February  2018, 15(1): 1-20. doi: 10.3934/mbe.2018001

The interplay between models and public health policies: Regional control for a class of spatially structured epidemics (think globally, act locally)

1. 

ADAMSS, Universitá degli Studi di Milano, 20133 MILANO, Italy

2. 

Faculty of Mathematics, "Alexandru Ioan Cuza" University of Iași, "Octav Mayer" Institute of Mathematics of the Romanian Academy, Iași 700506, Romania

* Corresponding author: Vincenzo Capasso

Received  January 12, 2016 Accepted  October 30, 2016 Published  May 2017

Fund Project: The first author wishes to dedicate this review to the late Enea Grosso, Professor of Public Health and Hygiene in Bari, who had inspired most of the work presented here on man-environment epidemic systems

A review is presented here of the research carried out, by a group including the authors, on the mathematical analysis of epidemic systems. Particular attention is paid to recent analysis of optimal control problems related to spatially structured epidemics driven by environmental pollution. A relevant problem, related to the possible eradication of the epidemic, is the so called zero stabilization. In a series of papers, necessary conditions, and sufficient conditions of stabilizability have been obtained. It has been proved that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by reducing the concentration of the pollutant in a nonempty and sufficiently large subset of the spatial domain. The stabilizability with a feedback control of harvesting type is related to the magnitude of the principal eigenvalue of a certain operator. The problem of finding the optimal position (by translation) of the support of the feedback stabilizing control is faced, in order to minimize both the infected population and the pollutant at a certain finite time.

Citation: Vincenzo Capasso, Sebastian AniȚa. The interplay between models and public health policies: Regional control for a class of spatially structured epidemics (think globally, act locally). Mathematical Biosciences & Engineering, 2018, 15 (1) : 1-20. doi: 10.3934/mbe.2018001
References:
[1]

H. Abbey, An examination of the Reed-Frost theory of epidemics, Human Biology, 24 (1952), 201-233.

[2]

S. Aniţa and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic model, Nonlinear Anal. Real World Appl., 3 (2002), 453-464. doi: 10.1016/S1468-1218(01)00025-6.

[3]

S. Aniţa and V. Capasso, A stabilization strategy for a reaction-diffusion system modelling a class of spatially structured epidemic systems (think globally, act locally), Nonlinear Anal. Real World Appl., 10 (2009), 2026-2035. doi: 10.1016/j.nonrwa.2008.03.009.

[4]

S. Aniţa and V. Capasso, On the stabilization of reaction-diffusion systems modelling a class of man-environment epidemics: A review, Mathematical Methods in Applied Sciences, 33 (2010), 1235-1244. doi: 10.1002/mma.1267.

[5]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control, Nonlinear Anal. Real World Appl., 13 (2012), 725-735. doi: 10.1016/j.nonrwa.2011.08.012.

[6]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling malaria transmission, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 1673-1684. doi: 10.3934/dcdsb.2012.17.1673.

[7]

S. Aniţa and V. Capasso, Regional control in optimal harvesting of population dynamics, Submitted, 2015.

[8]

V. ArnăutuV. Barbu and V. Capasso, Controlling the spread of a class of epidemics, Appl. Math. Optimiz., 20 (1989), 297-317. doi: 10.1007/BF01447658.

[9]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in Nonlinear Diffusion, (W. E. Fitzgibbon and A. F. Walker eds. ) Pitman, London, 1977, 1-23.

[10]

P. Babak, Nonlocal initial problems for coupled reaction-diffusion systems and their applications, Nonlinear Anal. RWA, 8 (2007), 980-996. doi: 10.1016/j.nonrwa.2006.05.001.

[11]

N. T. J. Bailey, A simple stochastic epidemic, Biometrika, 37 (1950), 193-202. doi: 10.1093/biomet/37.3-4.193.

[12]

M. S. Bartlett, Some evolutionary stochastic processes, J. Roy. Stat. Soc. Ser. B, 11 (1949), 211-229.

[13]

E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Computers and Mathematics in Applications, 12 (1986), 677-694. doi: 10.1016/0898-1221(86)90054-4.

