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doi: 10.3934/dcdsb.2022069
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Transmission dynamics of a general temporal-spatial vector-host epidemic model with an application to the dengue fever in Guangdong, China

1. 

College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China

2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St.John's, NL A1C5S7, Canada

*Corresponding author

Received  June 2021 Revised  February 2022 Early access March 2022

Due to the nature of the spread of vector-host epidemic disease, there are many factors affecting its dynamic behaviors. In this paper, a vector-host epidemic model with two seasonal development periods and awareness control of host is proposed to investigate the multi-effects of the spatial heterogeneity, seasonal development periods, temporal periodicity and awareness control. We first address the well-posedness of the model and then derive the basic reproduction number $ R_0 $. In the case where $ R_0<1 $, we establish the global attractivity of the disease-free periodic solution, and in the case where $ R_0>1 $, we show that the disease is uniformly persistent and the system admits at least one positive periodic endemic steady state, and further obtain the global attractivity of the positive endemic constant steady state for the model with constant coefficients. As a case study, we conduct numerical simulations for the dengue fever transmission in Guangdong, China, 2014. We find that the greater heterogeneity of the mosquito distribution and human population may increase the risk of disease transmission, and the stronger awareness control may lower the risk of disease transmission.

Citation: Yantao Luo, Zhidong Teng, Xiao-Qiang Zhao. Transmission dynamics of a general temporal-spatial vector-host epidemic model with an application to the dengue fever in Guangdong, China. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022069
References:
[1]

R. M. Anderson, The persistence of direct life cycle infectious diseases within populations of hosts, In S. A. Levin (Ed.), Lectures on Mathematics in the Life Sciences, Amer. Math. Soc., Providence, R.I., 12 (1979), 1–67.

[2]

Z. BaiR. Peng and X.-Q. Zhao, A reaction-diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77 (2018), 201-228.  doi: 10.1007/s00285-017-1193-7.

[3]

Y. CaiZ. DingB. YangZ. Peng and W. Wang, Transmission dynamics of Zika virus with spatial structure-A case study in Rio de Janeiro, Brazil, Phys. A., 514 (2019), 729-740.  doi: 10.1016/j.physa.2018.09.100.

[4]

L. CaiS. GuoX. Li and M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos Soliton. Fract., 42 (2009), 2297-2304.  doi: 10.1016/j.chaos.2009.03.130.

[5]

L. Cai and X. Li, Global analysis of a vector-host epidemic model with nonlinear incidences, Appl. Math. Comput., 217 (2010), 3531-3541.  doi: 10.1016/j.amc.2010.09.028.

[6]

L. CaiX. LiB. Fang and S. Ruan, Global properties of vector-host disease models with time delays., J. Math. Biol., 74 (2017), 1397-1423.  doi: 10.1007/s00285-016-1047-8.

[7]

Y. CaiK. Wang and W. Wang, Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92 (2019), 190-195.  doi: 10.1016/j.aml.2019.01.015.

[8]

V. Capasso and G. Serio, A generalisation of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[9]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser., vol.279, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992.

[10]

K. S. GaneshM. RajasekharM. S. Rao and B. K. Rao, Temperature dependent transmission potential model for Chikungunya in India, Sci. Total Environ., 647 (2019), 66-74. 

[11]

D. Gao, Y. Lou, D. He, T. Porco and Y. Kuang, et al., Prevention and control of zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis, Scientific Reports (Nature Publisher Group), 6 (2016), 28070. Web.

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.

[13]

X. HouK. LiuX. LiuG. Chang and et al, Nonlinear effects of climate factors on dengue epidemic in Guangdong province, China, Chin. J. Vector. Biol. & Control, 30 (2019), 25. 

[14]

G. HuangW. Ma and Y. Takeuchi, Global analysis for delay virus dynmaics model with Beddington-DeAngelis function response, Appl. Math. Lett., 24 (2011), 1199-1203.  doi: 10.1016/j.aml.2011.02.007.

[15]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a non-local periodic reaction-diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496-2516.  doi: 10.1137/070709761.

[16]

L. LambrechtsK. P. PaaijmansT. FansiriL. B. Carrington and et al., Impact of daily temperature fluctuations on dengue virus transmission by Aedes aegypti, Proc. Natl. Acad. Sci., 108 (2011), 7460-7465. 

[17]

X. LiangL. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations(with application to a spatial model for Lyme disease), J. Dyn. Differ. Equ., 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.

[18]

Z. Lin and H. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.  doi: 10.1007/s00285-017-1124-7.

[19]

W. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.

