one is locally asymptotically stable under some additional conditions but the other is unstable, which means there is backward bifurcation. In addition, the uniform persistence of this model is discussed. For the SEIR model with an antibody response, the basic reproduction number $ R_{0} $ is calculated, from which the disease-free equilibrium is globally asymptotically stable if $ R_0\leq1 $, and the unique endemic equilibrium is globally asymptotically stable if $ R_0 > 1 $. Antibody immunity in the host plays a great role in the control of disease transmission, especially when the diseases between the hosts are entirely extinct once antibody cells in the host reach a proper level. Finally, the main conclusions are illustrated by some special examples and numerical simulations.

References

[1]	A. Roumen, B. Jacek, B. Chelsea, L. Jean, O. Rachid, The big unknown: The asymptomatic spread of COVID-19, Biomath., 9 (2020), 2005103. https://doi.org/10.11145/j.biomath.2020.05.103 doi: 10.11145/j.biomath.2020.05.103
[2]	J. Deen, M. A. Mengel, J. D. Clemens, Epidemiology of cholera, Vaccine., 38 (2020), A31–A40. https://doi.org/10.1016/j.vaccine.2019.07.078 doi: 10.1016/j.vaccine.2019.07.078
[3]	A. Naheed, A study of spatio-temporal spread of infectious disease: SARS, B. Aust. Math. Soc., 94 (2016), 522–523. https://doi.org/10.1017/S0004972716000484 doi: 10.1017/S0004972716000484
[4]	E. R. Deyle, M. C. Maher, R. D. Hernandez, S. Basu, G. Sugihara, Global environmental drivers of influenza, P. Natl. Acad. Sci. USA., 113 (2016), 13081–13086. https://doi.org/10.1073/pnas.1607747113 doi: 10.1073/pnas.1607747113
[5]	P. Lawrence, N. Danet, O. Reynard, V. Volchkova, V. Volchkov, Human transmission of ebola viruses, Curr. Opin. Virol., 22 (2017), 51–58. https://doi.org/10.1016/j.coviro.2016.11.013 doi: 10.1016/j.coviro.2016.11.013
[6]	M. Tahir, S. Shah, G. Zaman, T. Khan, Stability behaviour of mathematical model MERS corona virus spread in population, Filomat., 33 (2019), 3947–3960. https://doi.org/10.2298/FIL1912947T doi: 10.2298/FIL1912947T
[7]	J. Lou, H. N. Zhou, D. Liang, Z. Jin, B. J. Song, The coupled within- and between-host dynamics in the evolution of HIV/AIDS in China, J. Appl. Anal. Comput., 5 (2015), 731–750. https://doi.org/10.11948/2015056 doi: 10.11948/2015056
[8]	A. E. S. Almocera, E. A. Hernandez-Vargas, Coupling multiscale within-host dynamics and between-host transmission with recovery (SIR) dynamic, Math. Biosci., 309 (2019), 34–41. https://doi.org/10.1016/j.mbs.2019.01.001 doi: 10.1016/j.mbs.2019.01.001
[9]	Z. L. Feng, J. X. Velasco-Hernandez, B. Tapia-Santos, A mathematical model for coupling within-host and between-host dynamics in an environmentally infectious disease, Math. Biosci., 241 (2013), 49–55. https://doi.org/10.1016/j.mbs.2012.09.004 doi: 10.1016/j.mbs.2012.09.004
[10]	A. E. S. Almocera, V. K. Nguyen, E. A. Hernandez-Vargas, Multiscale model within-host and between-host for viral infectious diseases, J. Math. Biol., 77 (2018), 1035–1057. https://doi.org/10.1007/s00285-018-1241-y doi: 10.1007/s00285-018-1241-y
[11]	Z. L. Feng, X. L. Cen, Y. L. Zhao, Jorge X. Velasco-Hernandez. Coupled within-host and between-host dynamoics and evolution of virulence, Math. Biosci., 270 (2015), 204–212. https://doi.org/10.1016/j.mbs.2015.02.012 doi: 10.1016/j.mbs.2015.02.012
[12]	X. L. Cen, Z. L. Feng, Y. L. Zhao, Emerging disease dynamics in a model coupling within-host and between-host systems, J. Theor. Biol., 361 (2014), 141–151. https://doi.org/10.1016/j.jtbi.2014.07.030 doi: 10.1016/j.jtbi.2014.07.030
[13]	B. Y. Wen, J. P. Wang, Z. D. Teng, A discrete-time analog for coupled within-host and between-host dynamics in environmentally driven infectious disease, Adv. Differ. Equ-Ny., 2018 (2018), 69. https://doi.org/10.1186/s13662-018-1522-1 doi: 10.1186/s13662-018-1522-1
[14]	N. Mideo, S. Alizon, T. Day, Linking within-host and between-host dynamics in the evolutionary epidemiology of infectious diseases, Trends. Ecol. Evol., 23 (2008), 511–517. https://doi.org/10.1016/j.tree.2008.05.009 doi: 10.1016/j.tree.2008.05.009
[15]	M. A. Gilchrist, D. Coombs, Evolution of virulence: interdependence, constraints, and selection using netsed models, Theor. Popul. Biol., 69 (2006), 145–153. https://doi.org/10.1016/j.tpb.2005.07.002 doi: 10.1016/j.tpb.2005.07.002
[16]	M. W. Shen, Y. N. Xiao, L. B. Rong, Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics, Math. Biosci., 263 (2015), 37–50. https://doi.org/10.1016/j.mbs.2015.02.003 doi: 10.1016/j.mbs.2015.02.003
[17]	Q. Tang, Z. D. Teng, X. Abdurahman, A new Lyapunov function for SIRS epidemic models, B. Malays. Math. Sci. So., 40 (2017), 237–258. https://doi.org/10.1007/s40840-016-0315-5 doi: 10.1007/s40840-016-0315-5
[18]	K. Hattaf, N. Yousfi, A. Tridane, Mathematical analysis of a viruses dynamics model with general incidence rate and cure rate, Nonlinear. Anal-Real., 13 (2012), 1866–1872. https://doi.org/10.1016/j.nonrwa.2011.12.015 doi: 10.1016/j.nonrwa.2011.12.015
[19]	A. Korobeinikov, Global properties of basic viruses dynamics models, B. Math. Biol., 66 (2004), 879–883. https://doi.org/10.1016/j.bulm.2004.02.001 doi: 10.1016/j.bulm.2004.02.001
[20]	J. J. Lu, Z. D. Teng, Y. K. Li, An age-structured model for coupling within-host and between-host dynamics in environmentally-driven infectious diseases, Chaos. Soliton. Fract., 139 (2020), 110024. https://doi.org/10.1016/j.chaos.2020.110024 doi: 10.1016/j.chaos.2020.110024
[21]	N. Wang, L. Zhang, Z. D. Teng, A reaction-diffusion model for nested within-host and between-host dynamics in an environmentally-driven infectious disease, J. Appl. Anal. Comput., 11 (2021), 1898–1926. https://doi.org/10.11948/20200273 doi: 10.11948/20200273
[22]	B. Y. Wen, Z. D. Teng, Dynamical behaviour in discrete coupled within-host and between-host epidemic model with environmentally driven and saturation incidence, J. Differ. Equ. Appl., 18 (2021), 1–22. https://doi.org/10.1080/10236198.2021.1929197 doi: 10.1080/10236198.2021.1929197
[23]	Y. Wang, M. M. Lu, J. Liu, Global stability of a delayed viruses model with latent infection and Beddington-DeAngelis infection function, Appl. Math. Lett., 107 (2020), 106463. https://doi.org/10.1016/j.aml.2020.106463 doi: 10.1016/j.aml.2020.106463
[24]	Y. X. Gao, W. P. Zhang, D. Liu, Y. J. Xiao, Bifurcation analysis for a delayed SEIR epidemic model with saturated incidence and saturated treatment function, J. Appl. Anal. Comput., 7 (2017), 1070–1094. https://doi.org/10.11948/2017067 doi: 10.11948/2017067
