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doi: 10.3934/dcdsb.2021004
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On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease

Nabahats Dib-Baghdadli 1,, , Rabah Labbas 2, , Tewfik Mahdjoub 1, and  Ahmed Medeghri 3,

1. 

NLAAML, Abou-Bekr Belkaid University, Tlemcen 13000, Algeria
2. 

Normandie Univ, UNIHAVRE, LMAH, FR-CNRS-3335, 76600 Le Havre, France
3. 

LMPA, Abdelhamid Ibn-Badis University, Mostaganem 27000, Algeria

* Corresponding author: Nabahats Dib-Baghdadli

Received  August 2020 Revised  October 2020 Early access December 2020

In this work, we study some reaction-diffusion equations set in two habitats which model the spatial dispersal of the triatomines, vectors of Chagas disease. We prove in particular that the dispersal operator generates an analytic semigroup in an adequate space and we prove the local existence of the solution for the corresponding Cauchy problem.

Keywords: Chagas disease, dispersal process, reaction-diffusion equations, skew brownian motion, sectorial operator.
Mathematics Subject Classification: Primary:35K57, 47D06;Secondary:92B05, 12H20.
Citation: Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021004
References:
[1]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10 (1960), 419-437.  doi: 10.2140/pjm.1960.10.419.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, Skew Brownian motion: A model for diffusion with interfaces, Proc. Int. Conf. Math. Model. Med. Heal. Sci., (1998), 73-78. Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operator with a bounded $H^\infty$ functional calculus, J. Austral. Math. Soc. Ser. A, 60 (1996), 51-89.  doi: 10.1017/S1446788700037393.  Google Scholar

[5]

B. Crawford, C. M. Kribs-Zaleta and G. Ambartsoumian, Invasion speed in cellular automaton models for T.cruzi Vector MIGration, Bull. Math. Biol., 75 (2013), 1051-1081.  doi: 10.1007/s11538-013-9840-7.  Google Scholar

[6]

H. Devillers, J. R. Lobry and F. Menu, An agent-based model for predicting the prevalence of Trypanosoma cruzi I and II in their host and vector populations, J. Theor. Biol., 255 (2008), 307-315.  doi: 10.1016/j.jtbi.2008.08.023.  Google Scholar

[7]

G. Dore, A. Favini, R. Labbas and K. Lemrabet, An abstract transmission problem in a thin layer, I: Sharp estimates, J. Funct. Anal., 261 (2011), 1865-1922.  doi: 10.1016/j.jfa.2011.05.021.  Google Scholar

[8]

G. Dore and S. Yakubov, Semigroup estimates and noncoercive boundary value problems, Semigroup Forum, 60 (2000), 93-121.   Google Scholar

[9]

N. El Saadi, A. Bah, T. Mahdjoub and C. Kribs, On the sylvatic transmission of T. cruzi, the parasite causing Chagas disease: A view from an agent-based model, Ecol. Modell., 423 (2020), 109001. Google Scholar

[10]

A. Favini, R. Labbas, S. Maingot and A. Thorel, Elliptic differential operator with an abstract Robin boundary condition containing two spectral parameters, study in a non commutative framework, To appear in 2020. Google Scholar

[11]

A. Favini, R. Labbas, A. Medeghri and A. Menad, Analytic semigroups generated by the dispersal process in two habitats incorporating individual behavior at the interface, J. Math. Anal. Appl., 471 (2019), 448-480.  doi: 10.1016/j.jmaa.2018.10.085.  Google Scholar

[12]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat. (6), 5 (1972), 657-729.   Google Scholar

[13]

M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[14]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1980. Google Scholar

[15]

C. Kribs-Zaleta, Vector consumption and contact process saturation in sylvatic transmission of T. cruzi, Math. Popul. Stud., 13 (2006), 135-152.  doi: 10.1080/08898480600788576.  Google Scholar

[16]

C. R. Lazzari, M. H. Pereira and M. G. Lorenzo, Behavioural biology of Chagas disease vectors, Mem. Inst. Oswaldo Cruz., 108 (2013), 34-47.   Google Scholar

[17]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.  Google Scholar

[18]

T. Mahdjoub and C. Kribs, Assessing the invasion speed of triatomine populations, Chagas disease vectors, Rev. Mat. Teor. Apl., 27 (2020), 73-92.   Google Scholar