[14]

D. Bernoulli, Réflexions sur les avantages de l'inoculation, Mercure de France, June (1760), 173-190.

[15]

D. J. Bradley, Epidemiological models Theory and reality, in The Population Dynamics of Infectious Diseases, (R. M. Anderson Ed. ) Chapman and Hall, London-New York, 2008, 320-333. doi: 10.1007/978-1-4899-2901-3_10.

[16]

F. Brauer, Some infectious disease models with population dynamics and general contact rates, Differential and Integral Equations, 3 (1990), 827-836.

[17]

J. Brownlee, The mathematical theory of random migration and epidemic distribution, Proc. Roy. Soc. Edinburgh, 31 (1912), 262-289. doi: 10.1017/S0370164600025116.

[18]

S. BusenbergK. L. Cooke and M. Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395. doi: 10.1137/0148085.

[19]

V. Capasso, Mathematical Structures of Epidemic Systems (corrected 2nd printing), Lecture Notes Biomath. , vol. 97, Springer-Verlag, Heidelberg, 2008.

[20]

V. Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284. doi: 10.1137/0135022.

[21]

V. Capasso, Asymptotic stability for an integro-differential reaction-diffusion system, J. Math. Anal. Appl., 103 (1984), 575-588. doi: 10.1016/0022-247X(84)90147-1.

[22] V. Capasso and D. Bakstein, An Introduction to Continuous-Time Stochastic Processes. Theory, Models, and Applications to Finance, Biology and Medicine, Third Edition, Birkhäuser, New York, 2015. doi: 10.1007/978-1-4939-2757-9.
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V. Capasso and B. Forte, Model building as an inverse problem in Biomathematics, in Frontiers in Mathematical Biology, (S. A. Levin Ed. ) Lecture Notes in Biomathematics, SpringerVerlag, Heidelberg, 100 (1994), 600-608. doi: 10.1007/978-3-642-50124-1_35.

[24]

V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Quarterly Appl. Math., 46 (1988), 431-450. doi: 10.1090/qam/963580.

[25]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981), 173-184. doi: 10.1007/BF00275212.

[26]

V. Capasso and L. Maddalena, Saddle point behaviour for a reaction-diffusion system: Application to a class of epidemic models, Math. Comput. Simulation, 24 (1982), 540-547. doi: 10.1016/0378-4754(82)90656-5.

[27]

V. Capasso and L. Maddalena, Periodic solutions for a reaction-diffusion system modelling the spread of a class of epidemics, SIAM J. Appl. Math., 43 (1983), 417-427. doi: 10.1137/0143027.

[28]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'Epidemiologie et de la Sante' Publique, 27 (1979), 121-132; Errata corrige, 28 (1980), p390.

[29]

V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.

[30]

V. Capasso and H. R. Thieme, A threshold theorem for a reaction-diffusion epidemic system, in Differential Equations and Applications (R. Aftabizadeh, ed. ), Ohio Univ. Press, Athens, OH, 1989,128-133.

[31]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modelling man-environment-man epidemics, SIAM J. Appl. Math., 57 (1997), 327-346. doi: 10.1137/S0036139995284681.

[32]

C. Castillo-ChavezK. L. CookeW. Huang and S. A. Levin, On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). Part 1: Single population models, J. Math. Biol., 27 (1989), 373-398. doi: 10.1007/BF00290636.

[33]

C. Castillo-Chavez, K. L. Cooke, W. Huang and S. A. Levin, On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). Part 2: Multiple group models, in Mathematical and Statistical Approaches to AIDS Epidemiology, (C. Castillo-Chavez ed. ) Lecture Notes in Biomathematics, Springer-Verlag, Heidelberg, 83 (1989), 200-217. doi: 10.1007/978-3-642-93454-4_9.

[34]

C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001), 1-14.

