B. Adams and D. D. Kapan, Man Bites Mosquito: Understanding the Contribution of Human Movement to Vector-Borne Disease Dynamics, PLoS ONE, vol.3, issue.8, p.6763, 2009.
DOI : 10.1371/journal.pone.0006763.s001

M. Alvim, A. Iggidr, J. Koiler, G. Sallet, M. L. Penna et al., Onset of a vector borne disease due to human circulation?uniform, local and network reproduction ratios, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00839351

G. Añez and M. Rios, Dengue in the united states of america: a worsening scenario? BioMed research international, 2013.

P. Auger, E. Kouokam, G. Sallet, M. Tchuente, and B. Tsanou, The Ross???Macdonald model in a patchy environment, Mathematical Biosciences, vol.216, issue.2, pp.123-131, 2008.
DOI : 10.1016/j.mbs.2008.08.010

N. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 1975.

B. Bollobás, Modern graph theory, 1998.
DOI : 10.1007/978-1-4612-0619-4

L. Cai, S. Guo, X. Li, and M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos, Solitons & Fractals, vol.42, issue.4, pp.2297-2304, 2009.
DOI : 10.1016/j.chaos.2009.03.130

F. R. Chung, Spectral graph theory, volume 92 of CBMS regional conference series in mathematics, 1996.

C. Cosner, J. Beier, R. Cantrell, D. Impoinvil, L. Kapitanski et al., The effects of human movement on the persistence of vector-borne diseases, Journal of Theoretical Biology, vol.258, issue.4, pp.550-560, 2009.
DOI : 10.1016/j.jtbi.2009.02.016

O. Diekmann and J. Heesterbeek, Mathematical epidemiology of infectious diseases: model building, analysis and interpretation . Wiley series in mathematical and computational biology, 2000.

O. Diekmann, J. A. Heesterbeek, and J. A. Metz, On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, vol.28, issue.4, pp.365-382, 1990.
DOI : 10.1007/BF00178324

K. Dietz, Transmission and control of arbovirus diseases, pp.104-121, 1975.

D. Ding, X. Wang, and X. Ding, Global Stability of Multigroup Dengue Disease Transmission Model, Journal of Applied Mathematics, vol.35, issue.3, 2012.
DOI : 10.1017/CBO9780511530043

J. Dushoff and S. Levin, The effects of population heterogeneity on disease invasion, Mathematical Biosciences, vol.128, issue.1-2, pp.25-40, 1995.
DOI : 10.1016/0025-5564(94)00065-8

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Mathematical Biosciences, vol.150, issue.2, pp.131-151, 1998.
DOI : 10.1016/S0025-5564(98)10003-2

A. Fall, A. Iggidr, G. Sallet, and J. J. Tewa, Epidemiological Models and Lyapunov Functions, Mathematical Modelling of Natural Phenomena, vol.2, issue.1, pp.55-73, 2007.
DOI : 10.1051/mmnp:2008011

URL : https://hal.archives-ouvertes.fr/inria-00596236

H. Freedman and J. So, Global stability and persistence of simple food chains, Mathematical Biosciences, vol.76, issue.1, pp.69-86, 1985.
DOI : 10.1016/0025-5564(85)90047-1

B. Goh, Global stability in a class of prey-predator models, Bulletin of Mathematical Biology, vol.7, issue.4, pp.525-533, 1978.
DOI : 10.1007/BF02460776

B. Goh, Stability in Models of Mutualism, The American Naturalist, vol.113, issue.2, pp.261-275, 1979.
DOI : 10.1086/283384

B. Goh, Management and analysis of biological populations, 1980.

B. S. Goh, Global Stability in Many-Species Systems, The American Naturalist, vol.111, issue.977, pp.135-143, 1977.
DOI : 10.1086/283144

D. Gubler, Dengue and dengue hemorrhagic fever, Clinical Microbiology Review, vol.11, pp.480-496, 1998.

D. J. Gubler, The changing epidemiology of yellow fever and dengue, 1900 to 2003: full circle?, Comparative Immunology, Microbiology and Infectious Diseases, vol.27, issue.5, pp.319-330, 1900.
DOI : 10.1016/j.cimid.2004.03.013

H. Guo, M. Y. Li, and Z. Shuai, Global stability of the endemic equilibrium of multigroup sir epidemic models, Can. Appl. Math. Q, vol.14, issue.3, pp.259-284, 2006.

H. Guo, M. Y. Li, and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proceedings of the, pp.2793-2802, 2008.
DOI : 10.1090/S0002-9939-08-09341-6

H. Guo, M. Y. Li, and Z. Shuai, Global Dynamics of a General Class of Multistage Models for Infectious Diseases, SIAM Journal on Applied Mathematics, vol.72, issue.1, pp.261-279, 2012.
DOI : 10.1137/110827028

G. W. Harrison, Global Stability of Food Chains, The American Naturalist, vol.114, issue.3, pp.455-457, 1979.
DOI : 10.1086/283493

G. W. Harrison, Global stability of predator-prey interactions, Journal of Mathematical Biology, vol.18, issue.2, pp.159-171, 1979.
DOI : 10.1007/BF00279719

G. Hasibeder and C. Dye, Population dynamics of mosquito-borne disease: Persistence in a completely heterogeneous environment, Theoretical Population Biology, vol.33, issue.1, pp.31-53, 1988.
DOI : 10.1016/0040-5809(88)90003-2

H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Mathematical Biosciences, vol.75, issue.2, pp.205-227, 1985.
DOI : 10.1016/0025-5564(85)90038-0

H. W. Hethcote and J. Yorke, Gonorrhea : transmission dynamics and control, Lect. Notes Biomath, vol.56, 1984.
DOI : 10.1007/978-3-662-07544-9

M. W. Hirsch and H. L. Smith, Monotone dynamical systems In Handbook of differential equations: ordinary differential equations, pp.239-357, 2005.

N. A. Honorio, R. M. Nogueira, C. T. Codeco, M. S. Carvalho, O. G. Cruz et al., Spatial Evaluation and Modeling of Dengue Seroprevalence and Vector Density in Rio de Janeiro, Brazil, PLoS Neglected Tropical Diseases, vol.72, issue.11, p.545, 2009.
DOI : 10.1371/journal.pntd.0000545.s001

N. A. Honório, W. D. Silva, P. J. Leite, J. M. Gonçalves, L. P. Lounibos et al., Dispersal of Aedes aegypti and Aedes albopictus (Diptera: Culicidae) in an urban endemic dengue area in the State of Rio de Janeiro, Brazil, Memórias do Instituto Oswaldo Cruz, pp.98191-198, 2003.
DOI : 10.1590/S0074-02762003000200005

S. Hsu, On global stability of a predator-prey system, Mathematical Biosciences, vol.39, issue.1-2, pp.1-10, 1978.
DOI : 10.1016/0025-5564(78)90025-1

S. Hsu, A survey of constructing lyapunov functions for mathematical models in population biology, Taiwanese Journal of Mathematics, vol.9, issue.2, p.151, 2005.

G. Huang, X. Liu, and Y. Takeuchi, Lyapunov Functions and Global Stability for Age-Structured HIV Infection Model, SIAM Journal on Applied Mathematics, vol.72, issue.1, pp.25-38, 2012.
DOI : 10.1137/110826588

A. Iggidr, J. Kamgang, G. Sallet, and J. Tewa, Global Analysis of New Malaria Intrahost Models with a Competitive Exclusion Principle, SIAM Journal on Applied Mathematics, vol.67, issue.1, pp.260-278, 2006.
DOI : 10.1137/050643271

URL : https://hal.archives-ouvertes.fr/inria-00552014

J. A. Jacquez and C. P. Simon, Qualitative Theory of Compartmental Systems, SIAM Review, vol.35, issue.1, pp.43-79, 1993.
DOI : 10.1137/1035003

C. Ji, D. Jiang, and N. Shi, Multigroup sir epidemic model with stochastic perturbation. Physica A: Statistical Mechanics and its Applications, pp.1747-1762, 2011.

D. E. Knuth, Fundamental Algorithms, volume 1 of The art of computer programming, 1997.

A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Applied Mathematics Letters, vol.14, issue.6, pp.697-699, 2001.
DOI : 10.1016/S0893-9659(01)80029-X

A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models, Mathematical Medicine and Biology, vol.21, issue.2, pp.75-83, 2004.
DOI : 10.1093/imammb/21.2.75

A. Korobeinikov, Lyapunov Functions and Global Stability for SIR and SIRS Epidemiological Models with Non-Linear Transmission, Bulletin of Mathematical Biology, vol.180, issue.8, pp.615-626, 2006.
DOI : 10.1007/s11538-005-9037-9

A. Korobeinikov, Global Properties of SIR and SEIR Epidemic Models with??Multiple Parallel Infectious Stages, Bulletin of Mathematical Biology, vol.180, issue.1, pp.75-83, 2009.
DOI : 10.1007/s11538-008-9352-z

A. Korobeinikov and P. K. Maini, A lyapunov function and global properties for sir and seir epidemiological models with nonlinear incidence, Mathematical Biosciences and Engineering, vol.1, issue.1, pp.57-60, 2004.

T. Kuniya, Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear Analysis: Real World Applications, vol.12, issue.5, pp.2640-2655, 2011.
DOI : 10.1016/j.nonrwa.2011.03.011

A. Lajmanovich and J. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Mathematical Biosciences, vol.28, issue.3-4, pp.221-236, 1976.
DOI : 10.1016/0025-5564(76)90125-5

L. Lambrechts, T. W. Scott, and D. J. Gubler, Consequences of the Expanding Global Distribution of Aedes albopictus for Dengue Virus Transmission, PLoS Neglected Tropical Diseases, vol.2, issue.5, p.646, 2010.
DOI : 10.1371/journal.pntd.0000646.s001

J. Li, Y. Xiao, F. Zhang, and Y. Yang, An algebraic approach to proving the global stability of a class of epidemic models, Nonlinear Analysis: Real World Applications, vol.13, issue.5, pp.2006-2016, 2012.
DOI : 10.1016/j.nonrwa.2011.12.022

M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, vol.160, issue.2, pp.191-213, 1999.
DOI : 10.1016/S0025-5564(99)00030-9

M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, Journal of Differential Equations, vol.248, issue.1, pp.1-20, 2010.
DOI : 10.1016/j.jde.2009.09.003

M. Y. Li, Z. Shuai, and C. Wang, Global stability of multi-group epidemic models with distributed delays, Journal of Mathematical Analysis and Applications, vol.361, issue.1, pp.38-47, 2010.
DOI : 10.1016/j.jmaa.2009.09.017

D. G. Luenberger, Introduction to dynamic systems. Theory, models, and applications, 1979.

P. Magal and C. Mccluskey, Two-Group Infection Age Model Including an Application to Nosocomial Infection, SIAM Journal on Applied Mathematics, vol.73, issue.2, pp.1058-1095, 2013.
DOI : 10.1137/120882056

H. Mckenzie, Y. Jin, J. Jacobsen, and M. Lewis, $R_0$ Analysis of a Spatiotemporal Model for a Stream Population, SIAM Journal on Applied Dynamical Systems, vol.11, issue.2, pp.567-596, 2012.
DOI : 10.1137/100802189

J. W. Moon, Counting labelled trees, canadian mathematical monographs, Canadian Mathematical Congress, 1970.

E. A. Mpolya, K. Yashima, H. Ohtsuki, and A. Sasaki, Epidemic dynamics of a vector-borne disease on a villages-and-city star network with commuters, Journal of Theoretical Biology, vol.343, 2013.
DOI : 10.1016/j.jtbi.2013.11.024

Y. Muroya, Y. Enatsu, and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Analysis: Real World Applications, vol.14, issue.3, pp.1693-1704, 2013.
DOI : 10.1016/j.nonrwa.2012.11.005

H. Nishiura, Mathematical and statistical analyses of the spread of dengue, Dengue Bulletin, vol.30, pp.51-67, 2006.

A. Nold, Heterogeneity in disease-transmission modeling, Mathematical Biosciences, vol.52, issue.3-4, pp.227-240, 1980.
DOI : 10.1016/0025-5564(80)90069-3

A. M. Powers, A. C. Brault, R. B. Tesh, and S. C. Weaver, Re-emergence of chikungunya and o???nyong-nyong viruses: evidence for distinct geographical lineages and distant evolutionary relationships, Journal of General Virology, vol.81, issue.2, pp.81471-81480, 2000.
DOI : 10.1099/0022-1317-81-2-471

R. Ross, The prevention of malaria, 1911.

S. Rushton and A. J. Mautner, THE DETERMINISTIC MODEL OF A SIMPLE EPIDEMIC FOR MORE THAN ONE COMMUNITY, Biometrika, vol.42, issue.1-2, pp.126-132, 1955.
DOI : 10.1093/biomet/42.1-2.126

Z. Shuai and P. Van-den-driessche, Impact of heterogeneity on the dynamics of an SEIR epidemic model, Mathematical Biosciences and Engineering, vol.9, issue.2, pp.393-411, 2012.
DOI : 10.3934/mbe.2012.9.393

Z. Shuai and P. Van-den-driessche, Global Stability of Infectious Disease Models Using Lyapunov Functions, SIAM Journal on Applied Mathematics, vol.73, issue.4, pp.1513-1532, 2013.
DOI : 10.1137/120876642

D. L. Smith, K. E. Battle, S. I. Hay, C. M. Barker, T. W. Scott et al., Ross, Macdonald, and a Theory for the Dynamics and Control of Mosquito-Transmitted Pathogens, PLoS Pathogens, vol.3, issue.4, pp.1002588-1002592
DOI : 10.1371/journal.ppat.1002588.s001

