doi: 10.3934/dcdsb.2021105

A general multipatch cholera model in periodic environments

1. 

Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan

2. 

Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

3. 

Washington State University, Department of Mathematics and Statistics, Pullman, WA 99164-3113, USA

* Corresponding author: Xueying Wang

Received  December 2020 Revised  February 2021 Early access  March 2021

Fund Project: The first author is supported in part by Ministry of Science and Technology, Taiwan; National Center for Theoretical Sciences (NCTS), National Taiwan University; and Chang Gung Memorial Hospital (BMRPD18, NMRPD5J0201 and CLRPG2H0041)

This paper is devoted to a general multipatch cholera epidemic model to investigate disease dynamics in a periodic environment. The basic reproduction number $ \mathcal{R}_0 $ is introduced and a threshold type of result is established in terms of $ \mathcal{R}_0 $. Specifically, we show that when $ \mathcal{R}_0<1 $, the disease-free steady state is globally attractive if either immigration of hosts is homogeneous or immunity loss of human hosts can be neglected; when $ \mathcal{R}_0>1 $, the disease is uniformly persistent and our system admits at least one positive periodic solution. Numerical simulations are carried out to illustrate the impact of asymptotic infections and population dispersal on the spread of cholera. Our result indicates that (a) neglecting asymptotic infections may underestimate the risk of infection; (b) travel can help the disease to become persistent (resp. eradicated) in the network, even though the disease dies out (resp. persists) in each isolated patch.

Citation: Feng-Bin Wang, Xueying Wang. A general multipatch cholera model in periodic environments. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021105
References:
[1]

J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: An epidemic model, Lancet, 377 (2011), 1248-1255.  doi: 10.1016/S0140-6736(11)60273-0.  Google Scholar

[2]

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Mathematical Biosciences, 38 (1978), 113-122.  doi: 10.1016/0025-5564(78)90021-4.  Google Scholar

[3]

E. BertuzzoR. CasagrandiM. GattoI. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010), 321-333.  doi: 10.1098/rsif.2009.0204.  Google Scholar

[4]

A. Camach et al. Cholera epidemic in Yemen, 2016-18: an analysis of surveillance data, The Lancet Global Health, 6 (2018), 680-690. doi: 10.1016/S2214-109X(18)30230-4.  Google Scholar

[5]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, Journal of Mathematical Biology, 13 (1981), 173-184.  doi: 10.1007/BF00275212.  Google Scholar

[6]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante, 27 (1979), 121-132.   Google Scholar

[7]

F. CaponeV. De Cataldis and R. De Luca, Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, Journal of Mathematical Biology, 71 (2015), 1107-1131.  doi: 10.1007/s00285-014-0849-9.  Google Scholar

[8]

A. Chowdhury, S. Tanveer and X. Wang, Nonlinear two-point boundary value problems: applications to a cholera epidemic model, Proceedings of the Royal Society A, 476 (2020), 20190673. doi: 10.1098/rspa.2019.0673.  Google Scholar

[9]

C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001). doi: 10.1186/1471-2334-1-1.  Google Scholar

[10]

R. R. ColwellP. BryatonD. HerringtonB. TallA. Huq and M. M. Levine, Viable but nonculturable vibrio cholerae revert to a cultivable state in the human intestine, World J. Microbiol. Biotechnol., 12 (1996), 28-31.  doi: 10.1007/BF00327795.  Google Scholar

[11]

E. Dangb$\rm\acute{e}$D. Ir$\rm\acute{e}$pranA. Perasso and D. B$\rm\acute{e}$koll$\rm\acute{e}$, Mathematical modelling and numerical simulations of the influence of hygiene and seasons on the spread of cholera, Mathematical Biosciences, 296 (2018), 60-70.  doi: 10.1016/j.mbs.2017.12.004.  Google Scholar

[12]

M. C. EisenbergG. KujbidaA. R. TuiteD. N. Fisman and J. H. Tien, Examining rainfall and cholera dynamics in Haiti using statistical and dynamic modeling approaches, Epidemics, 5 (2013), 197-207.  doi: 10.1016/j.epidem.2013.09.004.  Google Scholar

[13]

M. C. EisenbergZ. ShuaiJ. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement, Mathematical Bioscience, 246 (2013), 105-112.  doi: 10.1016/j.mbs.2013.08.003.  Google Scholar

[14]

M. Gatto, L. Mari, L. and A. Rinaldo, Leading eigenvalues and the spread of cholera, SIAM News, 46 (2013). Google Scholar

[15]

D. M. HartleyJ. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLOS Medicine, 3 (2006), 0063-0069.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[16]

D. HeX. WangD. Gao and J. Wang, Modeling the 2016-2017 Yemen cholera outbreak with the impact of limited medical resources, Journal of Theoretical Biology, 451 (2018), 80-85.  doi: 10.1016/j.jtbi.2018.04.041.  Google Scholar

[17]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM Journal on Mathematical Analysis, 16 (1985), 423-439.  doi: 10.1137/0516030.  Google Scholar

[18]

J. F. Jiang, On the global stability of cooperative systems, Bulletin of the London Mathematical Society, 26 (1994), 455-458. doi: 10.1112/blms/26.5.455.  Google Scholar

[19]

R. I. JohH. WangH. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold, Bulletin of Mathematical Biology, 71 (2009), 845-862.  doi: 10.1007/s11538-008-9384-4.  Google Scholar

[20]

A. A. KingE. L. IonidesM. Pascual and M. J. Bouma, Inapparent infections and cholera dynamics, Nature, 454 (2008), 877-880.  doi: 10.1038/nature07084.  Google Scholar

[21]

M. M. Levine, R. E. Black, M. L. Clements, D. R. Nalin, L. Cisneros and R. A. Finkelstein, Volunteer studies in development of vaccines against cholera and enterotoxigenic escherichia coli: A review, in Acute Enteric Infections in Children. New Prospects for Treatment and Prevention, (eds. T. Holme, J. Holmgren, M. H. Merson, R. Mollby), 1981. Elsevier/North-Holland Biomedical Press, Amsterdam, 443–459. doi: MR21415493.  Google Scholar

[22]

J. LuoJ. Wang and H. Wang, Seasonal forcing and exponential threshold incidence in cholera dynamics, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 2261-2290.  doi: 10.3934/dcdsb.2017095.  Google Scholar

[23]

Z. MukandavireS. LiaoJ. WangH. GaffD. L. Smith and J. G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences of the United States of America, 108 (2011), 8767-8772.  doi: 10.1073/pnas.1019712108.  Google Scholar

[24]

R. L. M. NeilanE. SchaeferH. GaffK. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, Bulletin of Mathematical Biology, 72 (2010), 2004-2018.  doi: 10.1007/s11538-010-9521-8.  Google Scholar

[25]

E. J. NelsonJ. B. HarrisJ. G. MorrisS. B. Calderwood and A. Camilli, Cholera transmission: the host, pathogen and bacteriophage dynamics, Nature Reviews Microbiology, 7 (2009), 693-702.  doi: 10.1038/nrmicro2204.  Google Scholar

[26]

R. Pastor-SatorrasC. CastellanoP. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics, 87 (2015), 925-979.  doi: 10.1103/RevModPhys.87.925.  Google Scholar

[27]

D. Posny and J. Wang, Modelling cholera in periodic environments, Journal of Biological Dynamics, 8 (2014), 1-19.  doi: 10.1080/17513758.2014.896482.  Google Scholar

[28]

Z. Shuai and P. van den Driessche, Modeling and control of cholera on networks with a common water source, Journal of Biological Dynamics, 9 (2015), 90-103.  doi: 10.1080/17513758.2014.944226.  Google Scholar

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41, American Mathematical Society Providence, RI, 1995. doi: 10.1090/surv/041.  Google Scholar

[30] H. L. Smith and P. Waltman., The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[31]

J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72 (2010), 1506-1533.  doi: 10.1007/s11538-010-9507-6.  Google Scholar

[32]

J. H. TienZ. ShuaiM. C. Eisenberg and P. Van den Driessche, Disease invasion on community networks with environmental pathogen movement, Journal of Mathematical Biology, 70 (2015), 1065-1092.  doi: 10.1007/s00285-014-0791-x.  Google Scholar

[33]

A. R. Tuite, J. Tien, M. Eisenberg, D. J. D. Earn, J. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010 – Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Annals of Internal Medicine, 154 (2011) 593–601. doi: 10.7326/0003-4819-154-9-201105030-00334.  Google Scholar

[34]

S. O. Wandiga, Climate Change and Induced Vulnerability to Malaria and Cholera in the Lake Victoria Region, AIACC Final Report, Project No. AF 91, Published by The International START Secretariat, Washington, DC, USA, 2006. Google Scholar

[35]

X. Wang and F.-B. Wang, Impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment, Journal of Mathematical Analysis and Applications, 480 (2019), 123407. doi: 10.1016/j.jmaa.2019.123407.  Google Scholar

[36]

X. Wang and J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, Journal of Biological Dynamics, 9 (2015), 233-261.  doi: 10.1080/17513758.2014.974696.  Google Scholar

[37]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[38]

X. WangX.-Q. Zhao and J. Wang, A cholera epidemic model in a spatiotemporally heterogeneous environment, Journal of Mathematical Analysis and Applications, 468 (2018), 893-912.  doi: 10.1016/j.jmaa.2018.08.039.  Google Scholar

[39]

WHO Yemen cholera situation reports, Available from: http://www.emro.who.int/yem/yemeninfocus/situation-reports.html Google Scholar

[40]

K. Yamazaki and X. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete & Continuous Dynamical Systems-B, 21 (2016), 1297-1316.  doi: 10.3934/dcdsb.2016.21.1297.  Google Scholar

[41]

C. YangX. WangD. Gao and J. Wang, Impact of awareness programs on cholera dynamics: two modeling approaches, Bulletin of Mathematical Biology, 79 (2017), 2109-2131.  doi: 10.1007/s11538-017-0322-1.  Google Scholar

[42]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, Journal of Mathematical Analysis and Applications, 325 (2007), 496-516.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[43]

X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Communications on Applied Nonlinear Analysis, 3 (1996), 43-66.   Google Scholar

[44]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

show all references

References:
[1]

J. R. Andrews and S. Basu, Transmission dynamics and control of cholera in Haiti: An epidemic model, Lancet, 377 (2011), 1248-1255.  doi: 10.1016/S0140-6736(11)60273-0.  Google Scholar

[2]

G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Mathematical Biosciences, 38 (1978), 113-122.  doi: 10.1016/0025-5564(78)90021-4.  Google Scholar

[3]

E. BertuzzoR. CasagrandiM. GattoI. Rodriguez-Iturbe and A. Rinaldo, On spatially explicit models of cholera epidemics, Journal of the Royal Society Interface, 7 (2010), 321-333.  doi: 10.1098/rsif.2009.0204.  Google Scholar

[4]

A. Camach et al. Cholera epidemic in Yemen, 2016-18: an analysis of surveillance data, The Lancet Global Health, 6 (2018), 680-690. doi: 10.1016/S2214-109X(18)30230-4.  Google Scholar

[5]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modeling the spatial spread of a class of bacterial and viral diseases, Journal of Mathematical Biology, 13 (1981), 173-184.  doi: 10.1007/BF00275212.  Google Scholar

[6]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. Epidemiol. Sante, 27 (1979), 121-132.   Google Scholar

[7]

F. CaponeV. De Cataldis and R. De Luca, Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic, Journal of Mathematical Biology, 71 (2015), 1107-1131.  doi: 10.1007/s00285-014-0849-9.  Google Scholar

[8]

A. Chowdhury, S. Tanveer and X. Wang, Nonlinear two-point boundary value problems: applications to a cholera epidemic model, Proceedings of the Royal Society A, 476 (2020), 20190673. doi: 10.1098/rspa.2019.0673.  Google Scholar

[9]

C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001). doi: 10.1186/1471-2334-1-1.  Google Scholar

[10]

R. R. ColwellP. BryatonD. HerringtonB. TallA. Huq and M. M. Levine, Viable but nonculturable vibrio cholerae revert to a cultivable state in the human intestine, World J. Microbiol. Biotechnol., 12 (1996), 28-31.  doi: 10.1007/BF00327795.  Google Scholar

[11]

E. Dangb$\rm\acute{e}$D. Ir$\rm\acute{e}$pranA. Perasso and D. B$\rm\acute{e}$koll$\rm\acute{e}$, Mathematical modelling and numerical simulations of the influence of hygiene and seasons on the spread of cholera, Mathematical Biosciences, 296 (2018), 60-70.  doi: 10.1016/j.mbs.2017.12.004.  Google Scholar

[12]

M. C. EisenbergG. KujbidaA. R. TuiteD. N. Fisman and J. H. Tien, Examining rainfall and cholera dynamics in Haiti using statistical and dynamic modeling approaches, Epidemics, 5 (2013), 197-207.  doi: 10.1016/j.epidem.2013.09.004.  Google Scholar

[13]

M. C. EisenbergZ. ShuaiJ. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement, Mathematical Bioscience, 246 (2013), 105-112.  doi: 10.1016/j.mbs.2013.08.003.  Google Scholar

[14]

M. Gatto, L. Mari, L. and A. Rinaldo, Leading eigenvalues and the spread of cholera, SIAM News, 46 (2013). Google Scholar

[15]

D. M. HartleyJ. G. Morris and D. L. Smith, Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics?, PLOS Medicine, 3 (2006), 0063-0069.  doi: 10.1371/journal.pmed.0030007.  Google Scholar

[16]

D. HeX. WangD. Gao and J. Wang, Modeling the 2016-2017 Yemen cholera outbreak with the impact of limited medical resources, Journal of Theoretical Biology, 451 (2018), 80-85.  doi: 10.1016/j.jtbi.2018.04.041.  Google Scholar

[17]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere, SIAM Journal on Mathematical Analysis, 16 (1985), 423-439.  doi: 10.1137/0516030.  Google Scholar

[18]

J. F. Jiang, On the global stability of cooperative systems, Bulletin of the London Mathematical Society, 26 (1994), 455-458. doi: 10.1112/blms/26.5.455.  Google Scholar

[19]

R. I. JohH. WangH. Weiss and J. S. Weitz, Dynamics of indirectly transmitted infectious diseases with immunological threshold, Bulletin of Mathematical Biology, 71 (2009), 845-862.  doi: 10.1007/s11538-008-9384-4.  Google Scholar

[20]

A. A. KingE. L. IonidesM. Pascual and M. J. Bouma, Inapparent infections and cholera dynamics, Nature, 454 (2008), 877-880.  doi: 10.1038/nature07084.  Google Scholar

[21]

M. M. Levine, R. E. Black, M. L. Clements, D. R. Nalin, L. Cisneros and R. A. Finkelstein, Volunteer studies in development of vaccines against cholera and enterotoxigenic escherichia coli: A review, in Acute Enteric Infections in Children. New Prospects for Treatment and Prevention, (eds. T. Holme, J. Holmgren, M. H. Merson, R. Mollby), 1981. Elsevier/North-Holland Biomedical Press, Amsterdam, 443–459. doi: MR21415493.  Google Scholar

[22]

J. LuoJ. Wang and H. Wang, Seasonal forcing and exponential threshold incidence in cholera dynamics, Discrete & Continuous Dynamical Systems-Series B, 22 (2017), 2261-2290.  doi: 10.3934/dcdsb.2017095.  Google Scholar

[23]

Z. MukandavireS. LiaoJ. WangH. GaffD. L. Smith and J. G. Morris, Estimating the reproductive numbers for the 2008-2009 cholera outbreaks in Zimbabwe, Proceedings of the National Academy of Sciences of the United States of America, 108 (2011), 8767-8772.  doi: 10.1073/pnas.1019712108.  Google Scholar

