Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles

Mathematical Models and Methods in Applied Sciences

Volumn 19, Issue SUPPL. 1, 2009, Pages 1405-1425

  De Lillo, S ^a,b   Delitala, M ^a   Salvatori, M C ^a  

^a UNIVERSITY OF PERUGIA   (Italy)

^b POLITECNICO DI TORINO   (Italy)

Author keywords

Active particles; Biology; Kinetic theory; Living systems; Nonlinearity.; Population dynamics

Indexed keywords

ACTIVE PARTICLES; HETEROGENEOUS DISTRIBUTIONS; IMMUNE SYSTEMS; LIVING SYSTEMS; MATHEMATICAL APPROACH; NONLINEARITY.; NUMERICAL SIMULATION; REAL SYSTEMS;

BIOLOGY; COMPUTER SIMULATION; EPIDEMIOLOGY; POPULATION DYNAMICS;

KINETIC THEORY;

EID: 69849114367     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202509003838     Document Type: Article

References (29)

1. L. Arlotti, N. Bellomo and E. De Angelis, Generalized kinetic (Boltzmann) models: Mathematical structures and applications, Math. Mod. Meth. Appl. Sci. 12 (2002) 567-591.
2. N. Bellomo, Modeling Complex Living Systems. A Mathematical Kinetic Theory and Stochastic Game Approach (Birkhäuser, 2008).
3. M. Delitala and A. Tosin, Mathematical modeling of vehicular traffc: A discrete kinetic theory approach, Math. Mod. Meth. Appl. Sci. 17 (2007) 901-932.
4. N. Bellomo and G. Forni, Complex multicellular systems and immune competition: New paradigms looking for a mathematical theory, Current Top. Develop. Biol. 81 (2008) 485-502.
5. M. Lo Schiavo, A dynamical model of electoral competition, Math. Comp. Model. 7 (2006) 1288-1309.
6. B. Carbonaro and C. Giordano, A second step towards mathematical models in psychology: A stochastic description of human feelings, Math. Comp. Model. 41 (2005) 587-614.
7. L. Bertotti and M. Delitala, On the existence of limit cycles in opinion formation processes under time-periodic iñuence of persuaders, Math. Mod. Meth. Appl. Sci. 18 (2008) 913-934.
8. B. Perthame, Mathematical tools for kinetic equations, Bull. Am. Math. Soc. 41 (2004) 205-244.
9. S. De Lillo, M. C. Salvatori and N. Bellomo, Mathematical tools of the kinetic theory of active particles with some reasoning on the modeling progression and heterogeneity, Math. Comp. Model. 45 (2007) 564-578.
10. A. Belloquid and M. Delitala, Modeling Complex Biological Systems: A Kinetic Theory Approach (Birkhäuser, 2006).
11. H. Moore and W. Gu, A mathematical model for treatment-resistant mutations of HIV, Math. Biosci. Eng. 2 (2005) 363-380.
12. M. Delitala, Generalized kinetic theory approach to modeling spread and evolution of epidemics, Math. Comp. Model. 39 (2004) 1-12.
13. O. Dieckmann and J. A. P. Heesterbeck, Mathematical Epidemiology of Infectious Diseases (Wiley, 2000).
14. H. R. Thieme, Mathematics in Population Biology (Princeton Univ. Press, 2003).
15. B. Perthame, Transport Equations in Biology (Birkhäuser, 2007).
16. A. D'Onofrio, Tumor-immune system interaction and immunotherapy: Modeling the tumor-stimulated proliferation of éectors, Math. Mod. Meth. Appl. Sci. 16 (2006) 1375-1401.
17. M. L. Hibid, E. Sanchez and R. Bravo de la Parra, State-dependent delays associated to threshold phenomena in structured populations, Math. Mod. Meth. Appl. Sci. 17 (2007) 877-900.
18. A. D'Onofrio and I. P. M. Tomlison, A nonlinear mathematical model of cell turnover, díerentiation and tumorigenesis in the intestinal crypt, J. Theor. Biol. 244 (2007) 367-374.
19. B. Perthame, From díerential equations to structured population dynamics, Frontiers Math. 1 (2007) 1-26.
20. P. C. Nowell, Tumor progression: A brief historical perspective, Semi. Cancer Biol. 12 (2002) 261-266.
21. N. Bellomo and M. Delitala, From the mathematical kinetic, and stochastic game theory for active particles to modeling mutations, onset, progression and immune competition of cancer cells, Phys. Life Rev. 5 (2008) 183-206.
22. C. R. Woese, A new biology for a new century, Microbiol. Mol. Biol. Rev. 68 (2004) 173-186.
23. E. De Angelis and P. E. Jabin, Mathematical models of therapeutical actions related to tumor and immune system competition, Math. Meth. Appl. Sci. 28 (2005) 2061-2083.
24. R. Erban, S. J. Chapman, K. D. Fisher, I. G. Kevrekidis and L. W. Seymour, Dynamics of polydispersed irreversible adsorption, Math. Mod. Meth. Appl. Sci. 17 (2007) 759-782.
25. H. G. Othmer and T. Hillen, The díusion limit of transport equations II: Chemotaxis equations, SIAM J. Appl. Math. 62 (2002) 1222-1250.
26. F. A. Chalub, P. A. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-díusion limits, Monat. Math. 142 (2004) 123-141.
27. N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multicellular growing systems: Hyperbolic limits towards macroscopic description, Math. Mod. Meth. Appl. Sci. 17 (2007) 1675-1692.
28. M. A. Nowak and K. Sigmund, Evolutionary dynamics of biological games, Science 303 (2004) 793-799.
29. N. Bellomo, N. Li and P. K. Maini, On the foundations of cancer modeling. Selected topics, speculations, and perspectives, Math. Mod. Meth. Appl. Sci. 18 (2008) 593-646.