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Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays
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Global dynamics of a chemotaxis model with signal-dependent diffusion and sensitivity
Stochastic and deterministic SIS patch model
Aix Marseille Univ, Marseille, France, CNRS, Centrale Marseille, I2M, Marseille, France, Univ. F. H. Boigny, UFR-MI, Abidjan, Côte d'Ivoire |
Here, we consider an SIS epidemic model where the individuals are distributed on several distinct patches. We construct a stochastic model and then prove that it converges to a deterministic model as the total population size tends to infinity. We next study the equilibria of the deterministic model. Our main contribution is a stability result of the endemic equilibrium in the case $ \mathcal{R}_0>1 $. Finally we compare the equilibria with those of the homogeneous model, and with those of isolated patches.
References:
[1] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai,
Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.
doi: 10.1137/060672522. |
[2] |
H. Andersson and T. Britton, Stochastic Epidemic Models and their Statistical Analysis, Vol. 151, Springer-Verlag, New York, 2000.
doi: 10.1007/978-1-4612-1158-7. |
[3] |
N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 2$^{nd}$ edition, Griffin, London, 1975. |
[4] |
M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner,
A clarification of transmission terms in host-microparasite models: Numbers, densities and areas, Epidemiology & Infection, 129 (2002), 147-153.
doi: 10.1017/S0950268802007148. |
[5] |
J. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1994.
doi: 10.1137/1.9781611971262. |
[6] |
D. Bichara, Y. Kang, C. Castillo-Chavez, R. Horan and C. Perrings,
SIS and SIR epidemic models under virtual dispersal, Bulletin of Mathematical Biology, 77 (2015), 2004-2034.
doi: 10.1007/s11538-015-0113-5. |
[7] |
T. Britton and E. Pardoux, Stochastic epidemic models with inference, in Lecture Notes in Math. (eds. F. Ball, C. Larédo, D. Sirl and V. C. Tran), 2255, Springer, (2019), 1–120.
doi: 10.1007/978-3-030-30900-8. |
[8] |
P. V. D. Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[9] |
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley & Sons, 1986.
doi: 10.1002/9780470316658. |
[10] |
A. Fall,
Epidemiological models and Lyapunov functions, Mathematical Modelling of Natural Phenomena, 2 (2007), 62-83.
doi: 10.1051/mmnp:2008011. |
[11] |
M. W. Hirsch,
The dynamical systems approach to differential equations, Bulletin of the American Mathematical Society, 11 (1984), 1-64.
doi: 10.1090/S0273-0979-1984-15236-4. |
[12] |
W. O. Kermack and A. G. Mckendrick,
A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, 115 (1927), 700-721.
|
[13] |
L. Michael and Z. Shuai,
Global stability of an epidemic model in a patchy environment, Canadian Applied Mathematics Quarterly, 17 (2009), 175-187.
|
[14] |
E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, J. Wiley & Sons, 2008.
doi: 10.1002/9780470721872. |
[15] |
J. Rebaza,
Global stability of a multipatch disease epidemics model, Chaos, Solitons & Fractals, 120 (2019), 56-61.
doi: 10.1016/j.chaos.2019.01.020. |
[16] |
R. Varga, Matrix Iterative Analysis, , Englewood Cliffs, NJ: Prentice-Hall, Inc, 1962. |
[17] |
M. Vidyasagar,
Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE Transactions Automatic Control, 25 (1980), 773-779.
doi: 10.1109/TAC.1980.1102422. |
[18] |
G. H. Weiss and M. Dishon,
On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Mathematical Biosciences, 11 (1971), 261-265.
doi: 10.1016/0025-5564(71)90087-3. |
[19] |
S. Zhisheng and P. Driessche,
Global stability of infectious disease models using Lyapunov functions, SIAM Journal on Applied Mathematics, 73 (2013), 1513-1532.
doi: 10.1137/120876642. |
show all references
References:
[1] |
L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai,
Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309.
doi: 10.1137/060672522. |
[2] |
H. Andersson and T. Britton, Stochastic Epidemic Models and their Statistical Analysis, Vol. 151, Springer-Verlag, New York, 2000.
doi: 10.1007/978-1-4612-1158-7. |
[3] |
N. T. J. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, 2$^{nd}$ edition, Griffin, London, 1975. |
[4] |
M. Begon, M. Bennett, R. G. Bowers, N. P. French, S. M. Hazel and J. Turner,
A clarification of transmission terms in host-microparasite models: Numbers, densities and areas, Epidemiology & Infection, 129 (2002), 147-153.
doi: 10.1017/S0950268802007148. |
[5] |
J. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Society for Industrial and Applied Mathematics, 1994.
doi: 10.1137/1.9781611971262. |
[6] |
D. Bichara, Y. Kang, C. Castillo-Chavez, R. Horan and C. Perrings,
SIS and SIR epidemic models under virtual dispersal, Bulletin of Mathematical Biology, 77 (2015), 2004-2034.
