September  2012, 17(6): 1673-1684. doi: 10.3934/dcdsb.2012.17.1673

Stabilization of a reaction-diffusion system modelling malaria transmission

1. 

Faculty of Mathematics, "Al.I. Cuza" University of Iaşi, Bd. Carol I nr. 11 and "Octav Mayer" Institute of Mathematics, Bd. Carol I nr. 8, Iaşi 700506, Romania

2. 

Dipartimento di Matematica, Universita di Milano, Via Saldini 50, 20133 Milano, Italy

Received  December 2011 Revised  January 2012 Published  May 2012

A two-component reaction-diffusion system modelling a class of spatially structured epidemic systems is considered. More specifically, the system describes the spread of malaria mediated by a population of infected mosquitoes. A relevant problem, related to the possible eradication of the epidemic, is the so called zero-stabilization. We prove that it is possible to diminish exponentially the epidemic process, in the whole habitat, just by acting on the segregation rate between the population of infected mosquitoes and the susceptible human population in a nonempty and sufficiently large subset of the spatial domain.
Citation: Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673
References:
[1]

B. Ainseba and S. Aniţa, Internal stabilizability for a reaction-diffusion problem modeling a predator-prey system,, Nonlinear Anal., 61 (2005), 491.  doi: 10.1016/j.na.2004.09.055.  Google Scholar

[2]

L.-I. Aniţa and S. Aniţa, Note on the stabilization of a reaction-diffusion model in epidemiology,, Nonlinear Anal. Real World Appl., 6 (2005), 537.   Google Scholar

[3]

S. Aniţa and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems,, Nonlinear Anal. Real World Appl., 3 (2002), 453.   Google Scholar

[4]

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.  doi: 10.1016/j.nonrwa.2008.03.009.  Google Scholar

[5]

S. Aniţa and V. Capasso, Stabilization for a reaction-diffusion system modelling a class of spatially structured epidemic systems. The periodic case,, in, (2011).   Google Scholar

[6]

S. Aniţa and V. Capasso, On the stabilization of reaction-diffusion systems modeling a class of man-environment epidemics: A review,, Math. Meth. Appl. Sci., 33 (2010), 1235.  doi: 10.1002/mma.1267.  Google Scholar

[7]

S. Aniţa, W. E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a class of reaction-diffusion systems posed on non coincident spatial domains,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 805.  doi: 10.3934/dcdsb.2009.11.805.  Google Scholar

[8]

J. L. Aron and R. M. May, The population dynamics of malaria,, in, (1982), 139.   Google Scholar

[9]

N. Bacaër and C. Sokhna, A reaction-diffusion system modeling the spread of resistance to an antimalarial drug,, Math. Biosci. Eng., 2 (2005), 227.   Google Scholar

[10]

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

[11]

V. Capasso, "Mathematical Structures of Epidemic Systems,'' With a foreword by Simon A. Levin, Corrected reprint of the 1993 original,, Lecture Notes Biomath., (2008).   Google Scholar

[12]

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.  doi: 10.1137/0143027.  Google Scholar

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F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmission,, Bull. Math. Biol., 73 (2011), 639.  doi: 10.1007/s11538-010-9545-0.  Google Scholar

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N. Chitnis, J. M. Cushing and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission,, SIAM Journal on Applied Mathematics, 67 (2006), 24.  doi: 10.1137/050638941.  Google Scholar

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N. Chitnis, T. A. Smith and R. W. Steketee, A mathematical model for the dynamics of malaria in mosquitoes feeding on a heterogeneous host population,, J. Biol. Dyn., 2 (2008), 259.   Google Scholar

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K. Deimling, "Nonlinear Functional Analysis,'', Springer-Verlag, (1985).   Google Scholar

[17]

K. Dietz, Mathematical models for transmission and control of malaria,, in, (1988), 1091.   Google Scholar

[18]

K. Dietz, T. Molineaux and A. Thomas, A malaria model tested in the African savannah,, Bull. World Health Org., 50 (1974), 347.   Google Scholar

[19]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964).   Google Scholar

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G. F. Killeen, F. F. McKenzie, B. D. Foy, C. Schieffelin, P. F. Billingsley and J. C. Beier, A simplified model for predicting malaria entomological inoculation rates based on entomologic and parasitologic parameters relevant to control,, Am. J. Trop. Med. Hyg., 62 (2000), 535.   Google Scholar