[19]

F. Menu, M. Ginoux, E. Rajon, C. R. Lazzari and J. E. Rabinovich, Adaptive developmental delay in chagas disease vectors: An evolutionary ecology approach, PLoS Negl. Trop. Dis., 4 (2010). Google Scholar

[20]

M. Mesk, T. Mahdjoub, S. Gourbière, J. E. Rabinovich and F. Menu, Invasion speeds of Triatoma dimidiata, vector of Chagas disease: An application of orthogonal polynomials method, J. Theor. Biol., 395 (2016), 126-143.  doi: 10.1016/j.jtbi.2016.01.017.  Google Scholar

[21]

P. Nouvellet, Z. M. Cucunubá and S. Gourbière, Chapter four - Ecology, evolution and control of chagas disease: A century of neglected modelling and a promising future, in Adv. Parasitol. (eds. M. A. Roy and B. Maria Gloria), Academic Press, (2015), 135-191. Google Scholar

[22]

V. Payet, M. J. Ramirez-Sierra, J. Rabinovich, F. Menu and E. Dumonteil, Variations in sex ratio, feeding, and fecundity of Triatoma dimidiata (Hemiptera: Reduviidae) among habitats in the Yucatan Peninsula, Mexico, Vector-Borne, Zoonotic Dis., 9 (2009), 243-251.   Google Scholar

[23]

R. Slimi, S. El Yacoubi, E. Dumonteil and S. Gourbière, A cellular automata model for Chagas disease, Appl. Math. Model., 33 (2009), 1072-1085.  doi: 10.1016/j.apm.2007.12.028.  Google Scholar

[24]

V. Steindorf and N. A. Maidana, Modeling the spatial spread of Chagas disease, Bull. Math. Biol., 81 (2019), 1687-1730.  doi: 10.1007/s11538-019-00581-5.  Google Scholar

show all references

References:
[1]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10 (1960), 419-437.  doi: 10.2140/pjm.1960.10.419.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, Skew Brownian motion: A model for diffusion with interfaces, Proc. Int. Conf. Math. Model. Med. Heal. Sci., (1998), 73-78. Google Scholar

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[4]

M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operator with a bounded $H^\infty$ functional calculus, J. Austral. Math. Soc. Ser. A, 60 (1996), 51-89.  doi: 10.1017/S1446788700037393.  Google Scholar

[5]

B. Crawford, C. M. Kribs-Zaleta and G. Ambartsoumian, Invasion speed in cellular automaton models for T.cruzi Vector MIGration, Bull. Math. Biol., 75 (2013), 1051-1081.  doi: 10.1007/s11538-013-9840-7.  Google Scholar

[6]

H. Devillers, J. R. Lobry and F. Menu, An agent-based model for predicting the prevalence of Trypanosoma cruzi I and II in their host and vector populations, J. Theor. Biol., 255 (2008), 307-315.  doi: 10.1016/j.jtbi.2008.08.023.  Google Scholar

[7]

G. Dore, A. Favini, R. Labbas and K. Lemrabet, An abstract transmission problem in a thin layer, I: Sharp estimates, J. Funct. Anal., 261 (2011), 1865-1922.  doi: 10.1016/j.jfa.2011.05.021.  Google Scholar

[8]

G. Dore and S. Yakubov, Semigroup estimates and noncoercive boundary value problems, Semigroup Forum, 60 (2000), 93-121.   Google Scholar

[9]

N. El Saadi, A. Bah, T. Mahdjoub and C. Kribs, On the sylvatic transmission of T. cruzi, the parasite causing Chagas disease: A view from an agent-based model, Ecol. Modell., 423 (2020), 109001. Google Scholar

[10]

A. Favini, R. Labbas, S. Maingot and A. Thorel, Elliptic differential operator with an abstract Robin boundary condition containing two spectral parameters, study in a non commutative framework, To appear in 2020. Google Scholar

[11]

A. Favini, R. Labbas, A. Medeghri and A. Menad, Analytic semigroups generated by the dispersal process in two habitats incorporating individual behavior at the interface, J. Math. Anal. Appl., 471 (2019), 448-480.  doi: 10.1016/j.jmaa.2018.10.085.  Google Scholar

[12]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat. (6), 5 (1972), 657-729.   Google Scholar

[13]

M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[14]