[20]

J. Liu-Helmersson, H. Stenlund, A. Wilder-Smith and J. Rocklöv, Vectorial capacity of Aedes aegypti: Effects of temperature and implications for global dengue epidemic potential, PLoS One, 9 (2014), Article e89783.

[21]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573-603.  doi: 10.1007/s00332-016-9344-3.

[22]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[23]

R. H. Martin and H. L. Smith, Abstract functional-differnential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[24]

T. MirskiM. Bartoszcze and A. Bielawska-Drozd, Impact of climate change on infectious diseases, Pol. J. Environ. Stud., 21 (2012), 525-532. 

[25]

T. Y. MiyaokaS. Lenhart and J. F. C. A. Meyer, Optimal control of vaccination in a vector-borne reaction-diffusion model applied to Zika virus, J. Math. Biol., 79 (2019), 1077-1104.  doi: 10.1007/s00285-019-01390-z.

[26]

D. NingJ. SunZ. Peng and et al., Epidemiological status and characteristics of dengue fever in Guangdong Province, S. China J. Prev. Med., 43 (2017), 368-372. 

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall. Englewood Cliffs, 1967. doi: 10.1007/978-1-4612-5282-5.

[28]

D. L. SmithJ. Dushoff and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biol., 2 (2004), 1957-1964. 

[29]

H. L. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs, vol.41, American Mathematical Society, Providence, RI, 1995.

[30]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[31]

L. D. ValdezaG. J. Sibonaa and C. A. Condata, Impact of rainfall on Aedes aegypti populations, Ecol. Model., 385 (2018), 96-105. 

[32]

J. Wang and Y. Chen, Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias, Appl. Math. Lett., 100 (2020), 106052, 7 pp. doi: 10.1016/j.aml.2019.106052.

[33]

L. Wang and H. Zhao, Dynamics analysis of a Zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement, Physica A., 522 (2019), 248-273.  doi: 10.1016/j.physa.2019.01.099.

[34]

M. Wang, Nonlinear Elliptic Equations, Science. Public., Beijing, 2010.

[35]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[36]

X. WangM. ShenY. Xiao and L. Rong, Optimal control and cost-effectiveness analysis of a Zika virus infection model with comprehensive interventions, Appl. Math. Comput., 359 (2019), 165-185.  doi: 10.1016/j.amc.2019.04.026.

[37]

Y. WangY. ZhouF. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune respomse incorporating abtiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.  doi: 10.1007/s00285-012-0580-3.

[38]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

R. Wu and X.-Q. Zhao, A reaction-diffusion model of vector-borne disease with periodic delays, J. Nonlinear Sci., 29 (2019), 29-64.  doi: 10.1007/s00332-018-9475-9.

[40]

L. ZhangZ. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period, J. Differ. Equations, 258 (2015), 3011-3036.  doi: 10.1016/j.jde.2014.12.032.

[41]

L. Zhao, Z. Wang and S. Ruan, Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period, Nonlinear Anal. Real World Appl., 51 (2020), 102966, 28 pp. doi: 10.1016/j.nonrwa.2019.102966.

[42]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

[43]

X.-Q. Zhao, Dynamical Systems in Population Biology, second edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.

[44]

B. ZhengL. Chen and Q. Sun, Analyzing the control of dengue by releasing Wolbachia-infected male mosquitoes through a delay differential equation model, Math. Biosci. Eng., 16 (2019), 5531-5550.  doi: 10.3934/mbe.2019275.

[45]

B. ZhengX. LiuM. TangZ. Xi and J. Yu, Use of age-stage structural models to seek optimal Wolbachia-infected male mosquito releases for mosquito-borne disease control, J. Theor. Biol., 472 (2019), 95-109.  doi: 10.1016/j.jtbi.2019.04.010.

[46]

T. Zheng and L. Nin, Modelling the transmission dynamics of two-strain Dengue in the presence awareness and vector control, J. Theor. Biol., 443 (2018), 82-91.  doi: 10.1016/j.jtbi.2018.01.017.

[47]

W. ZhouY. Xiao and J. M. Heffernan, A two-thresholds policy to interrupt transmission of West Nile Virus to birds, J. Theor. Biol., 463 (2019), 22-46.  doi: 10.1016/j.jtbi.2018.12.013.

[48]

X. Zhou and J. Cui, Global stability of the viral dynamics with delayed Beddington-DeAngelis response, Nonlinear Anal-Real., 15 (2011), 555-139. 

[49]

M. Zhu and Y. Xu, A time-periodic dengue fever model in a heterogrnrous environment, Math. Comput. Simulat., 155 (2019), 115-129.  doi: 10.1016/j.matcom.2017.12.008.