[24]

R. L. M. NeilanE. SchaeferH. GaffK. R. Fister and S. Lenhart, Modeling optimal intervention strategies for cholera, Bulletin of Mathematical Biology, 72 (2010), 2004-2018.  doi: 10.1007/s11538-010-9521-8.  Google Scholar

[25]

E. J. NelsonJ. B. HarrisJ. G. MorrisS. B. Calderwood and A. Camilli, Cholera transmission: the host, pathogen and bacteriophage dynamics, Nature Reviews Microbiology, 7 (2009), 693-702.  doi: 10.1038/nrmicro2204.  Google Scholar

[26]

R. Pastor-SatorrasC. CastellanoP. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Reviews of Modern Physics, 87 (2015), 925-979.  doi: 10.1103/RevModPhys.87.925.  Google Scholar

[27]

D. Posny and J. Wang, Modelling cholera in periodic environments, Journal of Biological Dynamics, 8 (2014), 1-19.  doi: 10.1080/17513758.2014.896482.  Google Scholar

[28]

Z. Shuai and P. van den Driessche, Modeling and control of cholera on networks with a common water source, Journal of Biological Dynamics, 9 (2015), 90-103.  doi: 10.1080/17513758.2014.944226.  Google Scholar

[29]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr., 41, American Mathematical Society Providence, RI, 1995. doi: 10.1090/surv/041.  Google Scholar

[30] H. L. Smith and P. Waltman., The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, 1995.  doi: 10.1017/CBO9780511530043.  Google Scholar
[31]

J. H. Tien and D. J. D. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bulletin of Mathematical Biology, 72 (2010), 1506-1533.  doi: 10.1007/s11538-010-9507-6.  Google Scholar

[32]

J. H. TienZ. ShuaiM. C. Eisenberg and P. Van den Driessche, Disease invasion on community networks with environmental pathogen movement, Journal of Mathematical Biology, 70 (2015), 1065-1092.  doi: 10.1007/s00285-014-0791-x.  Google Scholar

[33]

A. R. Tuite, J. Tien, M. Eisenberg, D. J. D. Earn, J. Ma and D. N. Fisman, Cholera epidemic in Haiti, 2010 – Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Annals of Internal Medicine, 154 (2011) 593–601. doi: 10.7326/0003-4819-154-9-201105030-00334.  Google Scholar

[34]

S. O. Wandiga, Climate Change and Induced Vulnerability to Malaria and Cholera in the Lake Victoria Region, AIACC Final Report, Project No. AF 91, Published by The International START Secretariat, Washington, DC, USA, 2006. Google Scholar

[35]

X. Wang and F.-B. Wang, Impact of bacterial hyperinfectivity on cholera epidemics in a spatially heterogeneous environment, Journal of Mathematical Analysis and Applications, 480 (2019), 123407. doi: 10.1016/j.jmaa.2019.123407.  Google Scholar

[36]

X. Wang and J. Wang, Analysis of cholera epidemics with bacterial growth and spatial movement, Journal of Biological Dynamics, 9 (2015), 233-261.  doi: 10.1080/17513758.2014.974696.  Google Scholar

[37]

W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, Journal of Dynamics and Differential Equations, 20 (2008), 699-717.  doi: 10.1007/s10884-008-9111-8.  Google Scholar

[38]

X. WangX.-Q. Zhao and J. Wang, A cholera epidemic model in a spatiotemporally heterogeneous environment, Journal of Mathematical Analysis and Applications, 468 (2018), 893-912.  doi: 10.1016/j.jmaa.2018.08.039.  Google Scholar

[39]

WHO Yemen cholera situation reports, Available from: http://www.emro.who.int/yem/yemeninfocus/situation-reports.html Google Scholar

[40]

K. Yamazaki and X. Wang, Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model, Discrete & Continuous Dynamical Systems-B, 21 (2016), 1297-1316.  doi: 10.3934/dcdsb.2016.21.1297.  Google Scholar

