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December  2020, 25(12): 4677-4701. doi: 10.3934/dcdsb.2020119

Effects of travel frequency on the persistence of mosquito-borne diseases

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

* Corresponding author: Daozhou Gao (dzgao@shnu.edu.cn)

Received  October 2019 Published  March 2020

Fund Project: This work was partially supported by National Natural Science Foundation of China grant 11601336, Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (TP2015050), and Shanghai Gaofeng Project for University Academic Development Program

Travel frequency of people varies widely with occupation, age, gender, ethnicity, income, climate and other factors. Meanwhile, the distribution of the numbers of times people in different regions or with different travel behaviors bitten by mosquitoes may be nonuniform. To reflect these two heterogeneities, we develop a multipatch model to study the impact of travel frequency and human biting rate on the spatial spread of mosquito-borne diseases. The human population in each patch is divided into four classes: susceptible unfrequent, infectious unfrequent, susceptible frequent, and infectious frequent. The basic reproduction number $ \mathcal{R}_0 $ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable if $ \mathcal{R}_0\leq 1 $, and there is a unique endemic equilibrium that is globally asymptotically stable if $ \mathcal{R}_0>1 $. A more detailed study is conducted on the single patch model. We use analytical and numerical methods to demonstrate that the model without considering the difference of humans in travel frequency mostly underestimates the risk of infection. Numerical simulations suggest that the greater the difference in travel frequency, the larger the underestimate of the transmission potential. In addition, the basic reproduction number $ \mathcal{R}_0 $ may decreasingly, or increasingly, or nonmonotonically vary when more people travel frequently.

Citation: Xianyun Chen, Daozhou Gao. Effects of travel frequency on the persistence of mosquito-borne diseases. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4677-4701. doi: 10.3934/dcdsb.2020119
References:
[1]

J. AlegreS. Mateo and L. Pou, Participation in tourism consumption and the intensity of participation: an analysis of their socio-demographic and economic determinants, Tourism Econo., 15 (2009), 531-546.  doi: 10.5367/000000009789036521.  Google Scholar

[2]

M. Anjomruz, M. A. Oshaghi, A. A. Pourfatollah, et al., Preferential feeding success of laboratory reared Anopheles stephensi mosquitoes according to ABO blood group status, Acta Trop., 140 (2014), 118-123. doi: 10.1016/j.actatropica.2014.08.012.  Google Scholar

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J. L. Aron, Mathematical modelling of immunity to malaria, Math. Biosci., 90 (1988), 385-396.  doi: 10.1016/0025-5564(88)90076-4.  Google Scholar

[4]

J. L. Aron and R. M. May, The population dynamics of malaria, in The Population Dynamics of Infectious Diseases: Theory and Applications (eds. R. M. Anderson), Springer, (1982), 139–179. doi: 10.1007/978-1-4899-2901-3.  Google Scholar

[5]

P. AugerE. KouokamG. SalletM. Tchuente and B. Tsanou, The Ross–Macdonald model in a patchy environment, Math. Biosci., 216 (2008), 123-131.  doi: 10.1016/j.mbs.2008.08.010.  Google Scholar

[6]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. E. Axelrod, M. Kimmel and M. Langlais), Wuerz, Winnipeg, Canada, (1995), 33–50. Google Scholar

[7]

N. ChitnisJ. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar

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C. Cosner, Models for the effects of host movement in vector-borne disease systems, Math. Biosci., 270 (2015), 192-197.  doi: 10.1016/j.mbs.2015.06.015.  Google Scholar

[9]

C. CosnerJ. C. BeierR. S. CantrellD. ImpoinvilL. KapitanskiM. D. PottsA. Troyo and S. Ruan, The effects of human movement on the persistence of vector-borne diseases, J. Theor. Biol., 258 (2009), 550-560.  doi: 10.1016/j.jtbi.2009.02.016.  Google Scholar

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J. M. Denstadli, Analysing air travel: a comparison of different survey methods and data collection procedures, J. Travel Res., 39 (2000), 4-10.  doi: 10.1177/004728750003900102.  Google Scholar

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O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