[35]

K. Dietz, Introduction to McKendrick (1926) Applications of mathematics to medical problems, in Breakthroughs in Statistics Volume Ⅲ, (S. Kotz and N. L. Johnson eds) SpringerVerlag, Heidelberg, 1997, 17-26.

[36]

K. Dietz, Mathematization in sciences epidemics: The fitting of the first dynamic models to data, J. Contemp. Math. Anal., 44 (2009), 97-104. doi: 10.3103/S1068362309020034.

[37]

A. d'OnofrioP. Manfredi and E. Salinelli, Vaccination behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoretical Population Biology, 71 (2007), 301-317.

[38]

J. L. Doob, Markoff chains-denumerable case, Trans. Am. Math. Society, 58 (1945), 455-473. doi: 10.2307/1990339.

[39]

En'ko, On the course of epidemics of some infectious diseases, (Translation from Russian by K. Dietz) Int. J. Epidemiology, 18 (1989), 749-755.

[40] S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.
[41]

W. Farr, Progress of epidemics, Second Report of the Registrar General, (1840), 91-98.

[42]

W. H. Frost, Some conceptions of epidemics in general, Am. J. Epidemiology, 103 (1976), 141-151.

[43]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comp. Physics, 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.

[44]

B. S. Goh, Global stability in a class of predator-prey models, Bull. Math. Biol., 40 (1978), 525-533. doi: 10.1007/BF02460776.

[45]

W. H. Hamer, Epidemic disease in England, Lancet, 1 (1906), 733-739.

[46] A. Henrot and M. Pierre, Variation et Optimisation de Formes. Une Analyse Géométrique, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.
[47]

O. A. van Herwaarden and J. Grasman, Stochastic epidemics: Major outbreaks and the duration of the endemic period, J. Math. Biol., 35 (1997), 793-813. doi: 10.1007/s002850050077.

[48]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539.

[49] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM, Philadelphia, 1975.
[50]

D. G. Kendall, Mathematical models of the spread of infection, in Mathematics and Computer Science in Biology and Medicine, H. M. S. O. , London, 1965,213-225.

[51]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London, Ser. A, 115 (1927), 700-721.

[52] M. A. Krasnoselkii, Positive Solutions of Operator Equations, Nordhooff, Groningen, 1964.
[53]

M. A. Krasnoselkii, Translation Along Trajectories of Differential Equations AMS, Providence, R. I. , 1968.

[54] J. L. Lions, Controlabilité Exacte, Stabilisation et Perturbation de Systémes Distribués, Masson, Paris, 1988.
[55]

W. M. LiuH. M. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[56]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rate upon the behaviour of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.

[57]

A. J. Lotka, Martini's equations for the epidemiology of immunizing diseases, Nature, 111 (1923), 633-634.

[58]

G. Macdonald, The analysis of malaria parasite rates in infants, Tropical Disease Bull., 47 (1950), 915-938.

[59] E. Martini, Berechnungen und Beobachtungen zur Epidemiologie und Bekämpfung der Malaria, Gente, Hamburg, 1921.
[60]

I. Näsell and W. M. Hirsch, The transmission dynamics of schistosomiasis, Comm. Pure Appl. Math., 26 (1973), 395-453. doi: 10.1002/cpa.3160260402.

[61]

A. Pugliese, An SEI epidemic model with varying population size, in Differential Equation Models in Biology, Epidemiology and Ecology, (S. Busenberg and M. Martelli eds. ) Lecture Notes in Biomathematics, Springer-Verlag, Heidelberg, 92 (1991), 121-138. doi: 10.1007/978-3-642-45692-3_9.

[62] M. Puma, Elementi per una Teoria Matematica del Contagio, Editoriale Aeronautica, Rome, 1939.
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[64]

R. E. Serfling, Historical review of epidemic theory, Human Biology, 24 (1952), 145-165.

[65]

N. C. Severo, Generalizations of some stochastic epidemic models, Math. Biosci., 4 (1969), 395-402. doi: 10.1016/0025-5564(69)90019-4.

