American Institute of Mathematical Sciences

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January  2022, 27(1): 393-420. doi: 10.3934/dcdsb.2021048

Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations

Naveen K. Vaidya 1, and  Feng-Bin Wang 2,

1. 

Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA, Computational Science Research Center, San Diego State University, San Diego, CA 92182, USA, Viral Information Institute, San Diego State University, San Diego, CA 92182, USA
2. 

Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan, Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

Received  August 2020 Revised  January 2021 Published  January 2022 Early access  February 2021

Dengue, a mosquito-borne disease, poses a tremendous burden to human health with about 390 million annual dengue infections worldwide. The environmental temperature plays a major role in the mosquito life-cycle as well as the mosquito-human-mosquito dengue transmission cycle. While previous studies have provided useful insights into the understanding of dengue diseases, there is little emphasis put on the role of environmental temperature variation, especially diurnal variation, in the mosquito vector and dengue dynamics. In this study, we develop a mathematical model to investigate the impact of seasonal and diurnal temperature variations on the persistence of mosquito vector and dengue. Importantly, using a threshold dynamical system approach to our model, we formulate the mosquito reproduction number and the infection invasion threshold, which completely determine the global threshold dynamics of mosquito population and dengue transmission, respectively. Our model predicts that both seasonal and diurnal variations of the environmental temperature can be determinant factors for the persistence of mosquito vector and dengue. In general, our numerical estimates of the mosquito reproduction number and the infection invasion threshold show that places with higher diurnal or seasonal temperature variations have a tendency to suffer less from the burden of mosquito population and dengue epidemics. Our results provide novel insights into the theoretical understanding of the role of diurnal temperature, which can be beneficial for the control of mosquito vector and dengue spread.

Keywords: Dengue epidemics, diurnal temperature, global stability, periodic models, seasonal temperature.
Mathematics Subject Classification: 34D20, 37N25, 92B05, 92D30, 92D40.
Citation: Naveen K. Vaidya, Feng-Bin Wang. Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 393-420. doi: 10.3934/dcdsb.2021048
References:
[1]

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

[2]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[3]

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, 11 (2011), 598-607.  doi: 10.1111/j.1365-3156.2011.02734.x.

[4]

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

[5]

O. J. Brady, P. W. Gething, S. Bhatt, J. P. Messina, J. S. Brownstein, et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus, PLoS Negl. Trop. Dis., 6 (2012), e1760. doi: 10.1371/journal.pntd.0001760.

[6]

G. Chowell, P. Diaz-Dueñas, 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.

[7]

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. 

[8]

T. P. Endy, A. Nisalak, S. Chunsuttiwat, D. H. Libraty and S. Green, Spatial and temporal circulation of dengue virus serotypes: A prospective study of primary school children in Kamphaeng Phet, Thailand, Am. J. Epidemiol., 156 (2002), 52-59.  doi: 10.1093/aje/kwf006.

[9]

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.

[10]

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. 

[11]

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.

[12]

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

[13]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439.  doi: 10.1137/0516030.

[14]

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. 

[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, PNAS, 108 (2011), 7460-7465.  doi: 10.1073/pnas.1101377108.

[18]

M.-T. Li, G.-Q. Sun, L. Yakob, H.-P. Zhu, Z. Jin and W.-Y. Zhang, The driving force for 2014 dengue outbreak in Guangdong, China, PLoS ONE, 11 (2016), e0166211. doi: 10.1371/journal.pone.0166211.

[19]

L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculousis model with seasonality, Bull. Math. Biol., 72 (2010), 931-952.  doi: 10.1007/s11538-009-9477-8.

[20]

A. Nisalak, T. P. Endy, S. Nimmannitya, S. Kalayanarooj and U. Thisayakorn, Serotype-specific dengue virus circulation and dengue disease in Bangkok, Thailand from 1973 to 1999, Am. J. Trop. Med. Hyg., 68 (2003), 191-202.  doi: 10.4269/ajtmh.2003.68.191.

[21]

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.

[22]

K. P. Paaijmans, A. F. Read and M. B. Thomas, Understanding the link between malaria risk and climate, PNAS, 106 (2009), 13844-13849.  doi: 10.1073/pnas.0903423106.

[23]

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 (2014), 225-233. 

[24]

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. 

[25]

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.