T. Kato, Perturbation Theory for Linear Operators, Springer, 1980. Google Scholar

[15]

C. Kribs-Zaleta, Vector consumption and contact process saturation in sylvatic transmission of T. cruzi, Math. Popul. Stud., 13 (2006), 135-152.  doi: 10.1080/08898480600788576.  Google Scholar

[16]

C. R. Lazzari, M. H. Pereira and M. G. Lorenzo, Behavioural biology of Chagas disease vectors, Mem. Inst. Oswaldo Cruz., 108 (2013), 34-47.   Google Scholar

[17]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, 1995.  Google Scholar

[18]

T. Mahdjoub and C. Kribs, Assessing the invasion speed of triatomine populations, Chagas disease vectors, Rev. Mat. Teor. Apl., 27 (2020), 73-92.   Google Scholar

[19]

F. Menu, M. Ginoux, E. Rajon, C. R. Lazzari and J. E. Rabinovich, Adaptive developmental delay in chagas disease vectors: An evolutionary ecology approach, PLoS Negl. Trop. Dis., 4 (2010). Google Scholar

[20]

M. Mesk, T. Mahdjoub, S. Gourbière, J. E. Rabinovich and F. Menu, Invasion speeds of Triatoma dimidiata, vector of Chagas disease: An application of orthogonal polynomials method, J. Theor. Biol., 395 (2016), 126-143.  doi: 10.1016/j.jtbi.2016.01.017.  Google Scholar

[21]

P. Nouvellet, Z. M. Cucunubá and S. Gourbière, Chapter four - Ecology, evolution and control of chagas disease: A century of neglected modelling and a promising future, in Adv. Parasitol. (eds. M. A. Roy and B. Maria Gloria), Academic Press, (2015), 135-191. Google Scholar

[22]

V. Payet, M. J. Ramirez-Sierra, J. Rabinovich, F. Menu and E. Dumonteil, Variations in sex ratio, feeding, and fecundity of Triatoma dimidiata (Hemiptera: Reduviidae) among habitats in the Yucatan Peninsula, Mexico, Vector-Borne, Zoonotic Dis., 9 (2009), 243-251.   Google Scholar

[23]

R. Slimi, S. El Yacoubi, E. Dumonteil and S. Gourbière, A cellular automata model for Chagas disease, Appl. Math. Model., 33 (2009), 1072-1085.  doi: 10.1016/j.apm.2007.12.028.  Google Scholar

[24]

V. Steindorf and N. A. Maidana, Modeling the spatial spread of Chagas disease, Bull. Math. Biol., 81 (2019), 1687-1730.  doi: 10.1007/s11538-019-00581-5.  Google Scholar

Figure 1.  Population density in two habitats
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Figure 2.  A schematic representation of the life cycle used in the triatomine's model
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Table 1.  The demographic and diffusion parameters of T.Dimidiata population
Parameter Definition Properties
$ s_{j}(t) $ Probability of survival of juveniles per unit of time $ 0\leq s_{j}(t)\leq 1 $
$ s_{a}(t) $ Probability of survival of adult per unit of time $ 0\leq s_{a}(t)\leq 1 $
$ w_{j}(t) $ Probability of transition from juvenile to adult $ 0\leq w_{j}(t)\leq 1 $
$ f_{a}(t) $ Female fertility per unit time $ f_{a}(t)\geq 0 $
$ d_{j-},d_{j+} $ diffusion coefficient of juveniles $ d_{j-},d_{j+}> 0 $
$ d_{a-},d_{a+} $ diffusion coefficient of adults $ d_{a-},d_{a+}> 0 $
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Parameter Definition Properties
$ s_{j}(t) $ Probability of survival of juveniles per unit of time $ 0\leq s_{j}(t)\leq 1 $
$ s_{a}(t) $ Probability of survival of adult per unit of time $ 0\leq s_{a}(t)\leq 1 $
$ w_{j}(t) $ Probability of transition from juvenile to adult $ 0\leq w_{j}(t)\leq 1 $
$ f_{a}(t) $ Female fertility per unit time $ f_{a}(t)\geq 0 $
$ d_{j-},d_{j+} $ diffusion coefficient of juveniles $ d_{j-},d_{j+}> 0 $
$ d_{a-},d_{a+} $ diffusion coefficient of adults $ d_{a-},d_{a+}> 0 $
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