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December  2017, 37(12): 6123-6138. doi: 10.3934/dcds.2017263

Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients

a. 

College of Science, National University of Defense Technology, Changsha, Hunan, China

b. 

School of Information Engineering, Nanchang Institute of Technology, Nanchang, Jiangxi, China

* Corresponding author: Xiao Wang

Received  October 2016 Revised  July 2017 Published  August 2017

It is extremely difficult to establish the existence of almost periodic solutions for delay differential equations via methods that need the compactness conditions such as Schauder's fixed point theorem. To overcome this difficulty, in this paper, we employ a novel technique to construct a contraction mapping, which enables us to establish the existence of almost periodic solution for a delay differential equation system with time-varying coefficients. When the system's coefficients are periodic, coincide degree theory is used to establish the existence of periodic solutions. Global stability results are also obtained by the method of Liapunov functionals.

Citation: Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263
References:
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G. Abramson, S. Goncalves and M. F. C. Gomes, Epidemic oscillations: Interaction between delays and seasonality, arXiv preprint, arXiv: 1303.3779, 2013-arxiv.orgGoogle Scholar

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I. Area, J. Losada and F. Nda$\ddot{i}$rou, et al. Mathematical Modeling of 2014 Ebola Outbreak, Mathematical Methods in the Applied Sciences, 2015. doi: 10.1002/mma.3794. Google Scholar

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J. Arino and S. Portet, Epidemiological implications of mobility between a large urban centre and smaller satellite cities, J. Math. Biol., 71 (2015), 1243-1265. doi: 10.1007/s00285-014-0854-z. Google Scholar

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L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients, Comput. Math. Appl., 51 (2006), 1-16. doi: 10.1016/j.camwa.2005.09.001. Google Scholar

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F. BrauerP. van den Driessche and L. Wang, Oscillations in a patchy environment disease model, Mathematical Biosciences, 215 (2008), 1-10. doi: 10.1016/j.mbs.2008.05.001. Google Scholar

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W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathemaics, 629, Springer-Verlag, Berlin, 1978. Google Scholar

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F. Cordova-LepeG. RobledoM. Pinto and E. Gonzalez-Olivares, Modeling pulse infectious events irrupting into a controlled context: A SIS disease with almost periodic parameters, Appl. Math. Model., 36 (2012), 1323-1337. doi: 10.1016/j.apm.2011.07.085. Google Scholar

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H. S. DingQ. L. Liu and J. J. Nieto, Existence of positive almost periodic solutions to a class of hematopoiesis model, Applied Mathematical Modelling, 40 (2016), 3289-3297. doi: 10.1016/j.apm.2015.10.020. Google Scholar

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A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, 377, Springer-Verlag, New York, 1974. Google Scholar

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D. Fisman, E. Khoo and A. TuiteVersion, Early Epidemic Dynamics of the West African 2014 Ebola Outbreak: Estimates Derived with a Simple Two-parameter Model, PLoS Curr. 2014. doi: 10.1371/currents.outbreaks.89c0d3783f36958d96ebbae97348d571. Google Scholar

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R. K. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. Google Scholar

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K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Press, Boston, 1992. doi: 10.1007/978-94-015-7920-9. Google Scholar

[14]

K. Gopalsamy and P. Weng, Global attractivity and level crossing in model of Hematopoiesis, Bulletin of the Institute of Mathematics, Academia Sinica, 22 (1994), 341-360. Google Scholar

[15]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar

[16]

C. Y. He, Almost periodic differential equations(In Chinese), Higher Education Press, Beijing, 1992.Google Scholar

[17]

Y. H. HsiehP. van den Driessche and L. Wang, Impact of travel between patches for spatial spread of disease, Bull. Math. Biol., 69 (2007), 1355-1375. doi: 10.1007/s11538-006-9169-6. Google Scholar

[18]