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C. Dye and G. Hasibeder, Population dynamics of mosquito-borne disease: effects of flies which bite some people more frequently than others, Trans. R. Soc. Trop. Med. Hyg., 80 (1986), 69-77.  doi: 10.1016/0035-9203(86)90199-9.  Google Scholar

[13]

J. L. Gallup and J. D. Sachs, The economic burden of malaria, Am. J. Trop. Med. Hyg., 64 (2001), 85-96.  doi: 10.4269/ajtmh.2001.64.85.  Google Scholar

[14]

D. Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math., 79 (2019), 1581-1606.  doi: 10.1137/18M1211957.  Google Scholar

[15]

D. GaoA. AmzaB. NassirouB. KadriN. Sippl-SwezeyF. LiuS. F. AckleyT. M. Lietman and T. C. Porco, Optimal seasonal timing of oral azithromycin for malaria, Am. J. Trop. Med. Hyg., 91 (2014), 936-942.  doi: 10.4269/ajtmh.13-0474.  Google Scholar

[16]

D. Gao and C. Dong, Fast diffusion inhibits disease outbreaks, Proc. Amer. Math. Soc., 148 (2020), 1709-1722.  doi: 10.1090/proc/14868.  Google Scholar

[17]

D. Gao, Y. Lou, D. He, T. C. Porco, Y. Kuang, G. Chowell and S. Ruan, Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Sci. Rep., 6 (2016), 28070. doi: 10.1038/srep28070.  Google Scholar

[18]

D. GaoY. Lou and S. Ruan, A periodic Ross–Macdonald model in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3133-3145.  doi: 10.3934/dcdsb.2014.19.3133.  Google Scholar

[19]

D. Gao and S. Ruan, A multipatch malaria model with logistic growth populations, SIAM J. Appl. Math., 72 (2012), 819-841.  doi: 10.1137/110850761.  Google Scholar

[20]

D. Gao and S. Ruan, Malaria models with spatial effects, in Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases (eds. D. Chen, B. Moulin and J. Wu), John Wiley & Sons, (2014), 109–136. doi: 10.1002/9781118630013.ch6.  Google Scholar

[21]

D. GaoP. van den Driessche and C. Cosner, Habitat fragmentation promotes malaria persistence, J. Math. Biol., 79 (2019), 2255-2280.  doi: 10.1007/s00285-019-01428-2.  Google Scholar

[22]

N. G. Gratz, Emerging and resurging vector-borne diseases, Annu. Rev. Entomol., 44 (1999), 51-75.  doi: 10.1146/annurev.ento.44.1.51.  Google Scholar

[23]

M. G. Guzman and E. Harris, Dengue, Lancet, 385 (2015), 453-465.  doi: 10.1016/S0140-6736(14)60572-9.  Google Scholar

[24]

G. Harrison Mosquitoes, Malaria and Man: a History of the Hostilities since 1880, John Murray, London, 1978. Google Scholar

[25]

G. Hasibeder and C. Dye, Population dynamics of mosquito-borne disease: persistence in a completely heterogeneous environment, Theor. Popu. Biol., 33 (1988), 31-53.  doi: 10.1016/0040-5809(88)90003-2.  Google Scholar

[26]

T. D. HollingsworthN. M. Ferguson and R. M. Anderson, Frequent travelers and rate of spread of epidemics, Emerg. Infect. Dis., 13 (2007), 1288-1294.  doi: 10.3201/eid1309.070081.  Google Scholar

[27] R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^nd$ edition, Cambridge University Press, New York, 2013.   Google Scholar
[28]

J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance, Malar. J., 2 (2003), 3. doi: 10.1186/1475-2875-2-3.  Google Scholar

[29]

R. S. Lanciotti, J. T. Roehrig, V. Deubel, et al., Origin of the West Nile virus responsible for an outbreak of encephalitis in the northeastern United States, Science, 286 (1999), 2333-2337. doi: 10.1126/science.286.5448.2333.  Google Scholar

[30]

S. LimJ. K. Lim and I. Yoon, An update on Zika virus in Asia, Infect. Chemother., 49 (2017), 91-100.  doi: 10.3947/ic.2017.49.2.91.  Google Scholar

[31]