doi: 10.1007/s11538-015-0113-5. |
[7] |
T. Britton and E. Pardoux, Stochastic epidemic models with inference, in Lecture Notes in Math. (eds. F. Ball, C. Larédo, D. Sirl and V. C. Tran), 2255, Springer, (2019), 1–120.
doi: 10.1007/978-3-030-30900-8. |
[8] |
P. V. D. Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
[9] |
S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, J. Wiley & Sons, 1986.
doi: 10.1002/9780470316658. |
[10] |
A. Fall,
Epidemiological models and Lyapunov functions, Mathematical Modelling of Natural Phenomena, 2 (2007), 62-83.
doi: 10.1051/mmnp:2008011. |
[11] |
M. W. Hirsch,
The dynamical systems approach to differential equations, Bulletin of the American Mathematical Society, 11 (1984), 1-64.
doi: 10.1090/S0273-0979-1984-15236-4. |
[12] |
W. O. Kermack and A. G. Mckendrick,
A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, 115 (1927), 700-721.
|
[13] |
L. Michael and Z. Shuai,
Global stability of an epidemic model in a patchy environment, Canadian Applied Mathematics Quarterly, 17 (2009), 175-187.
|
[14] |
E. Pardoux, Markov Processes and Applications: Algorithms, Networks, Genome and Finance, J. Wiley & Sons, 2008.
doi: 10.1002/9780470721872. |
[15] |
J. Rebaza,
Global stability of a multipatch disease epidemics model, Chaos, Solitons & Fractals, 120 (2019), 56-61.
doi: 10.1016/j.chaos.2019.01.020. |
[16] |
R. Varga, Matrix Iterative Analysis, , Englewood Cliffs, NJ: Prentice-Hall, Inc, 1962. |
[17] |
M. Vidyasagar,
Decomposition techniques for large-scale systems with nonadditive interactions: stability and stabilizability, IEEE Transactions Automatic Control, 25 (1980), 773-779.
doi: 10.1109/TAC.1980.1102422. |
[18] |
G. H. Weiss and M. Dishon,
On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Mathematical Biosciences, 11 (1971), 261-265.
doi: 10.1016/0025-5564(71)90087-3. |
[19] |
S. Zhisheng and P. Driessche,
Global stability of infectious disease models using Lyapunov functions, SIAM Journal on Applied Mathematics, 73 (2013), 1513-1532.
doi: 10.1137/120876642. |

1.5 | 2 | 1 | 1 | 0.0001 | 0.0001 | (0.332, 0.507) |
1.5 | 2 | 1 | 1 | 0.0001 | 0.0005 | (0.334, 0.497) |
1.5 | 2 | 1 | 1 | 0.001 | 0.0001 | (0.333, 0.497) |
1.5 | 2 | 1 | 1 | 0.0001 | 0.001 | (0.332, 0.497) |
3 | 2.5 | 1 | 1 | 0.0001 | 0.0001 | (0.667, 0.598) |
3 | 2.5 | 1 | 1 | 0.0007 | 0.0001 | (0.666, 0.599) |
3 | 2.5 | 1 | 1 | 0.001 | 0.0001 | (0.666, 0.598) |
3 | 2.5 | 1 | 1 | 0.0001 | 0.001 | (0.666, 0.598) |
1.5 | 1.2 | 01 | 1 | 0.0001 | 0.0001 | (0.332, 0.165) |
1.5 | 1.2 | 1 | 1 | 0.0001 | 0.0009 | (0.332, 0.165) |
1.5 | 1.2 | 1 | 1 | 0.001 | 0.0001 | (0.333, 0.165) |
1.5 | 1.2 | 1 | 1 | 0.0001 | 0.008 | (0.332, 0.165) |
1.5 | 2 | 1 | 1 | 0.0001 | 0.0001 | (0.332, 0.507) |
1.5 | 2 | 1 | 1 | 0.0001 | 0.0005 | (0.334, 0.497) |
1.5 | 2 | 1 | 1 | 0.001 | 0.0001 | (0.333, 0.497) |
1.5 | 2 | 1 | 1 | 0.0001 | 0.001 | (0.332, 0.497) |
3 | 2.5 | 1 | 1 | 0.0001 | 0.0001 | (0.667, 0.598) |
3 | 2.5 | 1 | 1 | 0.0007 | 0.0001 | (0.666, 0.599) |
3 | 2.5 | 1 | 1 | 0.001 | 0.0001 | (0.666, 0.598) |
3 | 2.5 | 1 | 1 | 0.0001 | 0.001 | (0.666, 0.598) |
1.5 | 1.2 | 01 | 1 | 0.0001 | 0.0001 | (0.332, 0.165) |
1.5 | 1.2 | 1 | 1 | 0.0001 | 0.0009 | (0.332, 0.165) |
1.5 | 1.2 | 1 | 1 | 0.001 | 0.0001 | (0.333, 0.165) |
1.5 | 1.2 | 1 | 1 | 0.0001 | 0.008 | (0.332, 0.165) |
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