D. L. Smith, F. E. Dushoff, and . Mckenzie, The Risk of a Mosquito-Borne Infectionin a Heterogeneous Environment, PLoS Biology, vol.94, issue.11, p.368, 2004.
DOI : 10.1371/journal.pbio.0020368.sd001

D. L. Smith, T. A. Perkins, R. C. Reiner, C. M. Barker, T. Niu et al., Recasting the theory of mosquito-borne pathogen transmission dynamics and control, Transactions of the Royal Society of Tropical Medicine and Hygiene, vol.108, issue.4, pp.185-197, 2014.
DOI : 10.1093/trstmh/tru026

H. L. Smith, Monotone dynamical systems, 1995.
DOI : 10.1090/surv/041/01

M. O. Souza, Multiscale analysis for a vector-borne epidemic model, Journal of Mathematical Biology, vol.137, issue.8, pp.1269-1293, 2014.
DOI : 10.1007/s00285-013-0666-6

M. O. Souza and J. P. Zubelli, Global Stability for a Class of Virus Models with??Cytotoxic T Lymphocyte Immune Response and??Antigenic Variation, Bulletin of Mathematical Biology, vol.48, issue.5???6, pp.609-625, 2011.
DOI : 10.1007/s11538-010-9543-2

S. T. Stoddard, A. C. Morrison, G. M. Vazquez-prokopec, V. Paz-soldan, T. J. Kochel et al., The Role of Human Movement in the Transmission of Vector-Borne Pathogens, PLoS Neglected Tropical Diseases, vol.82, issue.6, pp.481-488, 2009.
DOI : 10.1371/journal.pntd.0000481.s002

R. Sun and J. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates, Applied Mathematics and Computation, vol.218, issue.2, pp.280-286, 2011.
DOI : 10.1016/j.amc.2011.05.056

M. Teurlai, R. Huy, B. Cazelles, R. Duboz, C. Baehr et al., Can Human Movements Explain Heterogeneous Propagation of Dengue Fever in Cambodia?, PLoS Neglected Tropical Diseases, vol.123, issue.12, pp.1957-2012
DOI : 10.1371/journal.pntd.0001957.s009

URL : https://hal.archives-ouvertes.fr/hal-01495165

H. R. Thieme, Global asymptotic stability in epidemic models, Equadiff 82 number 1017 in Lectures Notes in Biomath, pp.608-615, 1982.
DOI : 10.1007/BF02320701

H. R. Thieme, Convergence results and a poincaré-bendixson trichotomy for asymptotically autonomous differential equations, Journal of mathematical biology, vol.30, issue.7, pp.755-763, 1992.

H. R. Thieme, Mathematics in population biology Princeton Series in Theoretical and Computational Biology, 2003.

H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators, Journal of Differential Equations, vol.250, issue.9, pp.3772-3801, 2011.
DOI : 10.1016/j.jde.2011.01.007

P. Van-den-driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, vol.180, issue.1-2, pp.29-48, 2002.
DOI : 10.1016/S0025-5564(02)00108-6

M. Vidyasagar, Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability, IEEE Transactions on Automatic Control, vol.25, issue.4, pp.773-779, 1980.
DOI : 10.1109/TAC.1980.1102422

J. Wang, J. Pang, and X. Liu, Modelling diseases with relapse and nonlinear incidence of infection: a multi-group epidemic model, Journal of Biological Dynamics, vol.3, issue.1, pp.99-116, 2014.
DOI : 10.1016/j.matcom.2008.02.007

M. J. Wonham, M. A. Lewis, J. Renclawowicz, and P. Van-den-driessche, Transmission assumptions generate conflicting predictions in host-vector disease models: a case study in West Nile virus, Ecology Letters, vol.4, issue.6, pp.706-725, 2006.
DOI : 10.1089/vbz.2005.5.40

Y. Xiao and X. Zou, Transmission dynamics for vector-borne diseases in a patchy environment, Journal of Mathematical Biology, vol.3, issue.1, pp.113-146, 2014.
DOI : 10.1007/s00285-013-0695-1

H. Yang, H. Wei, and X. Li, Global stability of an epidemic model for vector-borne disease, Journal of Systems Science and Complexity, vol.20, issue.2, pp.279-292, 2010.
DOI : 10.1007/s11424-010-8436-7

J. Yu, D. Jiang, and N. Shi, Global stability of two-group SIR model with random perturbation, Journal of Mathematical Analysis and Applications, vol.360, issue.1, pp.235-244, 2009.
DOI : 10.1016/j.jmaa.2009.06.050