N. LosadaE. AlénT. Domínguez and J. L. Nicolau, Travel frequency of seniors tourists, Tour. Manag., 53 (2016), 88-95.  doi: 10.1016/j.tourman.2015.09.013.  Google Scholar

[32]

Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 2384-2407.  doi: 10.1007/s11538-011-9628-6.  Google Scholar

[33] G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957.   Google Scholar
[34]

S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria–a review, Malar. J., 10 (2011), 202. doi: 10.1186/1475-2875-10-202.  Google Scholar

[35]

P. Martens and L. Hall, Malaria on the move: human population movement and malaria transmission, Emerg. Infect. Dis., 6 (2000), 103-109.  doi: 10.3201/eid0602.000202.  Google Scholar

[36]

P. E. Parham and E. Michael, Modeling the effects of weather and climate change on malaria transmission, Environ. Health Perspect., 118 (2010), 620-626.  doi: 10.1289/ehp.0901256.  Google Scholar

[37]

G. R. PortP. F. L. Boreham and J. H. Bryan, The relationship of host size to feeding by mosquitoes of the Anopheles gambiae Giles complex (Diptera: Culicidae), Bull. Entomol. Res., 70 (1980), 133-144.  doi: 10.1017/S0007485300009834.  Google Scholar

[38]

R. C. Reiner, T. A. Perkins, C. M. Barker, et al., A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010, J. R. Soc. Interface, 10 (2013), 20120921. doi: 10.1098/rsif.2012.0921.  Google Scholar

[39]

A. Robinson, A. O. Busula, M. A. Voets, et al., Plasmodium-associated changes in human odor attract mosquitoes, Proc. Natl. Acad. Sci. USA, 115 (2018), E4209–E4218. doi: 10.1073/pnas.1721610115.  Google Scholar

[40]

R. Ross, The Prevention of Malaria, John Murray, London, 1911. Google Scholar

[41]

S. RuanD. Xiao and J. C. Beier, On the delayed Ross–Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114.  doi: 10.1007/s11538-007-9292-z.  Google Scholar

[42]

H. L. Smith, Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative Systems, Vol 41, Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[43]

J. Sutcliffe, X. Ji and S. Yin, How many holes is too many? A prototype tool for estimating mosquito entry risk into damaged bed nets, Malar. J., 16 (2017), 304. doi: 10.1186/s12936-017-1951-4.  Google Scholar

[44]

A. J. Tatem and D. L. Smith, International population movements and regional Plasmodium falciparum malaria elimination strategies, Proc. Natl. Acad. Sci. USA, 107 (2010), 12222-12227.  doi: 10.1073/pnas.1002971107.  Google Scholar

[45]

S. Tilley and D. Houston, The gender turnaround: young women now travelling more than young men, J. Transp. Geogr., 54 (2016), 349-358.  doi: 10.1016/j.jtrangeo.2016.06.022.  Google Scholar

[46]

U.S. Department of Transportation–Federal Highway Administration, Summary of Travel Trends: 2017 National Household Travel Survey, 2018. Available from: https://nhts.ornl.gov/assets/2017-nhts-summary-travel-trends.pdf. Google Scholar

[47]

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

[48]

P. F. M. Verdonschot and A. A. Besse-Lototskaya, Flight distance of mosquitoes (Culicidae): a metadata analysis to support the management of barrier zones around rewetted and newly constructed wetlands, Limnologica, 45 (2014), 69-79.  doi: 10.1016/j.limno.2013.11.002.  Google Scholar

[49]

World Health Organization, Yellow Fever Situation Report, 2016. Available from: https://www.who.int/emergencies/yellow-fever/situation-reports/28-october-2016/en/. Google Scholar

[50]

World Health Organization, World Malaria Report 2018, 2018. Available from: http://www.who.int/malaria/publications/world-malaria-report-2018/en. Google Scholar

[51]

WorldAtlas, Countries that Travel the Most, 2019. Available from: https://www.worldatlas.com/articles/countries-whose-citizens-travel-the-most.html. Google Scholar

[52]

X.-Q. Zhao, Dynamical Systems in Population Biology, $2^nd$ edition, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

