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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Reprint of the 1980 edition, Springer-Verlag, Berlin, Heidelberg, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Reprint of the 1980 edition, Springer-Verlag, Berlin, Heidelberg, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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