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

[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

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

[1]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[2]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[3]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[4]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[5]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[6]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[7]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[8]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[9]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[10]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[11]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[12]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[13]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[14]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[15]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[16]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[17]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[18]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[19]

Lingju Kong, Roger Nichols. On principal eigenvalues of biharmonic systems. Communications on Pure & Applied Analysis, 2021, 20 (1) : 1-15. doi: 10.3934/cpaa.2020254

[20]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]