[53]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Can. Appl. Math. Quart., 4 (1996), 421-444.   Google Scholar

show all references

References:
[1]

J. AlegreS. Mateo and L. Pou, Participation in tourism consumption and the intensity of participation: an analysis of their socio-demographic and economic determinants, Tourism Econo., 15 (2009), 531-546.  doi: 10.5367/000000009789036521.  Google Scholar

[2]

M. Anjomruz, M. A. Oshaghi, A. A. Pourfatollah, et al., Preferential feeding success of laboratory reared Anopheles stephensi mosquitoes according to ABO blood group status, Acta Trop., 140 (2014), 118-123. doi: 10.1016/j.actatropica.2014.08.012.  Google Scholar

[3]

J. L. Aron, Mathematical modelling of immunity to malaria, Math. Biosci., 90 (1988), 385-396.  doi: 10.1016/0025-5564(88)90076-4.  Google Scholar

[4]

J. L. Aron and R. M. May, The population dynamics of malaria, in The Population Dynamics of Infectious Diseases: Theory and Applications (eds. R. M. Anderson), Springer, (1982), 139–179. doi: 10.1007/978-1-4899-2901-3.  Google Scholar

[5]

P. AugerE. KouokamG. SalletM. Tchuente and B. Tsanou, The Ross–Macdonald model in a patchy environment, Math. Biosci., 216 (2008), 123-131.  doi: 10.1016/j.mbs.2008.08.010.  Google Scholar

[6]

C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models, in Mathematical Population Dynamics: Analysis of Heterogeneity (eds. O. Arino, D. E. Axelrod, M. Kimmel and M. Langlais), Wuerz, Winnipeg, Canada, (1995), 33–50. Google Scholar

[7]

N. ChitnisJ. M. Hyman and J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol., 70 (2008), 1272-1296.  doi: 10.1007/s11538-008-9299-0.  Google Scholar

[8]

C. Cosner, Models for the effects of host movement in vector-borne disease systems, Math. Biosci., 270 (2015), 192-197.  doi: 10.1016/j.mbs.2015.06.015.  Google Scholar

[9]

C. CosnerJ. C. BeierR. S. CantrellD. ImpoinvilL. KapitanskiM. D. PottsA. Troyo and S. Ruan, The effects of human movement on the persistence of vector-borne diseases, J. Theor. Biol., 258 (2009), 550-560.  doi: 10.1016/j.jtbi.2009.02.016.  Google Scholar

[10]

J. M. Denstadli, Analysing air travel: a comparison of different survey methods and data collection procedures, J. Travel Res., 39 (2000), 4-10.  doi: 10.1177/004728750003900102.  Google Scholar

[11]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[12]

C. Dye and G. Hasibeder, Population dynamics of mosquito-borne disease: effects of flies which bite some people more frequently than others, Trans. R. Soc. Trop. Med. Hyg., 80 (1986), 69-77.  doi: 10.1016/0035-9203(86)90199-9.  Google Scholar

[13]

J. L. Gallup and J. D. Sachs, The economic burden of malaria, Am. J. Trop. Med. Hyg., 64 (2001), 85-96.  doi: 10.4269/ajtmh.2001.64.85.  Google Scholar

[14]

D. Gao, Travel frequency and infectious diseases, SIAM J. Appl. Math., 79 (2019), 1581-1606.  doi: 10.1137/18M1211957.  Google Scholar

[15]

D. GaoA. AmzaB. NassirouB. KadriN. Sippl-SwezeyF. LiuS. F. AckleyT. M. Lietman and T. C. Porco, Optimal seasonal timing of oral azithromycin for malaria, Am. J. Trop. Med. Hyg., 91 (2014), 936-942.  doi: 10.4269/ajtmh.13-0474.  Google Scholar

[16]

D. Gao and C. Dong, Fast diffusion inhibits disease outbreaks, Proc. Amer. Math. Soc., 148 (2020), 1709-1722.  doi: 10.1090/proc/14868.  Google Scholar

[17]

D. Gao, Y. Lou, D. He, T. C. Porco, Y. Kuang, G. Chowell and S. Ruan, Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Sci. Rep., 6 (2016), 28070. doi: 10.1038/srep28070.  Google Scholar