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J.-L. Lions, "Contrôlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués,'' Tome 1, (French) [Exact Controllability, Perturbations and Stabilization of Distributed Systems, Vol. 1], Contrôlabilité Exacte [Exact Controllability], Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 8,, Masson, (1988).   Google Scholar

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G. Macdonald, "The Epidemiology and Control of Malaria,'', Oxford University Press, (1957).   Google Scholar

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L. Molineaux and G. Gramiccia, "The Garki Project. Research on the Epidemiology and Control of Malaria in the Sudan Savanna of West Africa,'', World Health Organization, (1980).   Google Scholar

[24]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Corrected reprint of the 1967 original, (1967).   Google Scholar

[25]

R. Ross, "The Prevention of Malaria,'', 2nd edition, (1911).   Google Scholar

[26]

S. Ruan, D. Xiaob and J. C. Beierc, On the delayed Ross-Macdonald model for malaria transmission,, Bull. Math. Biol., 70 (2008), 1098.  doi: 10.1007/s11538-007-9292-z.  Google Scholar

[27]

T. Smith, G. Killeen, N. Maire, A. Ross, L. Molineaux, F. Tediosi, G. Hutton, J. Utzinger, K. Dietz and M. Tanner, Mathematical modeling of the impact of malaria vaccines on the clinical epidemiology and natural history of Plasmodium Falciparum malaria: Overview,, Am. J. Trop. Med. Hyg., 75 (2006), 1.   Google Scholar

[28]

T. Sochantha, S. Hewitt, C. Nguon, L. Okell, N. Alexander, S. Yeung, H. Vannara, M. Rowland and D. Socheat, Insecticide-treated bednets for the prevention of Plasmodium falciparum malaria in Cambodia: A cluster-randomized trial,, Trop. Med. Int. Health, 11 (2006), 1166.  doi: 10.1111/j.1365-3156.2006.01673.x.  Google Scholar

[29]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, (1994).   Google Scholar

[30]

L. J. White, R. J. Maude, W. Pongtavornpinyo, S. Saralamba, R. Aguas, T. Van Effelterre, N. P. J. Day and N. J. White, The role of simple mathematical models in malaria elimination strategy design,, Malaria Journal, 8 (2009).   Google Scholar

[31]

, "The Africa Malaria Report,'' WHO-UNICEF,, 2003., ().   Google Scholar

show all references

References:
[1]

B. Ainseba and S. Aniţa, Internal stabilizability for a reaction-diffusion problem modeling a predator-prey system,, Nonlinear Anal., 61 (2005), 491.  doi: 10.1016/j.na.2004.09.055.  Google Scholar

[2]

L.-I. Aniţa and S. Aniţa, Note on the stabilization of a reaction-diffusion model in epidemiology,, Nonlinear Anal. Real World Appl., 6 (2005), 537.   Google Scholar

[3]

S. Aniţa and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems,, Nonlinear Anal. Real World Appl., 3 (2002), 453.   Google Scholar

[4]

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.  doi: 10.1016/j.nonrwa.2008.03.009.  Google Scholar

[5]

S. Aniţa and V. Capasso, Stabilization for a reaction-diffusion system modelling a class of spatially structured epidemic systems. The periodic case,, in, (2011).   Google Scholar

[6]

S. Aniţa and V. Capasso, On the stabilization of reaction-diffusion systems modeling a class of man-environment epidemics: A review,, Math. Meth. Appl. Sci., 33 (2010), 1235.  doi: 10.1002/mma.1267.  Google Scholar

[7]

S. Aniţa, W. E. Fitzgibbon and M. Langlais, Global existence and internal stabilization for a class of reaction-diffusion systems posed on non coincident spatial domains,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 805.  doi: 10.3934/dcdsb.2009.11.805.  Google Scholar

[8]

J. L. Aron and R. M. May, The population dynamics of malaria,, in, (1982), 139.   Google Scholar

[9]

N. Bacaër and C. Sokhna, A reaction-diffusion system modeling the spread of resistance to an antimalarial drug,, Math. Biosci. Eng., 2 (2005), 227.   Google Scholar