Z-W. JiaG. Tang and Z. Jin, Modeling the impact of immigration on the epidemiology of tuberculosis, Theor. Popul. Biol., 73 (2008), 437-448. doi: 10.1016/j.tpb.2007.12.007. Google Scholar

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W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. Google Scholar

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L. LiY. Bai and Z. Jin, Periodic solutions of an epidemic model with saturated treatment, Nonlinear Dyn., 76 (2014), 1099-1108. doi: 10.1007/s11071-013-1193-0. Google Scholar

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R. P. Sigdel and C. C. McCluskey, Disease dynamics for the hometown of migrant workers, Math. Biosci. Eng., 11 (2014), 1175-1180. doi: 10.3934/mbe.2014.11.1175. Google Scholar

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L. StoneR. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. Google Scholar

[24]

M. StrandX. Wang and X. Duan, Presence and awareness of infectious disease among Chinese migrant workers, Int'l Quarterly of Community Health Education, 26 (2007), 379-395. Google Scholar

[25]

The National Migrant Workers Monitoring Report, 2017, http://www.stats.gov.cn/.Google Scholar

[26]

B. Wang and X. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J Dyn. Diff. Equat., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7. Google Scholar

[27]

L. Wang and X. Wang, Influence of temporary migration on the transmission of infectious diseases in a migrants' home village, J. Theoret. Biol., 300 (2012), 100-109. doi: 10.1016/j.jtbi.2012.01.004. Google Scholar

[28]

X. Wang and H. Zhang, A new approach to the existence, nonexistence and uniqueness of positive almost periodic solution for a model of Hematopoiesis, Nonlinear Analysis: Real World Applications, 11 (2010), 60-66. doi: 10.1016/j.nonrwa.2008.10.015. Google Scholar

[29]

M. E. Wilson, Travel and the emergence of infectious diseases, Emerg. Infect. Dis., 3 (1996), 51-66. doi: 10.1300/J096v03n01_05. Google Scholar

[30]

Y. Xiao, J. C. Beier, R. S. Cantrell, C. Cosner, L. D. DeAngelis and S. Ruan, Modelling the effects of seasonality and socioeconomic impact on the transmission of rift valley fever virus, PLOS Neglected Tropical Diseases, 9 (2015), e3388. doi: 10.1371/journal. pntd. 0003388. Google Scholar

[31]

R. XuD. C. Ekiert1J. C. KrauseR. HaiJ. E. Crowe Jr and I. A. Wilson, Structural basis of preexisting immunity to the 2009 H1N1 pandemic influenza virus, Science, 328 (2010), 357-360. doi: 10.1126/science.1186430. Google Scholar

[32]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Analysis: Real World Applications, 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040. Google Scholar

[33]

X. Zhou and J. Cui, Threshold dynamics for a cholera epidemic model with periodic transmission rate, Appl. Math. Model., 37 (2013), 3093-3101. doi: 10.1016/j.apm.2012.07.044. Google Scholar

[34]

Y. ZhouZ. Ma and F. Brauer, Discrete epidemic model for SARS transmission and control in China, Math. Comput. Model., 40 (2004), 1491-1506. doi: 10.1016/j.mcm.2005.01.007. Google Scholar

show all references

References:
[1]

G. Abramson, S. Goncalves and M. F. C. Gomes, Epidemic oscillations: Interaction between delays and seasonality, arXiv preprint, arXiv: 1303.3779, 2013-arxiv.orgGoogle Scholar

[2]

I. Area, J. Losada and F. Nda$\ddot{i}$rou, et al. Mathematical Modeling of 2014 Ebola Outbreak, Mathematical Methods in the Applied Sciences, 2015. doi: 10.1002/mma.3794. Google Scholar

[3]

J. Arino and S. Portet, Epidemiological implications of mobility between a large urban centre and smaller satellite cities, J. Math. Biol., 71 (2015), 1243-1265. doi: 10.1007/s00285-014-0854-z. Google Scholar

[4]

L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients, Comput. Math. Appl., 51 (2006), 1-16. doi: 10.1016/j.camwa.2005.09.001. Google Scholar