[18]

D. GaoY. Lou and S. Ruan, A periodic Ross–Macdonald model in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3133-3145.  doi: 10.3934/dcdsb.2014.19.3133.  Google Scholar

[19]

D. Gao and S. Ruan, A multipatch malaria model with logistic growth populations, SIAM J. Appl. Math., 72 (2012), 819-841.  doi: 10.1137/110850761.  Google Scholar

[20]

D. Gao and S. Ruan, Malaria models with spatial effects, in Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases (eds. D. Chen, B. Moulin and J. Wu), John Wiley & Sons, (2014), 109–136. doi: 10.1002/9781118630013.ch6.  Google Scholar

[21]

D. GaoP. van den Driessche and C. Cosner, Habitat fragmentation promotes malaria persistence, J. Math. Biol., 79 (2019), 2255-2280.  doi: 10.1007/s00285-019-01428-2.  Google Scholar

[22]

N. G. Gratz, Emerging and resurging vector-borne diseases, Annu. Rev. Entomol., 44 (1999), 51-75.  doi: 10.1146/annurev.ento.44.1.51.  Google Scholar

[23]

M. G. Guzman and E. Harris, Dengue, Lancet, 385 (2015), 453-465.  doi: 10.1016/S0140-6736(14)60572-9.  Google Scholar

[24]

G. Harrison Mosquitoes, Malaria and Man: a History of the Hostilities since 1880, John Murray, London, 1978. Google Scholar

[25]

G. Hasibeder and C. Dye, Population dynamics of mosquito-borne disease: persistence in a completely heterogeneous environment, Theor. Popu. Biol., 33 (1988), 31-53.  doi: 10.1016/0040-5809(88)90003-2.  Google Scholar

[26]

T. D. HollingsworthN. M. Ferguson and R. M. Anderson, Frequent travelers and rate of spread of epidemics, Emerg. Infect. Dis., 13 (2007), 1288-1294.  doi: 10.3201/eid1309.070081.  Google Scholar

[27] R. A. Horn and C. R. Johnson, Matrix Analysis, 2$^nd$ edition, Cambridge University Press, New York, 2013.   Google Scholar
[28]

J. C. Koella and R. Antia, Epidemiological models for the spread of anti-malarial resistance, Malar. J., 2 (2003), 3. doi: 10.1186/1475-2875-2-3.  Google Scholar

[29]

R. S. Lanciotti, J. T. Roehrig, V. Deubel, et al., Origin of the West Nile virus responsible for an outbreak of encephalitis in the northeastern United States, Science, 286 (1999), 2333-2337. doi: 10.1126/science.286.5448.2333.  Google Scholar

[30]

S. LimJ. K. Lim and I. Yoon, An update on Zika virus in Asia, Infect. Chemother., 49 (2017), 91-100.  doi: 10.3947/ic.2017.49.2.91.  Google Scholar

[31]

N. LosadaE. AlénT. Domínguez and J. L. Nicolau, Travel frequency of seniors tourists, Tour. Manag., 53 (2016), 88-95.  doi: 10.1016/j.tourman.2015.09.013.  Google Scholar

[32]

Y. Lou and X.-Q. Zhao, Modelling malaria control by introduction of larvivorous fish, Bull. Math. Biol., 73 (2011), 2384-2407.  doi: 10.1007/s11538-011-9628-6.  Google Scholar

[33] G. Macdonald, The Epidemiology and Control of Malaria, Oxford University Press, London, 1957.   Google Scholar
[34]

S. Mandal, R. R. Sarkar and S. Sinha, Mathematical models of malaria–a review, Malar. J., 10 (2011), 202. doi: 10.1186/1475-2875-10-202.  Google Scholar

[35]

P. Martens and L. Hall, Malaria on the move: human population movement and malaria transmission, Emerg. Infect. Dis., 6 (2000), 103-109.  doi: 10.3201/eid0602.000202.  Google Scholar

[36]

P. E. Parham and E. Michael, Modeling the effects of weather and climate change on malaria transmission, Environ. Health Perspect., 118 (2010), 620-626.  doi: 10.1289/ehp.0901256.  Google Scholar