[50]

L. ZouJ. ChenX. Feng and et al., Analysis of a dengue model with vertical transmission and application to the 2014 dengue outbreak in guangdong province china, Bull. Math. Biol., 80 (2018), 2633-2651.  doi: 10.1007/s11538-018-0480-9.

[51]

Available from: http://stats.gd.gov.cn/tjfx/content/post_1435240.html.

show all references

References:
[1]

R. M. Anderson, The persistence of direct life cycle infectious diseases within populations of hosts, In S. A. Levin (Ed.), Lectures on Mathematics in the Life Sciences, Amer. Math. Soc., Providence, R.I., 12 (1979), 1–67.

[2]

Z. BaiR. Peng and X.-Q. Zhao, A reaction-diffusion malaria model with seasonality and incubation period, J. Math. Biol., 77 (2018), 201-228.  doi: 10.1007/s00285-017-1193-7.

[3]

Y. CaiZ. DingB. YangZ. Peng and W. Wang, Transmission dynamics of Zika virus with spatial structure-A case study in Rio de Janeiro, Brazil, Phys. A., 514 (2019), 729-740.  doi: 10.1016/j.physa.2018.09.100.

[4]

L. CaiS. GuoX. Li and M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos Soliton. Fract., 42 (2009), 2297-2304.  doi: 10.1016/j.chaos.2009.03.130.

[5]

L. Cai and X. Li, Global analysis of a vector-host epidemic model with nonlinear incidences, Appl. Math. Comput., 217 (2010), 3531-3541.  doi: 10.1016/j.amc.2010.09.028.

[6]

L. CaiX. LiB. Fang and S. Ruan, Global properties of vector-host disease models with time delays., J. Math. Biol., 74 (2017), 1397-1423.  doi: 10.1007/s00285-016-1047-8.

[7]

Y. CaiK. Wang and W. Wang, Global transmission dynamics of a Zika virus model, Appl. Math. Lett., 92 (2019), 190-195.  doi: 10.1016/j.aml.2019.01.015.

[8]

V. Capasso and G. Serio, A generalisation of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[9]

D. Daners and P. Koch Medina, Abstract Evolution Equations, Periodic Problems and Applications, Pitman Res. Notes Math. Ser., vol.279, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992.

[10]

K. S. GaneshM. RajasekharM. S. Rao and B. K. Rao, Temperature dependent transmission potential model for Chikungunya in India, Sci. Total Environ., 647 (2019), 66-74. 

[11]

D. Gao, Y. Lou, D. He, T. Porco and Y. Kuang, et al., Prevention and control of zika as a mosquito-borne and sexually transmitted disease: A mathematical modeling analysis, Scientific Reports (Nature Publisher Group), 6 (2016), 28070. Web.

[12]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.

[13]

X. HouK. LiuX. LiuG. Chang and et al, Nonlinear effects of climate factors on dengue epidemic in Guangdong province, China, Chin. J. Vector. Biol. & Control, 30 (2019), 25. 

[14]

G. HuangW. Ma and Y. Takeuchi, Global analysis for delay virus dynmaics model with Beddington-DeAngelis function response, Appl. Math. Lett., 24 (2011), 1199-1203.  doi: 10.1016/j.aml.2011.02.007.

[15]

Y. Jin and X.-Q. Zhao, Spatial dynamics of a non-local periodic reaction-diffusion model with stage structure, SIAM J. Math. Anal., 40 (2009), 2496-2516.  doi: 10.1137/070709761.

[16]

L. LambrechtsK. P. PaaijmansT. FansiriL. B. Carrington and et al., Impact of daily temperature fluctuations on dengue virus transmission by Aedes aegypti, Proc. Natl. Acad. Sci., 108 (2011), 7460-7465. 

[17]

X. LiangL. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations(with application to a spatial model for Lyme disease), J. Dyn. Differ. Equ., 31 (2019), 1247-1278.  doi: 10.1007/s10884-017-9601-7.

[18]

Z. Lin and H. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.  doi: 10.1007/s00285-017-1124-7.

[19]

W. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.

[20]

J. Liu-Helmersson, H. Stenlund, A. Wilder-Smith and J. Rocklöv, Vectorial capacity of Aedes aegypti: Effects of temperature and implications for global dengue epidemic potential, PLoS One, 9 (2014), Article e89783.

[21]

Y. Lou and X.-Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573-603.  doi: 10.1007/s00332-016-9344-3.

[22]

P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.

