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

* Corresponding author

Received  September 2018 Revised  March 2019 Published  December 2019 Early access  July 2019

Fund Project: Our research were supported by National Natural Science Foundation of China (11571334 and 11801232) and Natural Science Foundation of Shandong Province (ZR2019QA006).

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 $.

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. DiekmannJ. 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. GuoF.-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. MckenzieY. JinJ. 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. DiekmannJ. 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. GuoF.-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. MckenzieY. JinJ. 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 1.  Comparison of results using two methods by ODEs
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
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