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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.org

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[21]

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

[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.

[23]

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

[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.

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

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.org

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[21]

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

[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.

[23]

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

[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.

[25]

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

[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.

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

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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