[23]

R. H. Martin and H. L. Smith, Abstract functional-differnential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.

[24]

T. MirskiM. Bartoszcze and A. Bielawska-Drozd, Impact of climate change on infectious diseases, Pol. J. Environ. Stud., 21 (2012), 525-532. 

[25]

T. Y. MiyaokaS. Lenhart and J. F. C. A. Meyer, Optimal control of vaccination in a vector-borne reaction-diffusion model applied to Zika virus, J. Math. Biol., 79 (2019), 1077-1104.  doi: 10.1007/s00285-019-01390-z.

[26]

D. NingJ. SunZ. Peng and et al., Epidemiological status and characteristics of dengue fever in Guangdong Province, S. China J. Prev. Med., 43 (2017), 368-372. 

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall. Englewood Cliffs, 1967. doi: 10.1007/978-1-4612-5282-5.

[28]

D. L. SmithJ. Dushoff and F. E. McKenzie, The risk of a mosquito-borne infection in a heterogeneous environment, PLoS Biol., 2 (2004), 1957-1964. 

[29]

H. L. Smith, Monotone Dynamical Systems, Mathematical Surveys and Monographs, vol.41, American Mathematical Society, Providence, RI, 1995.

[30]

H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.  doi: 10.1137/080732870.

[31]

L. D. ValdezaG. J. Sibonaa and C. A. Condata, Impact of rainfall on Aedes aegypti populations, Ecol. Model., 385 (2018), 96-105. 

[32]

J. Wang and Y. Chen, Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias, Appl. Math. Lett., 100 (2020), 106052, 7 pp. doi: 10.1016/j.aml.2019.106052.

[33]

L. Wang and H. Zhao, Dynamics analysis of a Zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement, Physica A., 522 (2019), 248-273.  doi: 10.1016/j.physa.2019.01.099.

[34]

M. Wang, Nonlinear Elliptic Equations, Science. Public., Beijing, 2010.

[35]

W. Wang and X.-Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.  doi: 10.1137/120872942.

[36]

X. WangM. ShenY. Xiao and L. Rong, Optimal control and cost-effectiveness analysis of a Zika virus infection model with comprehensive interventions, Appl. Math. Comput., 359 (2019), 165-185.  doi: 10.1016/j.amc.2019.04.026.

[37]

Y. WangY. ZhouF. Brauer and J. M. Heffernan, Viral dynamics model with CTL immune respomse incorporating abtiretroviral therapy, J. Math. Biol., 67 (2013), 901-934.  doi: 10.1007/s00285-012-0580-3.

[38]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

R. Wu and X.-Q. Zhao, A reaction-diffusion model of vector-borne disease with periodic delays, J. Nonlinear Sci., 29 (2019), 29-64.  doi: 10.1007/s00332-018-9475-9.

[40]

L. ZhangZ. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period, J. Differ. Equations, 258 (2015), 3011-3036.  doi: 10.1016/j.jde.2014.12.032.

[41]

L. Zhao, Z. Wang and S. Ruan, Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period, Nonlinear Anal. Real World Appl., 51 (2020), 102966, 28 pp. doi: 10.1016/j.nonrwa.2019.102966.

[42]

X.-Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Differ. Equ., 29 (2017), 67-82.  doi: 10.1007/s10884-015-9425-2.

[43]

X.-Q. Zhao, Dynamical Systems in Population Biology, second edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.

[44]

B. ZhengL. Chen and Q. Sun, Analyzing the control of dengue by releasing Wolbachia-infected male mosquitoes through a delay differential equation model, Math. Biosci. Eng., 16 (2019), 5531-5550.  doi: 10.3934/mbe.2019275.

[45]

B. ZhengX. LiuM. TangZ. Xi and J. Yu, Use of age-stage structural models to seek optimal Wolbachia-infected male mosquito releases for mosquito-borne disease control, J. Theor. Biol., 472 (2019), 95-109.  doi: 10.1016/j.jtbi.2019.04.010.

[46]

T. Zheng and L. Nin, Modelling the transmission dynamics of two-strain Dengue in the presence awareness and vector control, J. Theor. Biol., 443 (2018), 82-91.  doi: 10.1016/j.jtbi.2018.01.017.

[47]

W. ZhouY. Xiao and J. M. Heffernan, A two-thresholds policy to interrupt transmission of West Nile Virus to birds, J. Theor. Biol., 463 (2019), 22-46.  doi: 10.1016/j.jtbi.2018.12.013.

[48]

X. Zhou and J. Cui, Global stability of the viral dynamics with delayed Beddington-DeAngelis response, Nonlinear Anal-Real., 15 (2011), 555-139. 

