December  2021, 26(12): 6091-6115. doi: 10.3934/dcdsb.2021004

On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease

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 Published  December 2021 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.

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 and Continuous Dynamical Systems - B, 2021, 26 (12) : 6091-6115. 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.

[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.

[3]

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

[4]

M. CowlingI. DoustA. 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.

[5]

B. CrawfordC. 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.

[6]

H. DevillersJ. 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.

[7]

G. DoreA. FaviniR. 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.

[8]

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

[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.

[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.

[11]

A. FaviniR. LabbasA. 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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[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).

[20]

M. MeskT. MahdjoubS. GourbièreJ. 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.

[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.

[22]

V. PayetM. J. Ramirez-SierraJ. RabinovichF. 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. 

[23]

R. SlimiS. El YacoubiE. 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.

[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.

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.

[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.

[3]

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

[4]

M. CowlingI. DoustA. 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.

[5]

B. CrawfordC. 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.

[6]

H. DevillersJ. 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.

[7]

G. DoreA. FaviniR. 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.

[8]

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

[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.

[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.

[11]

A. FaviniR. LabbasA. 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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[17]

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

[18]

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

[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).

[20]

M. MeskT. MahdjoubS. GourbièreJ. 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.

[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.

[22]

V. PayetM. J. Ramirez-SierraJ. RabinovichF. 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. 

[23]

R. SlimiS. El YacoubiE. 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.

[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.

Figure 1.  Population density in two habitats
Figure 2.  A schematic representation of the life cycle used in the triatomine's model
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 $
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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