[41]

C. YangX. WangD. Gao and J. Wang, Impact of awareness programs on cholera dynamics: two modeling approaches, Bulletin of Mathematical Biology, 79 (2017), 2109-2131.  doi: 10.1007/s11538-017-0322-1.  Google Scholar

[42]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, Journal of Mathematical Analysis and Applications, 325 (2007), 496-516.  doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[43]

X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Communications on Applied Nonlinear Analysis, 3 (1996), 43-66.   Google Scholar

[44]

X.-Q. Zhao, Dynamical Systems in Population Biology, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

Figure 1.  The basic reproduction number $ \mathcal{R}_{0} $ of the 2-patch model as a function of $ p_1 $ and $ p_2 $. Here $ \mathcal{R}_0^{(1)} = \mathcal{R}_0^{(2)} $, and $ D_{12}^{u} = D_{21}^u = 1 $ for $ u = S, I_A, I_S, R, B $
Figure 2.  The basic reproduction number $ \mathcal{R}_{0} $ of the 2-patch model as a function of $ p_1 $ and $ p_2 $. Here $ \mathcal{R}_0^{(1)} = \mathcal{R}_0^{(2)} $, $ D_{12}^{u} = 10\, D_{21}^u $ and $ D_{12}^{u} = 1 $, for $ u = S, I_A, I_S, R, B $
Figure 3.  The basic reproduction number $ \mathcal{R}_{0} $ as a function of $ r_{u} $ of model (1) with $ \mathcal{R}_0^{(1)} = 0.8641 $ and $ \mathcal{R}_0^{(2)} = 1.1600 $, where $ u = S, I_A, I_S, B $
Figure 4.  The basic reproduction number $ \mathcal{R}_{0} $ of model (1) as a function of $ r_{I_S} $, where $ \mathcal{R}_0^{(1)} = 0.9592 $ and $ \mathcal{R}_0^{(2)} = 0.9604 $
Figure 5.  The basic reproduction number $ \mathcal{R}_{0} $ of model (1) as a function of $ r_{I_S} $, where $ \mathcal{R}_0^{(1)} = 1.1454 $ and $ \mathcal{R}_0^{(2)} = 1.0935 $
Table 1.  Definition of parameters of model (1)
Definition
$ \Lambda_i $ Recruitment rate of susceptible humans into patch $ i $
$ \sigma_i $ Immunity waning rate in path $ i $
$ \mu_i $ Natural death rate oh hosts in path $ i $
$ p_i $ Proportion of infections being asymptomatic in path $ i $, $ 0\le p_i\le 1 $
$ \mu_{A_i} $ Death rate of asymptomatic infected hosts in path $ i $
$ \gamma_{A_i} $ Recovery rate from asymptomatic infection in path $ i $
$ \mu_{S_i} $ Death rate of symptomatic infected hosts in path $ i $
$ \gamma_{S_i} $ Recovery rate from symptomatic infection in path $ i $
$ D_{ij}^u $ Immigration rate of population $ u $ from path $ i $ to patch $ j $,
with $ u=S, I_{A}, I_{S}, R, B $ and $ i\ne j $
Definition
$ \Lambda_i $ Recruitment rate of susceptible humans into patch $ i $
$ \sigma_i $ Immunity waning rate in path $ i $
$ \mu_i $ Natural death rate oh hosts in path $ i $
$ p_i $ Proportion of infections being asymptomatic in path $ i $, $ 0\le p_i\le 1 $
$ \mu_{A_i} $ Death rate of asymptomatic infected hosts in path $ i $
$ \gamma_{A_i} $ Recovery rate from asymptomatic infection in path $ i $
$ \mu_{S_i} $ Death rate of symptomatic infected hosts in path $ i $
$ \gamma_{S_i} $ Recovery rate from symptomatic infection in path $ i $
$ D_{ij}^u $ Immigration rate of population $ u $ from path $ i $ to patch $ j $,
with $ u=S, I_{A}, I_{S}, R, B $ and $ i\ne j $
[1]

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