[37]

G. R. PortP. F. L. Boreham and J. H. Bryan, The relationship of host size to feeding by mosquitoes of the Anopheles gambiae Giles complex (Diptera: Culicidae), Bull. Entomol. Res., 70 (1980), 133-144.  doi: 10.1017/S0007485300009834.  Google Scholar

[38]

R. C. Reiner, T. A. Perkins, C. M. Barker, et al., A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010, J. R. Soc. Interface, 10 (2013), 20120921. doi: 10.1098/rsif.2012.0921.  Google Scholar

[39]

A. Robinson, A. O. Busula, M. A. Voets, et al., Plasmodium-associated changes in human odor attract mosquitoes, Proc. Natl. Acad. Sci. USA, 115 (2018), E4209–E4218. doi: 10.1073/pnas.1721610115.  Google Scholar

[40]

R. Ross, The Prevention of Malaria, John Murray, London, 1911. Google Scholar

[41]

S. RuanD. Xiao and J. C. Beier, On the delayed Ross–Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098-1114.  doi: 10.1007/s11538-007-9292-z.  Google Scholar

[42]

H. L. Smith, Monotone Dynamical Systems: an Introduction to the Theory of Competitive and Cooperative Systems, Vol 41, Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[43]

J. Sutcliffe, X. Ji and S. Yin, How many holes is too many? A prototype tool for estimating mosquito entry risk into damaged bed nets, Malar. J., 16 (2017), 304. doi: 10.1186/s12936-017-1951-4.  Google Scholar

[44]

A. J. Tatem and D. L. Smith, International population movements and regional Plasmodium falciparum malaria elimination strategies, Proc. Natl. Acad. Sci. USA, 107 (2010), 12222-12227.  doi: 10.1073/pnas.1002971107.  Google Scholar

[45]

S. Tilley and D. Houston, The gender turnaround: young women now travelling more than young men, J. Transp. Geogr., 54 (2016), 349-358.  doi: 10.1016/j.jtrangeo.2016.06.022.  Google Scholar

[46]

U.S. Department of Transportation–Federal Highway Administration, Summary of Travel Trends: 2017 National Household Travel Survey, 2018. Available from: https://nhts.ornl.gov/assets/2017-nhts-summary-travel-trends.pdf. Google Scholar

[47]

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

[48]

P. F. M. Verdonschot and A. A. Besse-Lototskaya, Flight distance of mosquitoes (Culicidae): a metadata analysis to support the management of barrier zones around rewetted and newly constructed wetlands, Limnologica, 45 (2014), 69-79.  doi: 10.1016/j.limno.2013.11.002.  Google Scholar

[49]

World Health Organization, Yellow Fever Situation Report, 2016. Available from: https://www.who.int/emergencies/yellow-fever/situation-reports/28-october-2016/en/. Google Scholar

[50]

World Health Organization, World Malaria Report 2018, 2018. Available from: http://www.who.int/malaria/publications/world-malaria-report-2018/en. Google Scholar

[51]

WorldAtlas, Countries that Travel the Most, 2019. Available from: https://www.worldatlas.com/articles/countries-whose-citizens-travel-the-most.html. Google Scholar

[52]

X.-Q. Zhao, Dynamical Systems in Population Biology, $2^nd$ edition, Springer-Verlag, New York, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

[53]

X.-Q. Zhao and Z.-J. Jing, Global asymptotic behavior in some cooperative systems of functional differential equations, Can. Appl. Math. Quart., 4 (1996), 421-444.   Google Scholar

