American Institute of Mathematical Sciences

Advanced
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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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