[26]

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

[27]

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

[28]

D. J. Rogers and S. E. Randolph, Climate change and vector-borne diseases, Adv. Parasitol., 62 (2006), 345-381.  doi: 10.1016/S0065-308X(05)62010-6.

[29]

T. W. Scott, A. C. Morrison, L. H. Lorenz, G. G. Clark and D. Strickman, Longitudinal studies of aedes aegypti (diptera: Culicidae) in Thailand and puerto rico: Population dynamics, J. Med. Entomol., 37 (2000), 77-88. 

[30]

P. M. Sheppard, W. W. Macdonald, R. J. Tonnand and B. Grab, The dynamics of an adult population of aedes aegypti in relation to dengue haemorrhagic fever in bangkok, J. Anim. Ecol., 38 (1969), 661-702.  doi: 10.2307/3042.

[31]

C. P. Simmons, J. J. Farrar, N. van Vinh Chau and B. Wills, Dengue, J. Vector Borne Dis, 6 (2012), e1648. doi: 10.1056/NEJMra1110265.

[32]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society Providence, RI, 1995.

[33] H. L. Smith and P. Waltman, The Theory of the Chemostat, CCambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[34]

R. W. Sutherst, Global change and human vulnerability to vector-borne diseases, N. Engl. J. Med., 366 (2012), 1423-1432.  doi: 10.1128/CMR.17.1.136-173.2004.

[35]

M. Teurlai, C. E. Menkés, V. Cavarero, N. Degallier, E. Descloux, J.-P. Grangeon, et al., 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]

N. K. Vaidya, X. Li and F.-B. Wang, Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 321-349.  doi: 10.3934/dcdsb.2018099.

[37]

N. K. Vaidya and L. M. Wahl, Avian influenza dynamics under periodic environmental conditions, SIAM J. Appl. Math., 75 (2015), 443-467.  doi: 10.1137/140966642.

[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]

F.-B. Wang, S.-B. Hsu and W. Wang, Dynamics of harmful algae with seasonal temperature variations in the cove-main lake, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 313-335.  doi: 10.3934/dcdsb.2016.21.313.

[40]

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

[41]

WHO, Dengue Guidelines for Diagnosis, Treatment, Prevention, and Control, TDR: World Health Organization, (2009).

[42]

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

[43]

H. M. Yang, M. de L. da 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.

[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 dengue transmission, Epidemiol. Infect., 137 (2009), 1179-1187.  doi: 10.1017/S0950268809002052.

[45]

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.

[46]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.  doi: 10.1016/j.jmaa.2006.01.085.

[47]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

[48]

X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonlinear Anal., 3 (1996), 43-66. 

show all references

References:
[1]

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

[2]

N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco, J. Math. Biol., 53 (2006), 421-436.  doi: 10.1007/s00285-006-0015-0.

[3]

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, 11 (2011), 598-607.  doi: 10.1111/j.1365-3156.2011.02734.x.

[4]

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

[5]

O. J. Brady, P. W. Gething, S. Bhatt, J. P. Messina, J. S. Brownstein, et al., Refining the global spatial limits of dengue virus transmission by evidence-based consensus, PLoS Negl. Trop. Dis., 6 (2012), e1760. doi: 10.1371/journal.pntd.0001760.

[6]

G. Chowell, P. Diaz-Dueñas, 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.

[7]

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. 

[8]

T. P. Endy, A. Nisalak, S. Chunsuttiwat, D. H. Libraty and S. Green, Spatial and temporal circulation of dengue virus serotypes: A prospective study of primary school children in Kamphaeng Phet, Thailand, Am. J. Epidemiol., 156 (2002), 52-59.  doi: 10.1093/aje/kwf006.

[9]

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.

[10]

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. 

[11]

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.

[12]

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

[13]

M. W. Hirsch, Systems of differential equations that are competitive or cooperative Ⅱ: Convergence almost everywhere, SIAM J. Math. Anal., 16 (1985), 423-439.  doi: 10.1137/0516030.

[14]

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. 

[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, PNAS, 108 (2011), 7460-7465.  doi: 10.1073/pnas.1101377108.

[18]

M.-T. Li, G.-Q. Sun, L. Yakob, H.-P. Zhu, Z. Jin and W.-Y. Zhang, The driving force for 2014 dengue outbreak in Guangdong, China, PLoS ONE, 11 (2016), e0166211. doi: 10.1371/journal.pone.0166211.

