# A periodic SEIRS epidemic model with a time-dependent latent period

- 196 Downloads

## Abstract

Many infectious diseases have seasonal trends and exhibit variable periods of peak seasonality. