American Institute of Mathematical Sciences

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January  2019, 24(1): 321-349. doi: 10.3934/dcdsb.2018099

Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics

Naveen K. Vaidya 1, , Xianping Li 2, and  Feng-Bin Wang 3,4,

1. 

Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA
2. 

Department of Mathematics and Statistics, University of Missouri -Kansas City, Kansas City, MO 64110, USA
3. 

Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan
4. 

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

Received  January 2017 Revised  December 2017 Published  January 2019 Early access  March 2018

In recent years, the growing spatial spread of dengue, a mosquito-borne disease, has been a major international public health concern. In this paper, we propose a mathematical model to describe an impact of spatially heterogeneous temperature on the dynamics of dengue epidemics. We first consider homogeneous temperature profiles across space and study sensitivity of the basic reproduction number to the environmental temperature. We then introduce spatially heterogeneous temperature into the model and establish some important properties of dengue dynamics. In particular, we formulate two indices, mosquito reproduction number and infection invasion threshold, which completely determine the global threshold dynamics of the model. We also perform numerical simulations to explore the impact of spatially heterogeneous temperature on the disease dynamics. Our analytical and numerical results reveal that spatial heterogeneity of temperature can have significant impact on expansion of dengue epidemics. Our results, including threshold indices, may provide useful information for effective deployment of spatially targeted interventions.

Keywords: Dengue epidemics, diffusion, spatially heterogeneous temperature, basic reproductive number, spectral radius, principal eigenvalue, threshold dynamics.
Mathematics Subject Classification: 35K57, 37N25, 92D30.
Citation: Naveen K. Vaidya, Xianping Li, Feng-Bin Wang. Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 321-349. doi: 10.3934/dcdsb.2018099
References:
[1]

S. Banu, W. Hu, C. Hurst and S. Tong, Dengue transmission in the Asia-Pacific region: Impact of climate change and socio-environmental factors, Tropical Medicine and International Health, 16 (2011), 598-607.  doi: 10.1111/j.1365-3156.2011.02734.x.

[2]

S. Bhatt, P. W. Gething, O. J. Brady, J. P. Messina, A. W. Farlow and C. L. Moyes, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060.

[3]

G. Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse and J. M. Hyman, The Basic Reproduction Number $\mathcal{R}_0$ and Effectiveness of Reactive Interventions during Dengue Epidemics: The 2002 Dengue Outbreak in Easter Island, Chile, Math. Biosci. Eng., 10 (2013), 1455-1474.  doi: 10.3934/mbe.2013.10.1455.

[4]

G. Chowell, P. Diaz-Duenas, J. C. Miller, A. Alcazar-Velazco, J. M. Hyman, P. W. Fenimore and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data, Math. Biosci., 208 (2007), 571-589.  doi: 10.1016/j.mbs.2006.11.011.

[5]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. 

[6]

N. C. Dom, Z. A. Latif, A. H. Ahmad, R. Ismail and B. Pradhan, Manifestation of GIS tools for spatial pattern distribution analysis of dengue fever epidemic in the city of Subang Jaya, Malaysia, Environment Asia, 5 (2012), 82-92. 

[7]

D. A. Focks, E. Daniels, D. G. Haile and J. E. Keesling, A simulation model of the epidemiology of urban dengue fever: Literature analysis, model development, preliminary validation, and samples of simulation results, Am. J. Trop. Med. Hyg., 53 (1995), 489-506.  doi: 10.4269/ajtmh.1995.53.489.

[8]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Diff. Eq., 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.

[9]

A. K. Githeko, S. W. Lindsay, U. E. Confalonieri and J. A. Patz, Climate change and vector-borne diseases: a regional analysis, Bulletin of the World Health Organization, 78 (2000), 1136-1147. 

[10]

D. J. Gubler, Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century, Trends in Microbiology, 10 (2002), 100-103.  doi: 10.1016/S0966-842X(01)02288-0.

[11]

M. J. Hopp and J. A. Foley, Global-scale relationships between climate and the dengue fever vector, Aedes aegypti, Climatic Change, 48 (2001), 441-463. 

[12]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dynamics and Differential Equations, 23 (2011), 817-842.  doi: 10.1007/s10884-011-9224-3.

[13]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006.