[19]

L. Liu, X.-Q. Zhao and Y. Zhou, A tuberculousis model with seasonality, Bull. Math. Biol., 72 (2010), 931-952.  doi: 10.1007/s11538-009-9477-8.

[20]

A. Nisalak, T. P. Endy, S. Nimmannitya, S. Kalayanarooj and U. Thisayakorn, Serotype-specific dengue virus circulation and dengue disease in Bangkok, Thailand from 1973 to 1999, Am. J. Trop. Med. Hyg., 68 (2003), 191-202.  doi: 10.4269/ajtmh.2003.68.191.

[21]

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.

[22]

K. P. Paaijmans, A. F. Read and M. B. Thomas, Understanding the link between malaria risk and climate, PNAS, 106 (2009), 13844-13849.  doi: 10.1073/pnas.0903423106.

[23]

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 (2014), 225-233. 

[24]

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. 

[25]

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.

[26]

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

[27]

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

[28]

D. J. Rogers and S. E. Randolph, Climate change and vector-borne diseases, Adv. Parasitol., 62 (2006), 345-381.  doi: 10.1016/S0065-308X(05)62010-6.

[29]

T. W. Scott, A. C. Morrison, L. H. Lorenz, G. G. Clark and D. Strickman, Longitudinal studies of aedes aegypti (diptera: Culicidae) in Thailand and puerto rico: Population dynamics, J. Med. Entomol., 37 (2000), 77-88. 

[30]

P. M. Sheppard, W. W. Macdonald, R. J. Tonnand and B. Grab, The dynamics of an adult population of aedes aegypti in relation to dengue haemorrhagic fever in bangkok, J. Anim. Ecol., 38 (1969), 661-702.  doi: 10.2307/3042.

[31]

C. P. Simmons, J. J. Farrar, N. van Vinh Chau and B. Wills, Dengue, J. Vector Borne Dis, 6 (2012), e1648. doi: 10.1056/NEJMra1110265.

[32]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society Providence, RI, 1995.

[33] H. L. Smith and P. Waltman, The Theory of the Chemostat, CCambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511530043.
[34]

R. W. Sutherst, Global change and human vulnerability to vector-borne diseases, N. Engl. J. Med., 366 (2012), 1423-1432.  doi: 10.1128/CMR.17.1.136-173.2004.

[35]

M. Teurlai, C. E. Menkés, V. Cavarero, N. Degallier, E. Descloux, J.-P. Grangeon, et al., 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]

N. K. Vaidya, X. Li and F.-B. Wang, Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 321-349.  doi: 10.3934/dcdsb.2018099.

[37]

N. K. Vaidya and L. M. Wahl, Avian influenza dynamics under periodic environmental conditions, SIAM J. Appl. Math., 75 (2015), 443-467.  doi: 10.1137/140966642.

[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]

F.-B. Wang, S.-B. Hsu and W. Wang, Dynamics of harmful algae with seasonal temperature variations in the cove-main lake, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 313-335.  doi: 10.3934/dcdsb.2016.21.313.

[40]

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

[41]

WHO, Dengue Guidelines for Diagnosis, Treatment, Prevention, and Control, TDR: World Health Organization, (2009).

[42]

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

[43]

H. M. Yang, M. de L. da 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.

[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 dengue transmission, Epidemiol. Infect., 137 (2009), 1179-1187.  doi: 10.1017/S0950268809002052.

[45]

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.

[46]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516.  doi: 10.1016/j.jmaa.2006.01.085.

[47]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

[48]

X.-Q. Zhao, Asymptotic behavior for asymptotically periodic semiflows with applications, Comm. Appl. Nonlinear Anal., 3 (1996), 43-66. 