[5]

F. Brauer and P. van den Driessche, Models for transmission of disease with immigration of infectives, Math. Biosci., 171 (2001), 143-154. doi: 10.1016/S0025-5564(01)00057-8. Google Scholar

[6]

F. BrauerP. van den Driessche and L. Wang, Oscillations in a patchy environment disease model, Mathematical Biosciences, 215 (2008), 1-10. doi: 10.1016/j.mbs.2008.05.001. Google Scholar

[7]

W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathemaics, 629, Springer-Verlag, Berlin, 1978. Google Scholar

[8]

F. Cordova-LepeG. RobledoM. Pinto and E. Gonzalez-Olivares, Modeling pulse infectious events irrupting into a controlled context: A SIS disease with almost periodic parameters, Appl. Math. Model., 36 (2012), 1323-1337. doi: 10.1016/j.apm.2011.07.085. Google Scholar

[9]

H. S. DingQ. L. Liu and J. J. Nieto, Existence of positive almost periodic solutions to a class of hematopoiesis model, Applied Mathematical Modelling, 40 (2016), 3289-3297. doi: 10.1016/j.apm.2015.10.020. Google Scholar

[10]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, 377, Springer-Verlag, New York, 1974. Google Scholar

[11]

D. Fisman, E. Khoo and A. TuiteVersion, Early Epidemic Dynamics of the West African 2014 Ebola Outbreak: Estimates Derived with a Simple Two-parameter Model, PLoS Curr. 2014. doi: 10.1371/currents.outbreaks.89c0d3783f36958d96ebbae97348d571. Google Scholar

[12]

R. K. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977. Google Scholar

[13]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Press, Boston, 1992. doi: 10.1007/978-94-015-7920-9. Google Scholar

[14]

K. Gopalsamy and P. Weng, Global attractivity and level crossing in model of Hematopoiesis, Bulletin of the Institute of Mathematics, Academia Sinica, 22 (1994), 341-360. Google Scholar

[15]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar

[16]

C. Y. He, Almost periodic differential equations(In Chinese), Higher Education Press, Beijing, 1992.Google Scholar

[17]

Y. H. HsiehP. van den Driessche and L. Wang, Impact of travel between patches for spatial spread of disease, Bull. Math. Biol., 69 (2007), 1355-1375. doi: 10.1007/s11538-006-9169-6. Google Scholar

[18]

Z-W. JiaG. Tang and Z. Jin, Modeling the impact of immigration on the epidemiology of tuberculosis, Theor. Popul. Biol., 73 (2008), 437-448. doi: 10.1016/j.tpb.2007.12.007. Google Scholar

[19]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700-721. Google Scholar

[20]

L. LiY. Bai and Z. Jin, Periodic solutions of an epidemic model with saturated treatment, Nonlinear Dyn., 76 (2014), 1099-1108. doi: 10.1007/s11071-013-1193-0. Google Scholar

[21]

J. D. Murray, Mathematical Biology, I: An Introduction, Springer, 2002. Google Scholar

[22]

R. P. Sigdel and C. C. McCluskey, Disease dynamics for the hometown of migrant workers, Math. Biosci. Eng., 11 (2014), 1175-1180. doi: 10.3934/mbe.2014.11.1175. Google Scholar

[23]

L. StoneR. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. Google Scholar

[24]

M. StrandX. Wang and X. Duan, Presence and awareness of infectious disease among Chinese migrant workers, Int'l Quarterly of Community Health Education, 26 (2007), 379-395. Google Scholar

[25]

The National Migrant Workers Monitoring Report, 2017, http://www.stats.gov.cn/.Google Scholar

[26]

B. Wang and X. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J Dyn. Diff. Equat., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7. Google Scholar

[27]

L. Wang and X. Wang, Influence of temporary migration on the transmission of infectious diseases in a migrants' home village, J. Theoret. Biol., 300 (2012), 100-109. doi: 10.1016/j.jtbi.2012.01.004. Google Scholar