Figure 1.  Flowchart of the mosquito-borne disease model in patch $ i $
Figure 2.  The contour plot of $ \mathcal{R}_0-\hat{ \mathcal{R}}_0 $ versus the relative travel rates $ \tau_{12} $ and $ \tau_{21} $
Figure 3.  The contour plot of $ \mathcal{R}_0 $ versus the relative frequency change rates $ \tau_1 $ and $ \tau_2 $. (a) simultaneously decreases, (b) simultaneously increases, (c) increases in $ \tau_1 $ but decreases $ \tau_2 $, (d) non-monotonically varies with $ \tau_1 $ and decreases in $ \tau_2 $
Table 1.  Descriptions and ranges of parameters (time unit is day)
Description Range
$ a_i $ mosquito biting rate 0.1–1
$ b_i^u $ transmission probability from an infectious mosquito 0.01–0.8
to a susceptible unfrequent traveler per bite
$ b_i^f $ transmission probability from an infectious mosquito 0.01–0.8
to a susceptible frequent traveler per bite
$ c_i^u $ transmission probability from an infectious unfrequent 0.072–0.64
traveler to a susceptible mosquito per bite
$ c_i^f $ transmission probability from an infectious frequent 0.072–0.64
traveler to a susceptible mosquito per bite
$ \gamma_i^u $ recovery rate of infectious unfrequent humans 0.005–0.05
$ \gamma_i^f $ recovery rate of infectious frequent humans 0.005–0.05
$ \mu_i $ mosquito mortality rate 0.05–0.2
$ \sigma_i $ relative HBR of frequent travelers to unfrequent travelers 0.2–5
$ c_{ij}^f $ travel rate of frequent travelers from patches $ j $ to $ i $ 0.03–0.1
$ \tau_{ij} $ relative travel rate of unfrequent travelers 0–0.4
to frequent travelers
$ c_{ij}^u $ travel rate of unfrequent travelers from patches $ j $ to $ i $ $ c_{ij}^u=\tau_{ij}c_{ij}^f $
$ d_{ij} $ travel rate of mosquitoes from patch $ j $ to patch $ i $ 0.001–0.03
$ \phi_i^f $ change rate from frequent travelers to $ 2.7 \times 10^{-4} $
unfrequent travelers $ 9 \times 10^{-4} $
$ \tau_i $ relative change rate of unfrequent travelers 0.1–0.5
to frequent travelers
$ \phi_i^u $ change rate from unfrequent travelers to $ \phi_i^u=\tau_i\phi_i^f $
frequent travelers
$ V/H $ ratio of mosquitoes to humans 1–10
Description Range
$ a_i $ mosquito biting rate 0.1–1
$ b_i^u $ transmission probability from an infectious mosquito 0.01–0.8
to a susceptible unfrequent traveler per bite
$ b_i^f $ transmission probability from an infectious mosquito 0.01–0.8
to a susceptible frequent traveler per bite
$ c_i^u $ transmission probability from an infectious unfrequent 0.072–0.64
traveler to a susceptible mosquito per bite
$ c_i^f $ transmission probability from an infectious frequent 0.072–0.64
traveler to a susceptible mosquito per bite
$ \gamma_i^u $ recovery rate of infectious unfrequent humans 0.005–0.05
$ \gamma_i^f $ recovery rate of infectious frequent humans 0.005–0.05
$ \mu_i $ mosquito mortality rate 0.05–0.2
$ \sigma_i $ relative HBR of frequent travelers to unfrequent travelers 0.2–5
$ c_{ij}^f $ travel rate of frequent travelers from patches $ j $ to $ i $ 0.03–0.1