Figure 2.1.  A schematic diagram of the dengue transmission model. $ A, M_s, M_e, M_i $: aquatic, susceptible, exposed, and infected mosquitos. $ H_s, H_e, H_i, H_r $: susceptible, exposed, infected, and recovered humans. Solid arrows represent birth, maturation, infection, transfer, death, while dashed arrows indicate the effects of time dependent periodic environmental temperature, $ T(t) $
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Figure 2.2.  Best-fit curves provided by the experimental data [44] for $ \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)
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Figure 2.3.  Best-fit curve provided by the data generated from the previous estimates [17] for the transmission probability from human to mosquito, $ \beta_m (T) $
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Figure 2.4.  Temperature profile of $ T_m(t) $ [Left], $ \Psi_d(t) $ [Middle], and $ T(t) $ [Right]. Parameters used are $ T_0 = 25 ^oC, \epsilon_m = 5 ^oC, \tau_m $ = 365 day, $ \phi_m = 0 $, $ \epsilon_d = 5 ^oC $, $ \tau_d = 1 $ day, and $ \phi_d = 0 $
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Figure 4.1.  Mosquito reproduction number ($ \mathcal{R}^M $) [Left] and infection invasion threshold ($ \mathcal{R}^0 $) [right] for different values of the mean temperature ($ T_0 $) with amplitudes of seasonal temperature and diurnal temperature fixed at $ \epsilon_m = 5 $ $ ^oC $ and $ \epsilon_d = 5 $ $ ^oC $, respectively. For comparison purposes, $ \mathcal{R}^M $ and $ \mathcal{R}^0 $ for the constant temperature (i.e., $ \epsilon_m = \epsilon_d = 0 $ $ ^oC $) are also plotted
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Figure 4.2.  Mosquito reproduction number ($ \mathcal{R}^M $) [left column] and infection invasion threshold ($ \mathcal{R}^0 $) [right column] for different values of the amplitudes of seasonal temperature ($ \epsilon_m $) with the amplitude of diurnal temperature fixed at $ \epsilon_d = 5 $ $ ^oC $ and the mean temperature fixed at $ T_0 = 16 $ $ ^oC $ [top row], $ T_0 = 28 $ $ ^oC $ [middle row], and $ T_0 = 38 $ $ ^oC $ [bottom row]
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Figure 4.3.  Mosquito reproduction number ($ \mathcal{R}^M $) [left column] and infection invasion threshold ($ \mathcal{R}^0 $) [right column] for different values of the amplitudes of diurnal temperature ($ \epsilon_d $) with the amplitude of seasonal temperature fixed at $ \epsilon_m = 5 $ $ ^oC $ and the mean temperature fixed at $ T_0 = 16 $ $ ^oC $ [top row], $ T_0 = 28 $ $ ^oC $ [middle row], and $ T_0 = 38 $ $ ^oC $ [bottom row]
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Table 2.1.  Model parameters
Parameter Description Value Reference
$ k $ Fraction of female larvae from eggs 0.5 (0-1) [18,26]
$ b $ Per capita biting rate 0.1 [6,26]
$ \mu_h $ Natural death rate of humans 4.22$ \times 10^{-5} $ d$ ^{-1} $ Calculated, [16]
$ 1/\gamma_h $ Intrinsic period 10 days [6,16,18,26]
$ \alpha_h $ Human recovery rate 0.1 d$ ^{-1} $ [18,26]
$ \delta_m $ In $ \delta(t) $ 9.531 Data fitting
$ \delta_h $ In $ \delta(t) $ 22.55 Data fitting
$ N_{\delta} $ In $ \delta(t) $ 7.084 Data fitting
$ a_{\delta} $ In $ \delta(t) $ 0 Data fitting