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H. E. Soper, The interpretation of periodicity in disease prevalence, J. Roy. Stat. Soc., 92 (1929), 34-73. doi: 10.2307/2341437.

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E. B. Wilson and J. Worcester, The law of mass action in epidemiology, Proc. Nat. Acad. Sci., 31 (1945), 24-34. doi: 10.1073/pnas.31.1.24.

show all references

References:
[1]

H. Abbey, An examination of the Reed-Frost theory of epidemics, Human Biology, 24 (1952), 201-233.

[2]

S. Aniţa and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic model, Nonlinear Anal. Real World Appl., 3 (2002), 453-464. doi: 10.1016/S1468-1218(01)00025-6.

[3]

S. Aniţa and V. Capasso, A stabilization strategy for a reaction-diffusion system modelling a class of spatially structured epidemic systems (think globally, act locally), Nonlinear Anal. Real World Appl., 10 (2009), 2026-2035. doi: 10.1016/j.nonrwa.2008.03.009.

[4]

S. Aniţa and V. Capasso, On the stabilization of reaction-diffusion systems modelling a class of man-environment epidemics: A review, Mathematical Methods in Applied Sciences, 33 (2010), 1235-1244. doi: 10.1002/mma.1267.

[5]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling a class of spatially structured epidemic systems via feedback control, Nonlinear Anal. Real World Appl., 13 (2012), 725-735. doi: 10.1016/j.nonrwa.2011.08.012.

[6]

S. Aniţa and V. Capasso, Stabilization of a reaction-diffusion system modelling malaria transmission, Discrete and Continuous Dynamical Systems, Series B, 17 (2012), 1673-1684. doi: 10.3934/dcdsb.2012.17.1673.

[7]

S. Aniţa and V. Capasso, Regional control in optimal harvesting of population dynamics, Submitted, 2015.

[8]

V. ArnăutuV. Barbu and V. Capasso, Controlling the spread of a class of epidemics, Appl. Math. Optimiz., 20 (1989), 297-317. doi: 10.1007/BF01447658.

[9]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, in Nonlinear Diffusion, (W. E. Fitzgibbon and A. F. Walker eds. ) Pitman, London, 1977, 1-23.

[10]

P. Babak, Nonlocal initial problems for coupled reaction-diffusion systems and their applications, Nonlinear Anal. RWA, 8 (2007), 980-996. doi: 10.1016/j.nonrwa.2006.05.001.

[11]

N. T. J. Bailey, A simple stochastic epidemic, Biometrika, 37 (1950), 193-202. doi: 10.1093/biomet/37.3-4.193.

[12]

M. S. Bartlett, Some evolutionary stochastic processes, J. Roy. Stat. Soc. Ser. B, 11 (1949), 211-229.

[13]

E. Beretta and V. Capasso, On the general structure of epidemic systems. Global asymptotic stability, Computers and Mathematics in Applications, 12 (1986), 677-694. doi: 10.1016/0898-1221(86)90054-4.

[14]

D. Bernoulli, Réflexions sur les avantages de l'inoculation, Mercure de France, June (1760), 173-190.

[15]

D. J. Bradley, Epidemiological models Theory and reality, in The Population Dynamics of Infectious Diseases, (R. M. Anderson Ed. ) Chapman and Hall, London-New York, 2008, 320-333. doi: 10.1007/978-1-4899-2901-3_10.

[16]

F. Brauer, Some infectious disease models with population dynamics and general contact rates, Differential and Integral Equations, 3 (1990), 827-836.

[17]

J. Brownlee, The mathematical theory of random migration and epidemic distribution, Proc. Roy. Soc. Edinburgh, 31 (1912), 262-289. doi: 10.1017/S0370164600025116.

