Geodesic
Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches
Vincent Calvez1 ; Susely Figueroa Iglesias2 ; Hélène Hivert3 ; Sylvie Méléard4 ; Anna Melnykova5 ; Samuel Nordmann6
1CNRS & Institut Camille Jordan, Lyon
2Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse, UPS, IMT, F-31062 Toulouse Cedex 9, France
3Univ Lyon, École centrale de Lyon, CNRS UMR 5208, Institut Camille Jordan, F-69134 Écully, France
4CMAP, UMR 7641, Ecole polytechnique, France
5Université de Cergy-Pontoise, AGM CNRS UMR 8088 and Université de Grenoble Alpes, LJK CNRS UMR 5224
6CAMS, EHESS, PSL Université, Paris
ESAIM. Proceedings, Tome 67 (2020), pp. 135-160
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Horizontal gene Transfer (HT) denotes the transmission of genetic material between two living organisms, while the vertical transmission refers to a DNA transfer from parents to their offspring. Consistent experimental evidence report that this phenomenon plays an essential role in the evolution of certain bacterias. In particular, HT is believed to be the main instrument of developing the antibiotic resistance. In this work, we consider several models which describe this phenomenon: a stochastic jump process (individual-based) and the deterministic nonlinear integrodifferential equation obtained as a limit for large populations. We also consider a Hamilton-Jacobi equation, obtained as a limit of the deterministic model under the assumption of small mutations. The goal of this paper is to compare these models with the help of numerical simulations. More specifically, our goal is to understand to which extent the Hamilton-Jacobi model reproduces the qualitative behavior of the stochastic model and the phenomenon of evolutionary rescue in particular.
DOI : 10.1051/proc/202067009

Vincent Calvez  1   ; Susely Figueroa Iglesias  2   ; Hélène Hivert  3   ; Sylvie Méléard  4   ; Anna Melnykova  5   ; Samuel Nordmann  6

1 CNRS & Institut Camille Jordan, Lyon
2 Institut de Mathématiques de Toulouse; UMR 5219, Université de Toulouse, UPS, IMT, F-31062 Toulouse Cedex 9, France
3 Univ Lyon, École centrale de Lyon, CNRS UMR 5208, Institut Camille Jordan, F-69134 Écully, France
4 CMAP, UMR 7641, Ecole polytechnique, France
5 Université de Cergy-Pontoise, AGM CNRS UMR 8088 and Université de Grenoble Alpes, LJK CNRS UMR 5224
6 CAMS, EHESS, PSL Université, Paris 
@article{EP_2020_67_a9,
     author = {Vincent Calvez and Susely Figueroa Iglesias and H\'el\`ene Hivert and Sylvie M\'el\'eard and Anna Melnykova and Samuel Nordmann},
     title = {Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches},
     journal = {ESAIM. Proceedings},
     pages = {135--160},
     year = {2020},
     volume = {67},
     doi = {10.1051/proc/202067009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/202067009/}
}
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Vincent Calvez; Susely Figueroa Iglesias; Hélène Hivert; Sylvie Méléard; Anna Melnykova; Samuel Nordmann. Horizontal gene transfer: numerical comparison between stochastic and deterministic approaches. ESAIM. Proceedings, Tome 67 (2020), pp. 135-160. doi: 10.1051/proc/202067009

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