$ \epsilon_{\delta} $ In $ \delta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_a} $ In $ \mu_a(t) $ 2.914 Data fitting
$ a_{1\mu_a} $ In $ \mu_a(t) $ -0.4986 Data fitting
$ a_{2\mu_a} $ In $ \mu_a(t) $ 0.03099 Data fitting
$ a_{3\mu_a} $ In $ \mu_a(t) $ -0.0008236 Data fitting
$ a_{4\mu_a} $ In $ \mu_a(t) $ 7.975$ \times 10^{-6} $ Data fitting
$ a_{0\theta} $ In $ \theta(t) $ 8.044$ \times 10^{-5} $ Data fitting
$ a_{1\theta} $ In $ \theta(t) $ 11.386 Data fitting
$ a_{2\theta} $ In $ \theta(t) $ 40.1461 Data fitting
$ \epsilon_{\theta} $ In $ \theta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_m} $ In $ \mu_m(t) $ 0.1901 Data fitting
$ a_{1\mu_m} $ In $ \mu_m(t) $ -0.0134 Data fitting
$ a_{2\mu_m} $ In $ \mu_m(t) $ 2.739$ \times 10^{-4} $ Data fitting
$ a_{0\gamma_m} $ In $ \gamma_m(t) $ 5$ \times 10^{4/3} $ Data fitting
$ a_{1\gamma_m} $ In $ \gamma_m(t) $ 0.0768 Data fitting
$ \beta_{mh} $ In $ \beta_m(t) $ 18.9871 Data fitting
$ N_{\beta_m} $ In $ \beta_m(t) $ 7 Data fitting
$ \epsilon_{\beta m} $ In $ \beta_m(t) $ $ 10^{-6} $ Data fitting
$ a_{\beta m} $ In $ \beta_m(t) $ 0 Data fitting
$ a_{0\beta_h} $ In $ \beta_h(t) $ 1.044$ \times 10^{-3} $ Data fitting
$ a_{1\beta_h} $ In $ \beta_h(t) $ 12.286 Data fitting
$ a_{2\beta_h} $ In $ \beta_h(t) $ 32.461 Data fitting
$ \epsilon_{\beta h} $ In $ \beta_h(t) $ $ 10^{-6} $ Data fitting
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Parameter Description Value Reference
$ k $ Fraction of female larvae from eggs 0.5 (0-1) [18,26]
$ b $ Per capita biting rate 0.1 [6,26]
$ \mu_h $ Natural death rate of humans 4.22$ \times 10^{-5} $ d$ ^{-1} $ Calculated, [16]
$ 1/\gamma_h $ Intrinsic period 10 days [6,16,18,26]
$ \alpha_h $ Human recovery rate 0.1 d$ ^{-1} $ [18,26]
$ \delta_m $ In $ \delta(t) $ 9.531 Data fitting
$ \delta_h $ In $ \delta(t) $ 22.55 Data fitting
$ N_{\delta} $ In $ \delta(t) $ 7.084 Data fitting
$ a_{\delta} $ In $ \delta(t) $ 0 Data fitting
$ \epsilon_{\delta} $ In $ \delta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_a} $ In $ \mu_a(t) $ 2.914 Data fitting
$ a_{1\mu_a} $ In $ \mu_a(t) $ -0.4986 Data fitting
$ a_{2\mu_a} $ In $ \mu_a(t) $ 0.03099 Data fitting
$ a_{3\mu_a} $ In $ \mu_a(t) $ -0.0008236 Data fitting
$ a_{4\mu_a} $ In $ \mu_a(t) $ 7.975$ \times 10^{-6} $ Data fitting
$ a_{0\theta} $ In $ \theta(t) $ 8.044$ \times 10^{-5} $ Data fitting
$ a_{1\theta} $ In $ \theta(t) $ 11.386 Data fitting
$ a_{2\theta} $ In $ \theta(t) $ 40.1461 Data fitting
$ \epsilon_{\theta} $ In $ \theta(t) $ $ 10^{-6} $ Data fitting
$ a_{0\mu_m} $ In $ \mu_m(t) $ 0.1901 Data fitting
$ a_{1\mu_m} $ In $ \mu_m(t) $ -0.0134 Data fitting
$ a_{2\mu_m} $ In $ \mu_m(t) $ 2.739$ \times 10^{-4} $ Data fitting
$ a_{0\gamma_m} $ In $ \gamma_m(t) $ 5$ \times 10^{4/3} $ Data fitting
$ a_{1\gamma_m} $ In $ \gamma_m(t) $ 0.0768 Data fitting
$ \beta_{mh} $ In $ \beta_m(t) $ 18.9871 Data fitting
$ N_{\beta_m} $ In $ \beta_m(t) $ 7 Data fitting
$ \epsilon_{\beta m} $ In $ \beta_m(t) $ $ 10^{-6} $ Data fitting
$ a_{\beta m} $ In $ \beta_m(t) $ 0 Data fitting
$ a_{0\beta_h} $ In $ \beta_h(t) $ 1.044$ \times 10^{-3} $ Data fitting
$ a_{1\beta_h} $ In $ \beta_h(t) $ 12.286 Data fitting
$ a_{2\beta_h} $ In $ \beta_h(t) $ 32.461 Data fitting
$ \epsilon_{\beta h} $ In $ \beta_h(t) $ $ 10^{-6} $ Data fitting
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