[28]

X. Wang and H. Zhang, A new approach to the existence, nonexistence and uniqueness of positive almost periodic solution for a model of Hematopoiesis, Nonlinear Analysis: Real World Applications, 11 (2010), 60-66. doi: 10.1016/j.nonrwa.2008.10.015. Google Scholar

[29]

M. E. Wilson, Travel and the emergence of infectious diseases, Emerg. Infect. Dis., 3 (1996), 51-66. doi: 10.1300/J096v03n01_05. Google Scholar

[30]

Y. Xiao, J. C. Beier, R. S. Cantrell, C. Cosner, L. D. DeAngelis and S. Ruan, Modelling the effects of seasonality and socioeconomic impact on the transmission of rift valley fever virus, PLOS Neglected Tropical Diseases, 9 (2015), e3388. doi: 10.1371/journal. pntd. 0003388. Google Scholar

[31]

R. XuD. C. Ekiert1J. C. KrauseR. HaiJ. E. Crowe Jr and I. A. Wilson, Structural basis of preexisting immunity to the 2009 H1N1 pandemic influenza virus, Science, 328 (2010), 357-360. doi: 10.1126/science.1186430. Google Scholar

[32]

Z. Yuan and L. Wang, Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear Analysis: Real World Applications, 11 (2010), 995-1004. doi: 10.1016/j.nonrwa.2009.01.040. Google Scholar

[33]

X. Zhou and J. Cui, Threshold dynamics for a cholera epidemic model with periodic transmission rate, Appl. Math. Model., 37 (2013), 3093-3101. doi: 10.1016/j.apm.2012.07.044. Google Scholar

[34]

Y. ZhouZ. Ma and F. Brauer, Discrete epidemic model for SARS transmission and control in China, Math. Comput. Model., 40 (2004), 1491-1506. doi: 10.1016/j.mcm.2005.01.007. Google Scholar

Figure 1.  Distributions of holidays in China for the period of 2012-2016.
Figure 2.  Numerical solutions of (1.2) with $\lambda_0=0.5$, $\lambda_1=0.2$, $\beta_0=0.0003$, $\beta_1=0.01$, $m_0=0.4$, $m_1=0.01$, $\omega_1=\frac{\pi}{3.2}$, $\omega_2=1$, $\mu_S=\mu_R=\frac{1}{72}$, $\gamma=0.05$, $\delta=\frac{1}{50}$, $\tau=0.78$, $p=\frac{2\tau}{100+2\tau}$. Three sets of initial conditions IV1, IV2 and IV3 are used
Figure 3.  Numerical solutions of (1.2) with the same parameter values as in Figure 2 except that $\lambda_0=1.5$ and $m_1=0.2$
Figure 4.  The numerical solution to system (1.2) with $\lambda_0=0.5$, $\lambda_1=0.2$, $\beta_0=0.001$, $\beta_1=0.01$, $m_0=0.4$, $m_1=0.2, $ $\omega_1=1$, $\omega_2=0$, $\mu_S=\mu_R=\frac{1}{72}$, $\gamma=0.05$, $\delta=\frac{1}{50}$, $\tau=0.78$ and $p=\frac{2\tau}{100+2\tau}$ and the initial conditions as $I(0)=2$, $S(\theta)=5(1+0.1\cos(\omega_1\theta))$ and $R(\theta)=1$ for $\theta\in[-\tau, 0]$
Figure 5.  Numerical solutions of (1.2) with $\lambda_0=0.5$, $\lambda_1=0.2$, $\beta_0=0.001$, $\beta_1=0.01$, $m_0=0.3$, $m_1=0.3$, $\omega_1=1$, $\omega_2=0$, $\mu_S=\mu_R=\frac{1}{72}$, $\gamma=0.05$, $\delta=\frac{1}{50}$, $\tau=0.78$ and $p=\frac{2\tau}{100+2\tau}$
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