$ \tau_{ij} $ relative travel rate of unfrequent travelers 0–0.4
to frequent travelers
$ c_{ij}^u $ travel rate of unfrequent travelers from patches $ j $ to $ i $ $ c_{ij}^u=\tau_{ij}c_{ij}^f $
$ d_{ij} $ travel rate of mosquitoes from patch $ j $ to patch $ i $ 0.001–0.03
$ \phi_i^f $ change rate from frequent travelers to $ 2.7 \times 10^{-4} $
unfrequent travelers $ 9 \times 10^{-4} $
$ \tau_i $ relative change rate of unfrequent travelers 0.1–0.5
to frequent travelers
$ \phi_i^u $ change rate from unfrequent travelers to $ \phi_i^u=\tau_i\phi_i^f $
frequent travelers
$ V/H $ ratio of mosquitoes to humans 1–10
Table 2.  Parameter settings for Figure 3
Symbol Figure 3a Figure 3b Figure 3c Figure 3d
$ a_1 $ 0.438 0.380 0.535 0.161
$ a_2 $ 0.402 0.100 0.147 0.051
$ b_1 $ 0.548 0.686 0.373 0.656
$ b_2 $ 0.642 0.159 0.178 0.716
$ c_1 $ 0.398 0.324 0.608 0.586
$ c_2 $ 0.636 0.085 0.618 0.403
$ \gamma_1 $ 0.049 0.044 0.044 0.009
$ \gamma_2 $ 0.026 0.029 0.014 0.020
$ \mu_1 $ 0.135 0.066 0.094 0.053
$ \mu_2 $ 0.165 0.194 0.174 0.103
$ \sigma_1 $ 1 1 1 1
$ \sigma_2 $ 1 1 1 1
$ c_{12}^u $ 0.021 0.006 0.0128 0.0028
$ c_{21}^u $ 0.018 0.002 0.0182 0.0005
$ c_{12}^f $ 0.094 0.060 0.0534 0.0550
$ c_{21}^f $ 0.092 0.076 0.0943 0.0497
$ d_{12} $ 0 0 0 0
$ d_{21} $ 0 0 0 0
$ \phi_1^u $ 7.73 $ \times 10^{-5} $ 2.80$ \times 10^{-4} $ 1.40$ \times 10^{-4} $ 1.50 $ \times 10^{-4} $
$ \phi_2^u $ 5.11$ \times 10^{-5} $ 3.46$ \times 10^{-4} $ 2.10$ \times 10^{-4} $ 1.50$ \times 10^{-4} $
$ \phi_1^f $ 7.16 $ \times 10^{-4} $ 7.58$ \times 10^{-4} $ 5.35$ \times 10^{-4} $ 4.82$ \times 10^{-4} $
$ \phi_2^f $ 3.24$ \times 10^{-4} $ 7.62$ \times 10^{-4} $ 4.67$ \times 10^{-4} $ 5.55$ \times 10^{-4} $
$ V_1 $ 5859 23633 6065 13838
$ V_2 $ 18790 29698 17129 19918
$ H $ 10000 10000 10000 10000
Symbol Figure 3a Figure 3b Figure 3c Figure 3d
$ a_1 $ 0.438 0.380 0.535 0.161
$ a_2 $ 0.402 0.100 0.147 0.051
$ b_1 $ 0.548 0.686 0.373 0.656
$ b_2 $ 0.642 0.159 0.178 0.716
$ c_1 $ 0.398 0.324 0.608 0.586
$ c_2 $ 0.636 0.085 0.618 0.403
$ \gamma_1 $ 0.049 0.044 0.044 0.009
$ \gamma_2 $ 0.026 0.029 0.014 0.020
$ \mu_1 $ 0.135 0.066 0.094 0.053
$ \mu_2 $ 0.165 0.194 0.174 0.103
$ \sigma_1 $ 1 1 1 1
$ \sigma_2 $ 1 1 1 1
$ c_{12}^u $ 0.021 0.006 0.0128 0.0028
$ c_{21}^u $ 0.018 0.002 0.0182 0.0005
$ c_{12}^f $ 0.094 0.060 0.0534 0.0550
$ c_{21}^f $ 0.092 0.076 0.0943 0.0497
$ d_{12} $ 0 0 0 0
$ d_{21} $ 0 0 0 0
$ \phi_1^u $ 7.73 $ \times 10^{-5} $ 2.80$ \times 10^{-4} $ 1.40$ \times 10^{-4} $ 1.50 $ \times 10^{-4} $
$ \phi_2^u $ 5.11$ \times 10^{-5} $ 3.46$ \times 10^{-4} $ 2.10$ \times 10^{-4} $ 1.50$ \times 10^{-4} $
$ \phi_1^f $ 7.16 $ \times 10^{-4} $ 7.58$ \times 10^{-4} $ 5.35$ \times 10^{-4} $ 4.82$ \times 10^{-4} $
$ \phi_2^f $ 3.24$ \times 10^{-4} $ 7.62$ \times 10^{-4} $ 4.67$ \times 10^{-4} $ 5.55$ \times 10^{-4} $
$ V_1 $ 5859 23633 6065 13838
$ V_2 $ 18790 29698 17129 19918
$ H $ 10000 10000 10000 10000
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