[14]

T. W. Hwang and F.-B. Wang, Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation, Discrete and Continuous Dynamical System Series-B, 18 (2013), 147-161. 

[15]

S. Karl, N. Halder, J. K. Kelso, S. A. Ritchie and G. J. Milne, A spatial simulation model for dengue virus infection in urban areas, BMC Infec. Dis., 14 (2014), p447.  doi: 10.1186/1471-2334-14-447.

[16]

A. Khan, M. Hassan and M. Imran, Estimating the basic reproduction number for single-strain dengue fever epidemics, Infectious Diseases of Poverty, 3 (2014), p12.  doi: 10.1186/2049-9957-3-12.

[17]

L. Lambrechts, K. P. Paaijmans, T. Fansiri, L. B. Carrington, L. D. Kramer, M. B. Thomas and T. W. Scott, Impact of daily temperature fluctuations on dengue virus transmission by Aedes aegypti, Proc. Nat. Acad. Sci., 108 (2011), 7460-7465.  doi: 10.1073/pnas.1101377108.

[18]

M. Li, G. Sun, L. Yakob, H. Zhu, Z. Jin and W. Zhang, The Driving Force for 2014 Dengue Outbreak in Guangdong, China, PLoS ONE, 11 (2016), e0166211.  doi: 10.1371/journal.pone.0166211.

[19]

Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[20]

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.

[21]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A.M.S., 321 (1990), 1-44. 

[22]

R. D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in Fixed Point Theory, Lecture Notes in Mathematics (eds. E. Fadell, G. Fournier), 886, Springer, New York/Berlin, (1981), 309–330.

[23]

M. Oki and T. Yamamoto, Climate change, population immunity, and hyperendemicity in the transmission threshold of dengue, PLoS ONE, 7 (2010), e48258.  doi: 10.1371/journal.pone.0048258.

[24]

A. Pakhare, Y. Sabde, A. Joshi, R. Jain, A. Kokane and R. Joshi, A study of spatial and meteorological determinants of dengue outbreak in Bhopal City in 2014, PLoS Negl. Trop. Dis., 53 (2016), 225-233. 

[25]

W. G. Panhuisa, M. Choisyb, X. Xionga, N. S. Choka and P. Akarasewid, Region-wide synchrony and traveling waves of dengue across eight countries in Southeast Asia, Proc. Nat. Acad. Sci., 112 (2015), 13069-13074. 

[26]

J. A. Patz, D. Campbell-Lendrum, T. Holloway and J. A. Foley, Impact of regional climate change on human health, Nature, 438 (2005), 310-317.  doi: 10.1038/nature04188.

[27]

S. T. R. Pinho, C. P. Ferreira, L. Esteva, F. R. Barreto, V. C. M. e Silva and M. G. L. Teixeira, Modelling the dynamics of dengue real epidemics, Phil. Trans. R. Soc. A, 368 (2010), 5679-5693.  doi: 10.1098/rsta.2010.0278.

[28] M. H. Protter and M. F. Weinberger, Maximum Principles in Differential Equations, Springer Verlag, 1984. 
[29]

V. Racloz, R. Ramsey, S. Tong and W. Hu, Surveillance of dengue fever virus: A review of epidemiological models and early warning systems, J. Vector Borne Dis., 6 (2012), e1648.  doi: 10.1371/journal.pntd.0001648.

[30]

C. P. Simmons, J. J. Farrar, N. van Vinh Chau and B. Wills, Dengue, N. Engl. J. Med., 366 (2012), 1423-1432.  doi: 10.1056/NEJMra1110265.

[31]

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.

[32]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[33]

A. K. Supriatna, Estimating the basic reproduction number of dengue transmission during 2002-2007 outbreaks in Bandung, Indonesia, Dengue Bulletin, 33 (2009), 21-32. 

[34]

R. W. Sutherst, Global change and human vulnerability to vector-borne diseases, Clin. Microbiol. Rev., 17 (2004), 136-173.  doi: 10.1128/CMR.17.1.136-173.2004.

[35]

M. Teurlai, C. E. Menkes, V. Cavarero, N. Degallier, E. Descloux and J. Grangeon, Socio-economic and climate factors associated with dengue fever spatial heterogeneity: A worked example in New Caledonia, PLoS Negl. Trop. Dis., 9 (2015), e0004211.  doi: 10.1371/journal.pntd.0004211.