[18]

S. BusenbergK. L. Cooke and M. Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395. doi: 10.1137/0148085.

[19]

V. Capasso, Mathematical Structures of Epidemic Systems (corrected 2nd printing), Lecture Notes Biomath. , vol. 97, Springer-Verlag, Heidelberg, 2008.

[20]

V. Capasso, Global solution for a diffusive nonlinear deterministic epidemic model, SIAM J. Appl. Math., 35 (1978), 274-284. doi: 10.1137/0135022.

[21]

V. Capasso, Asymptotic stability for an integro-differential reaction-diffusion system, J. Math. Anal. Appl., 103 (1984), 575-588. doi: 10.1016/0022-247X(84)90147-1.

[22] V. Capasso and D. Bakstein, An Introduction to Continuous-Time Stochastic Processes. Theory, Models, and Applications to Finance, Biology and Medicine, Third Edition, Birkhäuser, New York, 2015. doi: 10.1007/978-1-4939-2757-9.
[23]

V. Capasso and B. Forte, Model building as an inverse problem in Biomathematics, in Frontiers in Mathematical Biology, (S. A. Levin Ed. ) Lecture Notes in Biomathematics, SpringerVerlag, Heidelberg, 100 (1994), 600-608. doi: 10.1007/978-3-642-50124-1_35.

[24]

V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Quarterly Appl. Math., 46 (1988), 431-450. doi: 10.1090/qam/963580.

[25]

V. Capasso and L. Maddalena, Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol., 13 (1981), 173-184. doi: 10.1007/BF00275212.

[26]

V. Capasso and L. Maddalena, Saddle point behaviour for a reaction-diffusion system: Application to a class of epidemic models, Math. Comput. Simulation, 24 (1982), 540-547. doi: 10.1016/0378-4754(82)90656-5.

[27]

V. Capasso and L. Maddalena, Periodic solutions for a reaction-diffusion system modelling the spread of a class of epidemics, SIAM J. Appl. Math., 43 (1983), 417-427. doi: 10.1137/0143027.

[28]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Revue d'Epidemiologie et de la Sante' Publique, 27 (1979), 121-132; Errata corrige, 28 (1980), p390.

[29]

V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.

[30]

V. Capasso and H. R. Thieme, A threshold theorem for a reaction-diffusion epidemic system, in Differential Equations and Applications (R. Aftabizadeh, ed. ), Ohio Univ. Press, Athens, OH, 1989,128-133.

[31]

V. Capasso and R. E. Wilson, Analysis of a reaction-diffusion system modelling man-environment-man epidemics, SIAM J. Appl. Math., 57 (1997), 327-346. doi: 10.1137/S0036139995284681.

[32]

C. Castillo-ChavezK. L. CookeW. Huang and S. A. Levin, On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). Part 1: Single population models, J. Math. Biol., 27 (1989), 373-398. doi: 10.1007/BF00290636.

[33]

C. Castillo-Chavez, K. L. Cooke, W. Huang and S. A. Levin, On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS). Part 2: Multiple group models, in Mathematical and Statistical Approaches to AIDS Epidemiology, (C. Castillo-Chavez ed. ) Lecture Notes in Biomathematics, Springer-Verlag, Heidelberg, 83 (1989), 200-217. doi: 10.1007/978-3-642-93454-4_9.

[34]

C. T. Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001), 1-14.

[35]

K. Dietz, Introduction to McKendrick (1926) Applications of mathematics to medical problems, in Breakthroughs in Statistics Volume Ⅲ, (S. Kotz and N. L. Johnson eds) SpringerVerlag, Heidelberg, 1997, 17-26.

[36]

K. Dietz, Mathematization in sciences epidemics: The fitting of the first dynamic models to data, J. Contemp. Math. Anal., 44 (2009), 97-104. doi: 10.3103/S1068362309020034.

[37]

A. d'OnofrioP. Manfredi and E. Salinelli, Vaccination behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoretical Population Biology, 71 (2007), 301-317.

