Figure 1.
Comparison of results using two methods by ODEs
-
Previous Article
Bifurcation analysis of an enzyme-catalyzed reaction system with branched sink
- DCDS-B Home
- This Issue
-
Next Article
Semidefinite approximations of invariant measures for polynomial systems
December
2019, 24(12): 6771-6782.
doi: 10.3934/dcdsb.2019166
Remarks on basic reproduction ratios for periodic abstract functional differential equations
| 1. | School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China |
| 2. | Department of Mathematics, Harbin Institute of Technology in Weihai, Weihai, Shandong 264209, China |
In this paper, we extend the theory of basic reproduction ratios $ \mathcal{R}_0 $ in [Liang, Zhang, Zhao, JDDE], which concerns with abstract functional differential systems in a time-periodic environment. We prove the threshold dynamics, that is, the sign of $ \mathcal{R}_0-1 $ determines the dynamics of the associated linear system. We also propose a direct and efficient numerical method to calculate $ \mathcal{R}_0 $.
Keywords:
Basic reproduction ratio,
abstract functional differential system,
time-periodic,
threshold dynamics,
numerical method.
Mathematics Subject Classification:
Primary: 34K20, 35K57; Secondary: 37B55.
Citation:
Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B,
2019, 24
(12)
: 6771-6782.
doi: 10.3934/dcdsb.2019166
![]()
References:
| [1] |
N. Bacaër and E. H. Ait Dads,
Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762.
doi: 10.1007/s00285-010-0354-8. |
| [2] |
N. Bacaër and E. H. Ait Dads,
On the biological interpretation of a definition for the parameter R0 in periodic population models, J. Math. Biol., 65 (2012), 601-621.
doi: 10.1007/s00285-011-0479-4. |
| [3] |
N. Bacaër and S. Guernaoui,
The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.
doi: 10.1007/s00285-006-0015-0. |
| [4] |
L. Burlando,
Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math. (Basel), 56 (1991), 49-57.
doi: 10.1007/BF01190081. |
| [5] |
D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, vol. 279 of Pitman Res. Notes Math. Ser., Longman Scientific & Technical, Harlow, UK, 1992. |
| [6] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985.
doi: 10.1007/978-3-662-00547-7. |
| [7] |
O. Diekmann, J. Heesterbeek and J. A. Metz,
On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
| [8] |
Z. Guo, F.-B. Wang and X. Zou,
Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.
doi: 10.1007/s00285-011-0500-y. |
| [9] |
H. Inaba,
On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.
doi: 10.1007/s00285-011-0463-z. |
| [10] |
T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Reprint of the 1980 edition, Springer-Verlag, Berlin, Heidelberg, 1995. |
| [11] |
X. Liang, L. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for lyme disease), J. Dynam. Differential Equations
doi: 10.1007/s10884-017-9601-7. |
| [12] |
Y. Lou and X.-Q. Zhao,
A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
| [13] |
R. Martin and H. Smith,
Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
| [14] |
H. Mckenzie, Y. Jin, J. Jacobsen and M. Lewis,
R0 analysis of a spatiotemporal model for a stream population, SIAM J. Appl. Dyn. Syst., 11 (2012), 567-596.
doi: 10.1137/100802189. |
| [15] |
D. Posny and J. Wang,
Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments, Appl. Math. Comput., 242 (2014), 473-490.
doi: 10.1016/j.amc.2014.05.079. |
| [16] |
H. R. Thieme,
Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
doi: 10.1137/080732870. |
| [17] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
| [18] |
B.-G. Wang and X.-Q. Zhao,
Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dynam. Differential Equations, 25 (2013), 535-562.
doi: 10.1007/s10884-013-9304-7. |
| [19] |
W. Wang and X.-Q. Zhao,
Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717.
doi: 10.1007/s10884-008-9111-8. |
| [20] |
W. Wang and X.-Q. Zhao,
Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
| [21] |
X. Yu and X.-Q. Zhao,
A nonlocal spatial model for Lyme disease, J. Differential Equations, 261 (2016), 340-372.
doi: 10.1016/j.jde.2016.03.014. |
| [22] |
Y. Zhang and X.-Q. Zhao,
A reaction-diffusion Lyme disease model with seasonality, SIAM J. Appl. Math., 73 (2013), 2077-2099.
doi: 10.1137/120875454. |
| [23] |
X.-Q. Zhao,
Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Differential Equations, 29 (2017), 67-82.
doi: 10.1007/s10884-015-9425-2. |
show all references
References:
| [1] |
N. Bacaër and E. H. Ait Dads,
Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 62 (2011), 741-762.
doi: 10.1007/s00285-010-0354-8. |
| [2] |
N. Bacaër and E. H. Ait Dads,
On the biological interpretation of a definition for the parameter R0 in periodic population models, J. Math. Biol., 65 (2012), 601-621.
doi: 10.1007/s00285-011-0479-4. |
| [3] |
N. Bacaër and S. Guernaoui,
The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436.
doi: 10.1007/s00285-006-0015-0. |
| [4] |
L. Burlando,
Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math. (Basel), 56 (1991), 49-57.
doi: 10.1007/BF01190081. |
| [5] |
D. Daners and P. K. Medina, Abstract Evolution Equations, Periodic Problems and Applications, vol. 279 of Pitman Res. Notes Math. Ser., Longman Scientific & Technical, Harlow, UK, 1992. |
| [6] |
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985.
doi: 10.1007/978-3-662-00547-7. |
| [7] |
O. Diekmann, J. Heesterbeek and J. A. Metz,
On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
| [8] |
Z. Guo, F.-B. Wang and X. Zou,
Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. Math. Biol., 65 (2012), 1387-1410.