[36]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. 

[37]

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.

[38]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[39]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM. J Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.

[40]

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.

[41]

WHO, Dengue guidelines for giagnosis, treatment, prevention, and control, Geneva: TDR: World Health Organization, 2009.

[42]

R. E. Woodruff and T. McMichael, Climate change and human health: All affected bit some more than others, Social Alternatives, 23 (2004), 17-22. 

[43]

H. M. Yang, M. L. G. Macoris, K. C. Galvani, M. T. M. Andrighetti and D. M. V. Wanderley, Assessing the effects of temperature on dengue transmission, Epidemiol. Infect., 137 (2009), 1179-1187.  doi: 10.1017/S0950268809002052.

[44]

H. M. Yang, M. L. G. Macoris, K. C. Galvani, M. T. M. Andrighetti and D. M. V. Wanderley, Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue, Epidemiol. Infect., 137 (2009), 1188-1202.  doi: 10.1017/S0950268809002040.

[45]

H. M. Yang, M. L. G. Macoris, K. C. Galvani and M. T. M. Andrighetti, Follow up estimation of Aedes aegypti entomological parameters and mathematical modellings, Biosystems, 103 (2011), 360-371.  doi: 10.1016/j.biosystems.2010.11.002.

[46] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. 

show all references

References:
[1]

S. Banu, W. Hu, C. Hurst and S. Tong, Dengue transmission in the Asia-Pacific region: Impact of climate change and socio-environmental factors, Tropical Medicine and International Health, 16 (2011), 598-607.  doi: 10.1111/j.1365-3156.2011.02734.x.

[2]

S. Bhatt, P. W. Gething, O. J. Brady, J. P. Messina, A. W. Farlow and C. L. Moyes, The global distribution and burden of dengue, Nature, 496 (2013), 504-507.  doi: 10.1038/nature12060.

[3]

G. Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse and J. M. Hyman, The Basic Reproduction Number $\mathcal{R}_0$ and Effectiveness of Reactive Interventions during Dengue Epidemics: The 2002 Dengue Outbreak in Easter Island, Chile, Math. Biosci. Eng., 10 (2013), 1455-1474.  doi: 10.3934/mbe.2013.10.1455.

[4]

G. Chowell, P. Diaz-Duenas, J. C. Miller, A. Alcazar-Velazco, J. M. Hyman, P. W. Fenimore and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data, Math. Biosci., 208 (2007), 571-589.  doi: 10.1016/j.mbs.2006.11.011.

[5]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in the models for infectious disease in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. 

[6]

N. C. Dom, Z. A. Latif, A. H. Ahmad, R. Ismail and B. Pradhan, Manifestation of GIS tools for spatial pattern distribution analysis of dengue fever epidemic in the city of Subang Jaya, Malaysia, Environment Asia, 5 (2012), 82-92. 

[7]

D. A. Focks, E. Daniels, D. G. Haile and J. E. Keesling, A simulation model of the epidemiology of urban dengue fever: Literature analysis, model development, preliminary validation, and samples of simulation results, Am. J. Trop. Med. Hyg., 53 (1995), 489-506.  doi: 10.4269/ajtmh.1995.53.489.

[8]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Diff. Eq., 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.

[9]

A. K. Githeko, S. W. Lindsay, U. E. Confalonieri and J. A. Patz, Climate change and vector-borne diseases: a regional analysis, Bulletin of the World Health Organization, 78 (2000), 1136-1147. 

[10]

D. J. Gubler, Epidemic dengue/dengue hemorrhagic fever as a public health, social and economic problem in the 21st century, Trends in Microbiology, 10 (2002), 100-103.  doi: 10.1016/S0966-842X(01)02288-0.

[11]

M. J. Hopp and J. A. Foley, Global-scale relationships between climate and the dengue fever vector, Aedes aegypti, Climatic Change, 48 (2001), 441-463. 

[12]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dynamics and Differential Equations, 23 (2011), 817-842.  doi: 10.1007/s10884-011-9224-3.

[13]

S. B. Hsu, F. B. Wang and X.-Q. Zhao, Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differential Equations, 255 (2013), 265-297.  doi: 10.1016/j.jde.2013.04.006.

[14]

T. W. Hwang and F.-B. Wang, Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation, Discrete and Continuous Dynamical System Series-B, 18 (2013), 147-161. 