[10]

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

[11]

V. Capasso, "Mathematical Structures of Epidemic Systems,'' With a foreword by Simon A. Levin, Corrected reprint of the 1993 original,, Lecture Notes Biomath., (2008).   Google Scholar

[12]

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.  doi: 10.1137/0143027.  Google Scholar

[13]

F. Chamchod and N. F. Britton, Analysis of a vector-bias model on malaria transmission,, Bull. Math. Biol., 73 (2011), 639.  doi: 10.1007/s11538-010-9545-0.  Google Scholar

[14]

N. Chitnis, J. M. Cushing and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission,, SIAM Journal on Applied Mathematics, 67 (2006), 24.  doi: 10.1137/050638941.  Google Scholar

[15]

N. Chitnis, T. A. Smith and R. W. Steketee, A mathematical model for the dynamics of malaria in mosquitoes feeding on a heterogeneous host population,, J. Biol. Dyn., 2 (2008), 259.   Google Scholar

[16]

K. Deimling, "Nonlinear Functional Analysis,'', Springer-Verlag, (1985).   Google Scholar

[17]

K. Dietz, Mathematical models for transmission and control of malaria,, in, (1988), 1091.   Google Scholar

[18]

K. Dietz, T. Molineaux and A. Thomas, A malaria model tested in the African savannah,, Bull. World Health Org., 50 (1974), 347.   Google Scholar

[19]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964).   Google Scholar

[20]

G. F. Killeen, F. F. McKenzie, B. D. Foy, C. Schieffelin, P. F. Billingsley and J. C. Beier, A simplified model for predicting malaria entomological inoculation rates based on entomologic and parasitologic parameters relevant to control,, Am. J. Trop. Med. Hyg., 62 (2000), 535.   Google Scholar

[21]

J.-L. Lions, "Contrôlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués,'' Tome 1, (French) [Exact Controllability, Perturbations and Stabilization of Distributed Systems, Vol. 1], Contrôlabilité Exacte [Exact Controllability], Recherches en Mathématiques Appliquées [Research in Applied Mathematics], 8,, Masson, (1988).   Google Scholar

[22]

G. Macdonald, "The Epidemiology and Control of Malaria,'', Oxford University Press, (1957).   Google Scholar

[23]

L. Molineaux and G. Gramiccia, "The Garki Project. Research on the Epidemiology and Control of Malaria in the Sudan Savanna of West Africa,'', World Health Organization, (1980).   Google Scholar

[24]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Corrected reprint of the 1967 original, (1967).   Google Scholar

[25]

R. Ross, "The Prevention of Malaria,'', 2nd edition, (1911).   Google Scholar

[26]

S. Ruan, D. Xiaob and J. C. Beierc, On the delayed Ross-Macdonald model for malaria transmission,, Bull. Math. Biol., 70 (2008), 1098.  doi: 10.1007/s11538-007-9292-z.  Google Scholar

[27]

T. Smith, G. Killeen, N. Maire, A. Ross, L. Molineaux, F. Tediosi, G. Hutton, J. Utzinger, K. Dietz and M. Tanner, Mathematical modeling of the impact of malaria vaccines on the clinical epidemiology and natural history of Plasmodium Falciparum malaria: Overview,, Am. J. Trop. Med. Hyg., 75 (2006), 1.   Google Scholar

[28]

T. Sochantha, S. Hewitt, C. Nguon, L. Okell, N. Alexander, S. Yeung, H. Vannara, M. Rowland and D. Socheat, Insecticide-treated bednets for the prevention of Plasmodium falciparum malaria in Cambodia: A cluster-randomized trial,, Trop. Med. Int. Health, 11 (2006), 1166.  doi: 10.1111/j.1365-3156.2006.01673.x.  Google Scholar

[29]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", 2nd edition, (1994).   Google Scholar

[30]

L. J. White, R. J. Maude, W. Pongtavornpinyo, S. Saralamba, R. Aguas, T. Van Effelterre, N. P. J. Day and N. J. White, The role of simple mathematical models in malaria elimination strategy design,, Malaria Journal, 8 (2009).   Google Scholar

[31]

, "The Africa Malaria Report,'' WHO-UNICEF,, 2003., ().   Google Scholar

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