[49]

M. Zhu and Y. Xu, A time-periodic dengue fever model in a heterogrnrous environment, Math. Comput. Simulat., 155 (2019), 115-129.  doi: 10.1016/j.matcom.2017.12.008.

[50]

L. ZouJ. ChenX. Feng and et al., Analysis of a dengue model with vertical transmission and application to the 2014 dengue outbreak in guangdong province china, Bull. Math. Biol., 80 (2018), 2633-2651.  doi: 10.1007/s11538-018-0480-9.

[51]

Available from: http://stats.gd.gov.cn/tjfx/content/post_1435240.html.

Figure 1.  The diagram of the vector-host disease transmission
Figure 2.  The temporal-spatial periodic evolution solution of system (12) with with initial value (44) and the Nuemann boundary condition when $ \mathcal{R}_0 = 3.0391>1 $
Figure 3.  The temporal-spatial periodic evolution solution of system (12) with with initial value (44) and the Nuemann boundary condition when $ \mathcal{R}_0 = 0.9701<1 $
Figure 4.  (a) The basic reproduction number $ R_0 $ decreases with $ m $. (b): The basic reproduction number $ R_0 $ increases with $ \eta $;
Figure 5.  The basic reproduction number $ R_0 $ increases with $ \xi $
Table 1.  Parameter values
Value Definition Reference
$ N_h $ 603(km$ ^2 $)$ ^{-1} $ Total human see text
population density
$ A_{h} $ $ 8.96\times10^{-4} $(km$ ^2 $ month)$ ^{-1} $ Human birth rate see text
$ \sigma $ $ 3.90\times 10^{-4} $(km$ ^2 $ month)$ ^{-1} $ Human death rate see text
$ m $ [0–1] conscious control rate [46]
of the susceptible human
$ u $ $ 6.18\times10^{-10} $(km$ ^2 $ month)$ ^{-1} $ Disease-related mortality [26]
$ \alpha $ $ 30.4/6 $ month $ ^{-1} $ Human constant recovery rate [50]
$ \tau_p $ $ [3 / 30.4,14 / 30.4] $ month $ ^{-1} $ Development time of [44]
dengue virus in human
$ A_w $ to be estimated Recruitment rate see text
of mosquitoes
$ \tau_w $ to be evaluated Development time see text
of dengue virus
in mosquitoes
$ l $ 0.1-0.75 Transmission probability from [11]
mosquitoes to hosts per bite
$ p $ 0.3-0.75 Transmission probability from [11]
hosts to mosquitoes per bite
$ \delta $ to be evaluated Death rate see text
of mosquitoes
Value Definition Reference
$ N_h $ 603(km$ ^2 $)$ ^{-1} $ Total human see text
population density
$ A_{h} $ $ 8.96\times10^{-4} $(km$ ^2 $ month)$ ^{-1} $ Human birth rate see text
$ \sigma $ $ 3.90\times 10^{-4} $(km$ ^2 $ month)$ ^{-1} $ Human death rate see text
$ m $ [0–1] conscious control rate [46]
of the susceptible human
$ u $ $ 6.18\times10^{-10} $(km$ ^2 $ month)$ ^{-1} $ Disease-related mortality [26]
$ \alpha $ $ 30.4/6 $ month $ ^{-1} $ Human constant recovery rate [50]
$ \tau_p $ $ [3 / 30.4,14 / 30.4] $ month $ ^{-1} $ Development time of [44]
dengue virus in human
$ A_w $ to be estimated Recruitment rate see text
of mosquitoes
$ \tau_w $ to be evaluated Development time see text
of dengue virus
in mosquitoes
$ l $ 0.1-0.75 Transmission probability from [11]
mosquitoes to hosts per bite
$ p $ 0.3-0.75 Transmission probability from [11]
hosts to mosquitoes per bite
$ \delta $ to be evaluated Death rate see text
of mosquitoes
Table 2.  Monthly mean temperature Guangdong Province (in $ \left.^{\circ} \mathrm{C}\right) $
Month Jul Aug Sep Oct Nov Dec
Temperature 32.7 32.5 30.4 27.6 23.6 18.4
Month Jan Feb Mar Apr May June
Temperature 16.7 17.2 20.7 25.6 29.4 30.7
Month Jul Aug Sep Oct Nov Dec
Temperature 32.7 32.5 30.4 27.6 23.6 18.4
Month Jan Feb Mar Apr May June
Temperature 16.7 17.2 20.7 25.6 29.4 30.7
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