[15]

S. Karl, N. Halder, J. K. Kelso, S. A. Ritchie and G. J. Milne, A spatial simulation model for dengue virus infection in urban areas, BMC Infec. Dis., 14 (2014), p447.  doi: 10.1186/1471-2334-14-447.

[16]

A. Khan, M. Hassan and M. Imran, Estimating the basic reproduction number for single-strain dengue fever epidemics, Infectious Diseases of Poverty, 3 (2014), p12.  doi: 10.1186/2049-9957-3-12.

[17]

L. Lambrechts, K. P. Paaijmans, T. Fansiri, L. B. Carrington, L. D. Kramer, M. B. Thomas and T. W. Scott, Impact of daily temperature fluctuations on dengue virus transmission by Aedes aegypti, Proc. Nat. Acad. Sci., 108 (2011), 7460-7465.  doi: 10.1073/pnas.1101377108.

[18]

M. Li, G. Sun, L. Yakob, H. Zhu, Z. Jin and W. Zhang, The Driving Force for 2014 Dengue Outbreak in Guangdong, China, PLoS ONE, 11 (2016), e0166211.  doi: 10.1371/journal.pone.0166211.

[19]

Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.  doi: 10.1007/s00285-010-0346-8.

[20]

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.

[21]

R. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. of A.M.S., 321 (1990), 1-44. 

[22]

R. D. Nussbaum, Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in Fixed Point Theory, Lecture Notes in Mathematics (eds. E. Fadell, G. Fournier), 886, Springer, New York/Berlin, (1981), 309–330.

[23]

M. Oki and T. Yamamoto, Climate change, population immunity, and hyperendemicity in the transmission threshold of dengue, PLoS ONE, 7 (2010), e48258.  doi: 10.1371/journal.pone.0048258.

[24]

A. Pakhare, Y. Sabde, A. Joshi, R. Jain, A. Kokane and R. Joshi, A study of spatial and meteorological determinants of dengue outbreak in Bhopal City in 2014, PLoS Negl. Trop. Dis., 53 (2016), 225-233. 

[25]

W. G. Panhuisa, M. Choisyb, X. Xionga, N. S. Choka and P. Akarasewid, Region-wide synchrony and traveling waves of dengue across eight countries in Southeast Asia, Proc. Nat. Acad. Sci., 112 (2015), 13069-13074. 

[26]

J. A. Patz, D. Campbell-Lendrum, T. Holloway and J. A. Foley, Impact of regional climate change on human health, Nature, 438 (2005), 310-317.  doi: 10.1038/nature04188.

[27]

S. T. R. Pinho, C. P. Ferreira, L. Esteva, F. R. Barreto, V. C. M. e Silva and M. G. L. Teixeira, Modelling the dynamics of dengue real epidemics, Phil. Trans. R. Soc. A, 368 (2010), 5679-5693.  doi: 10.1098/rsta.2010.0278.

[28] M. H. Protter and M. F. Weinberger, Maximum Principles in Differential Equations, Springer Verlag, 1984. 
[29]

V. Racloz, R. Ramsey, S. Tong and W. Hu, Surveillance of dengue fever virus: A review of epidemiological models and early warning systems, J. Vector Borne Dis., 6 (2012), e1648.  doi: 10.1371/journal.pntd.0001648.

[30]

C. P. Simmons, J. J. Farrar, N. van Vinh Chau and B. Wills, Dengue, N. Engl. J. Med., 366 (2012), 1423-1432.  doi: 10.1056/NEJMra1110265.

[31]

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.

[32]

H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal., 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.

[33]

A. K. Supriatna, Estimating the basic reproduction number of dengue transmission during 2002-2007 outbreaks in Bandung, Indonesia, Dengue Bulletin, 33 (2009), 21-32. 

[34]

R. W. Sutherst, Global change and human vulnerability to vector-borne diseases, Clin. Microbiol. Rev., 17 (2004), 136-173.  doi: 10.1128/CMR.17.1.136-173.2004.

[35]

M. Teurlai, C. E. Menkes, V. Cavarero, N. Degallier, E. Descloux and J. Grangeon, Socio-economic and climate factors associated with dengue fever spatial heterogeneity: A worked example in New Caledonia, PLoS Negl. Trop. Dis., 9 (2015), e0004211.  doi: 10.1371/journal.pntd.0004211.