[25]	X. Y. Zhou, J. G. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear. Sci., 16 (2011), 4438–4450. https://doi.org/10.1016/j.cnsns.2011.03.026 doi: 10.1016/j.cnsns.2011.03.026
[26]	J. J. Jiao, Z. Z. Liu, S. H. Cai, Dynamics of an SEIR model with infectivity in incubation period and homestead-isolation on the susceptible, Appl. Math. Lett., 107 (2020), 106442. https://doi.org/10.1016/j.aml.2020.106442 doi: 10.1016/j.aml.2020.106442
[27]	S. F. Wang, D. Y. Zou, Global stability of in-host viral models eith humoral immunity and intracellular delays, Appl. Math. Model., 36 (2012), 1313–1322. https://doi.org/10.1016/j.apm.2011.07.086 doi: 10.1016/j.apm.2011.07.086
[28]	A. M. Elaiw, Global stability analysis of humoral immunity viruses dynamics model including latently infected cells, J. Biol. Dynam., 9 (2015), 215–228. https://doi.org/10.1080/17513758.2015.1056846 doi: 10.1080/17513758.2015.1056846
[29]	A. Gandolf, A. Pugliese, C. Sinisgalli, Epidemic dynamics and host immune response: a nested approach, J. Math. Biol., 70 (2015), 399–435. https://doi.org/10.1007/s00285-014-0769-8 doi: 10.1007/s00285-014-0769-8
[30]	S. T. Tang, Z. D. Teng, H. Miao, Global dynamics of a reaction-diffusion viruses infection model with humoral immunity and nonlinear incidence, Comput. Math. Appl., 78 (2019), 786–806. https://doi.org/10.1016/j.camwa.2019.03.004 doi: 10.1016/j.camwa.2019.03.004
[31]	J. A. Deans, S. Cohen, Immunology of malaria, Annu. Rev. Microbiol., 37 (1983), 25–49. https://doi.org/10.1146/annurev.mi.37.100183.000325 doi: 10.1146/annurev.mi.37.100183.000325
[32]	A. Murase, T. Sasaki, T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics, J. Math. Biol., 51 (2005), 247–267. https://doi.org/10.1007/s00285-005-0321-y doi: 10.1007/s00285-005-0321-y
[33]	J. P. LaSalle, The Stability of Dynamics Systems, SIAM Philadelphia, 1976.
[34]	M. Y. Li, J. S. Muldowney, A geometric approach to global-stability problems, SIAM. J. Math. Anal., 27 (1996), 1070–1083. https://doi.org/10.1137/S0036141094266449 doi: 10.1137/S0036141094266449
[35]	E. X. DeJesus, C. Kaufman, Routh-Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A., 35 (1987), 5288–5290. https://doi.org/10.1103/PhysRevA.35.5288 doi: 10.1103/PhysRevA.35.5288
[36]	M. A. Nowak, R. M. May, Viruses Dynamics: Mathmetical Principles of Immunology and Virology, Oxford University, Oxford, 2000. https://doi.org/10.1016/S0168-1702(01)00293-3
[37]	X. M. Feng, Z. D. Teng, K. Wang, F. Q. Zhang, Backward bifurcation and global stability in an epidemic model with treatment and vaccination, Discrete. Cont. Dyn-B., 19 (2014), 999–1025. https://doi.org/10.3934/dcdsb.2014.19.999 doi: 10.3934/dcdsb.2014.19.999
[38]	J. Arino, C. C. Mccluskey, P. V. D. Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM. J. Appl. Math., 64 (2003), 260–276. https://doi.org/10.1137/S0036139902413829 doi: 10.1137/S0036139902413829
[39]	F. Dumortier, J. Llibre, J. C. Arts, Qualitative Theory of Planar Differential Systems, Springer Berlin, Heidelberg, 2006. https://doi.org/10.1007/978-3-540-32902-2

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)