[38]

J. L. Doob, Markoff chains-denumerable case, Trans. Am. Math. Society, 58 (1945), 455-473. doi: 10.2307/1990339.

[39]

En'ko, On the course of epidemics of some infectious diseases, (Translation from Russian by K. Dietz) Int. J. Epidemiology, 18 (1989), 749-755.

[40] S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.
[41]

W. Farr, Progress of epidemics, Second Report of the Registrar General, (1840), 91-98.

[42]

W. H. Frost, Some conceptions of epidemics in general, Am. J. Epidemiology, 103 (1976), 141-151.

[43]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, J. Comp. Physics, 22 (1976), 403-434. doi: 10.1016/0021-9991(76)90041-3.

[44]

B. S. Goh, Global stability in a class of predator-prey models, Bull. Math. Biol., 40 (1978), 525-533. doi: 10.1007/BF02460776.

[45]

W. H. Hamer, Epidemic disease in England, Lancet, 1 (1906), 733-739.

[46] A. Henrot and M. Pierre, Variation et Optimisation de Formes. Une Analyse Géométrique, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.
[47]

O. A. van Herwaarden and J. Grasman, Stochastic epidemics: Major outbreaks and the duration of the endemic period, J. Math. Biol., 35 (1997), 793-813. doi: 10.1007/s002850050077.

[48]

H. W. Hethcote and P. van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539.

[49] F. Hoppensteadt, Mathematical Theories of Populations: Demographics, Genetics and Epidemics, SIAM, Philadelphia, 1975.
[50]

D. G. Kendall, Mathematical models of the spread of infection, in Mathematics and Computer Science in Biology and Medicine, H. M. S. O. , London, 1965,213-225.

[51]

W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. London, Ser. A, 115 (1927), 700-721.

[52] M. A. Krasnoselkii, Positive Solutions of Operator Equations, Nordhooff, Groningen, 1964.
[53]

M. A. Krasnoselkii, Translation Along Trajectories of Differential Equations AMS, Providence, R. I. , 1968.

[54] J. L. Lions, Controlabilité Exacte, Stabilisation et Perturbation de Systémes Distribués, Masson, Paris, 1988.
[55]

W. M. LiuH. M. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380. doi: 10.1007/BF00277162.

[56]

W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rate upon the behaviour of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.

[57]

A. J. Lotka, Martini's equations for the epidemiology of immunizing diseases, Nature, 111 (1923), 633-634.

[58]

G. Macdonald, The analysis of malaria parasite rates in infants, Tropical Disease Bull., 47 (1950), 915-938.

[59] E. Martini, Berechnungen und Beobachtungen zur Epidemiologie und Bekämpfung der Malaria, Gente, Hamburg, 1921.
[60]

I. Näsell and W. M. Hirsch, The transmission dynamics of schistosomiasis, Comm. Pure Appl. Math., 26 (1973), 395-453. doi: 10.1002/cpa.3160260402.

[61]

A. Pugliese, An SEI epidemic model with varying population size, in Differential Equation Models in Biology, Epidemiology and Ecology, (S. Busenberg and M. Martelli eds. ) Lecture Notes in Biomathematics, Springer-Verlag, Heidelberg, 92 (1991), 121-138. doi: 10.1007/978-3-642-45692-3_9.

[62] M. Puma, Elementi per una Teoria Matematica del Contagio, Editoriale Aeronautica, Rome, 1939.
[63] R. Ross, The Prevention of Malaria, London, Murray, 1911.
[64]

R. E. Serfling, Historical review of epidemic theory, Human Biology, 24 (1952), 145-165.

[65]

N. C. Severo, Generalizations of some stochastic epidemic models, Math. Biosci., 4 (1969), 395-402. doi: 10.1016/0025-5564(69)90019-4.

[66]

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Figure 1.  The transfer diagram for an SEIR compartmental model including the susceptible class S, the exposed, but not yet infective, class E, the infective class I, and the removed class R
Figure 2.  Nonlinear forces of infection [29]
Figure 3.  Think Globally, Act Locally
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