[36]

H. R. Thieme, Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763. 

[37]

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.

[38]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[39]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM. J Appl. Math., 71 (2011), 147-168.  doi: 10.1137/090775890.

[40]

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.

[41]

WHO, Dengue guidelines for giagnosis, treatment, prevention, and control, Geneva: TDR: World Health Organization, 2009.

[42]

R. E. Woodruff and T. McMichael, Climate change and human health: All affected bit some more than others, Social Alternatives, 23 (2004), 17-22. 

[43]

H. M. Yang, M. L. G. Macoris, K. C. Galvani, M. T. M. Andrighetti and D. M. V. Wanderley, Assessing the effects of temperature on dengue transmission, Epidemiol. Infect., 137 (2009), 1179-1187.  doi: 10.1017/S0950268809002052.

[44]

H. M. Yang, M. L. G. Macoris, K. C. Galvani, M. T. M. Andrighetti and D. M. V. Wanderley, Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue, Epidemiol. Infect., 137 (2009), 1188-1202.  doi: 10.1017/S0950268809002040.

[45]

H. M. Yang, M. L. G. Macoris, K. C. Galvani and M. T. M. Andrighetti, Follow up estimation of Aedes aegypti entomological parameters and mathematical modellings, Biosystems, 103 (2011), 360-371.  doi: 10.1016/j.biosystems.2010.11.002.