doi: 10.1007/s00285-011-0500-y. |
| [9] |
H. Inaba,
On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 65 (2012), 309-348.
doi: 10.1007/s00285-011-0463-z. |
| [10] |
T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Reprint of the 1980 edition, Springer-Verlag, Berlin, Heidelberg, 1995. |
| [11] |
X. Liang, L. Zhang and X.-Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for lyme disease), J. Dynam. Differential Equations
doi: 10.1007/s10884-017-9601-7. |
| [12] |
Y. Lou and X.-Q. Zhao,
A reaction–diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
| [13] |
R. Martin and H. Smith,
Abstract functional-differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
| [14] |
H. Mckenzie, Y. Jin, J. Jacobsen and M. Lewis,
R0 analysis of a spatiotemporal model for a stream population, SIAM J. Appl. Dyn. Syst., 11 (2012), 567-596.
doi: 10.1137/100802189. |
| [15] |
D. Posny and J. Wang,
Computing the basic reproductive numbers for epidemiological models in nonhomogeneous environments, Appl. Math. Comput., 242 (2014), 473-490.
doi: 10.1016/j.amc.2014.05.079. |
| [16] |
H. R. Thieme,
Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211.
doi: 10.1137/080732870. |
| [17] |
P. van den Driessche and J. Watmough,
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.
doi: 10.1016/S0025-5564(02)00108-6. |
| [18] |
B.-G. Wang and X.-Q. Zhao,
Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dynam. Differential Equations, 25 (2013), 535-562.
doi: 10.1007/s10884-013-9304-7. |
| [19] |
W. Wang and X.-Q. Zhao,
Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), 699-717.
doi: 10.1007/s10884-008-9111-8. |
| [20] |
W. Wang and X.-Q. Zhao,
Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673.
doi: 10.1137/120872942. |
| [21] |
X. Yu and X.-Q. Zhao,
A nonlocal spatial model for Lyme disease, J. Differential Equations, 261 (2016), 340-372.
doi: 10.1016/j.jde.2016.03.014. |
| [22] |
Y. Zhang and X.-Q. Zhao,
A reaction-diffusion Lyme disease model with seasonality, SIAM J. Appl. Math., 73 (2013), 2077-2099.
doi: 10.1137/120875454. |
| [23] |
X.-Q. Zhao,
Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Differential Equations, 29 (2017), 67-82.
doi: 10.1007/s10884-015-9425-2. |
Figure 2.
Comparison of results using two methods by Reaction-Diffusion systems
Figure 3.
Comparison of results using two methods by DDEs
Figure 4.
Comparison of results using two methods by Reaction-Diffusion systems with time-delay
Table 1.
Mean values and relative errors under different partitions
| m | Mean numerical value | Relative error(%) |
| 500 | 1.7599 | 0.5681 |
| 1000 | 1.7550 | 0.2838 |
| 2000 | 1.7525 | 0.1419 |
| 8000 | 1.7506 | 0.0351 |
| m | Mean numerical value | Relative error(%) |
| 500 | 1.7599 | 0.5681 |
| 1000 | 1.7550 | 0.2838 |
| 2000 | 1.7525 | 0.1419 |
| 8000 | 1.7506 | 0.0351 |
| [1] |
Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete and Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267 |
| [2] |
Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 |
| [3] |
Jean-Jérôme Casanova. Existence of time-periodic strong solutions to a fluid–structure system. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3291-3313. doi: 10.3934/dcds.2019136 |
| [4] |
Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure and Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133 |
| [5] |
Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170 |
| [6] |
Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure and Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014 |
| [7] |
Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080 |
| [8] |
Yijun Lou, Xiao-Qiang Zhao. Threshold dynamics in a time-delayed periodic SIS epidemic model. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 169-186. doi: 10.3934/dcdsb.2009.12.169 |
| [9] |
Xin Jiang, Zhikun She, Shigui Ruan. Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1967-1990. doi: 10.3934/dcdsb.2020041 |
| [10] |
Seiji Ukai. Time-periodic solutions of the Boltzmann equation. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 579-596. doi: 10.3934/dcds.2006.14.579 |
| [11] |
Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160 |
| [12] |
Martin Heida, Alexander Mielke. Averaging of time-periodic dissipation potentials in rate-independent processes. Discrete and Continuous Dynamical Systems - S, 2017, 10 (6) : 1303-1327. doi: 10.3934/dcdss.2017070 |
| [13] |
Peter Giesl, Holger Wendland. Approximating the basin of attraction of time-periodic ODEs by meshless collocation. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1249-1274. doi: 10.3934/dcds.2009.25.1249 |
| [14] |
Yi Wang, Dun Zhou. Transversality for time-periodic competitive-cooperative tridiagonal systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1821-1830. doi: 10.3934/dcdsb.2015.20.1821 |
| [15] |
Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 |
| [16] |
Suobing Zhang, Zhen-Hui Bu, Zhi-Cheng Wang. Periodic curved fronts in reaction-diffusion equations with ignition time-periodic nonlinearity. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022185 |
| [17] |
Jitai Liang, Ben Niu, Junjie Wei. Linearized stability for abstract functional differential equations subject to state-dependent delays with applications. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 6167-6188. doi: 10.3934/dcdsb.2019134 |
| [18] |
Marcos Lizana, Julio Marín. On the dynamics of a ratio dependent Predator-Prey system with diffusion and delay. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1321-1338. doi: 10.3934/dcdsb.2006.6.1321 |
| [19] |
Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 |
| [20] |
Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]