[46] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. 
Figure 2.1.  Functional curves $\delta (T)$, oviposition rate, $\mu_a(T)$, aquatic phase mortality rate, $\theta (T)$, mosquito emergence rate from acuatic phase, and $\mu_m(T)$, mosquito mortality rate, fitted to the experimental data [44]
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Figure 2.2.  Functional curve $\beta_m (T)$, the transmission probability from human to mosquito, fitted to the data generated from the previous estimates [17]
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Figure 3.1.  The basic reproduction number, $\bar{\mathcal{R}}_0$, vs. environmental temperature, $T$, for different values of carrying capacity, $C$.
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Figure 4.1.  Spatio-temporal distribution of prevalence (left) and new infection (right) during an epidemic. Here, $T_m = 22.5$℃ and $\Delta T = 25$℃, and $D_M = D_H = 0.0001$.
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Figure 4.2.  Distribution of prevalence (left) and new infection (middle) at different times and the total infection during an epidemic (right). Here, $T_m = 22.5$℃ and $\Delta T = 25$℃, and $D_M = D_H = 0.0001$.
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Figure 4.3.  Distribution of prevalence (left) and new infection (middle) at different times and the total infection during an epidemic (right) for the end point temperature difference $\Delta T = 15$℃ (upper panel) and $\Delta T = 35$℃ (lower panel). Here $T_m = 22.5$℃ and $D_M = D_H = 0.0001$.
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Figure 4.4.  Distribution of prevalence (left) and new infection (middle) at different times and the total infection during an epidemic (right) for the mean temperature $T_m = 15$℃ (upper panel) and $T_m = 30$℃ (lower panel). Here $\Delta T = 25$℃ and $D_M = D_H = 0.0001$.
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Figure 4.5.  Distribution of prevalence (left) and new infection (middle) at different times and the total infection during an epidemic (right) for $D_M/D_H = 0.1$ (upper panel) and $D_M/D_H = 10$ (lower panel). Here $T_m = 22.5$℃ and $\Delta T = 25$℃.
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Table 2.1.  Model parameters
Parameter Description Value Reference
$k$ fraction of female larvae from eggs 0.5 (0-1) [18,27]
$b$ per capita biting rate 0.1 [4,27]
$\mu_h$ Natural death rate of humans 4.22$\times 10^{-5}$ d$^{-1}$ Calculated, [16]
$1/\gamma_h$ Intrinsic period 10 days [4,16,18,27]
$\alpha_h$ Human recovery rate 0.1 d$^{-1}$ [18,27]
$D_M, D_H$ Diffusion coefficients - varied
$\delta_m$ In $\delta(x)$ 9.531 Data fitting
$\delta_h$ In $\delta(x)$ 22.55 Data fitting
$N_{\delta}$ In $\delta(x)$ 7.084 Data fitting
$a_{0\mu_a}$ In $\mu_a(x)$ 2.914 Data fitting
$a_{1\mu_a}$ In $\mu_a(x)$ -0.4986 Data fitting
$a_{2\mu_a}$ In $\mu_a(x)$ 0.03099 Data fitting
$a_{3\mu_a}$ In $\mu_a(x)$ -0.0008236 Data fitting
$a_{4\mu_a}$ In $\mu_a(x)$ 7.975$\times 10^{-6}$ Data fitting
$a_{0\theta}$ In $\theta(x)$ 8.044$\times 10^{-5}$ Data fitting
$a_{1\theta}$ In $\theta(x)$ 11.386 Data fitting
$a_{2\theta}$ In $\theta(x)$ 40.1461 Data fitting
$a_{0\mu_m}$ In $\mu_m(x)$ 0.1901 Data fitting
$a_{1\mu_m}$ In $\mu_m(x)$ -0.0134 Data fitting
$a_{2\mu_m}$ In $\mu_m(x)$ 2.739$\times 10^{-4}$ Data fitting
$a_{0\gamma_m}$ In $\gamma_m(x)$ 5$\times 10^{4/3}$ Data fitting
$a_{1\gamma_m}$ In $\gamma_m(x)$ 0.0768 Data fitting
$\beta_{mh}$ In $\beta_m(x)$ 18.9871 Data fitting
$N_{\beta_m}$ In $\beta_m(x)$ 7 Data fitting
$a_{0\beta_h}$ In $\beta_h(x)$ 1.044$\times 10^{-3}$ Data fitting
$a_{1\beta_h}$ In $\beta_h(x)$ 12.286 Data fitting
$a_{2\beta_h}$ In $\beta_h(x)$ 32.461 Data fitting
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Parameter Description Value Reference
$k$ fraction of female larvae from eggs 0.5 (0-1) [18,27]
$b$ per capita biting rate 0.1 [4,27]
$\mu_h$ Natural death rate of humans 4.22$\times 10^{-5}$ d$^{-1}$ Calculated, [16]
$1/\gamma_h$ Intrinsic period 10 days [4,16,18,27]
$\alpha_h$ Human recovery rate 0.1 d$^{-1}$ [18,27]
$D_M, D_H$ Diffusion coefficients - varied
$\delta_m$ In $\delta(x)$ 9.531 Data fitting
$\delta_h$ In $\delta(x)$ 22.55 Data fitting
$N_{\delta}$ In $\delta(x)$ 7.084 Data fitting
$a_{0\mu_a}$ In $\mu_a(x)$ 2.914 Data fitting
$a_{1\mu_a}$ In $\mu_a(x)$ -0.4986 Data fitting
$a_{2\mu_a}$ In $\mu_a(x)$ 0.03099 Data fitting
$a_{3\mu_a}$ In $\mu_a(x)$ -0.0008236 Data fitting
$a_{4\mu_a}$ In $\mu_a(x)$ 7.975$\times 10^{-6}$ Data fitting
$a_{0\theta}$ In $\theta(x)$ 8.044$\times 10^{-5}$ Data fitting
$a_{1\theta}$ In $\theta(x)$ 11.386 Data fitting
$a_{2\theta}$ In $\theta(x)$ 40.1461 Data fitting
$a_{0\mu_m}$ In $\mu_m(x)$ 0.1901 Data fitting
$a_{1\mu_m}$ In $\mu_m(x)$ -0.0134 Data fitting
$a_{2\mu_m}$ In $\mu_m(x)$ 2.739$\times 10^{-4}$ Data fitting
$a_{0\gamma_m}$ In $\gamma_m(x)$ 5$\times 10^{4/3}$ Data fitting
$a_{1\gamma_m}$ In $\gamma_m(x)$ 0.0768 Data fitting
$\beta_{mh}$ In $\beta_m(x)$ 18.9871 Data fitting
$N_{\beta_m}$ In $\beta_m(x)$ 7 Data fitting
$a_{0\beta_h}$ In $\beta_h(x)$ 1.044$\times 10^{-3}$ Data fitting
$a_{1\beta_h}$ In $\beta_h(x)$ 12.286 Data fitting
$a_{2\beta_h}$ In $\beta_h(x)$ 32.461 Data fitting
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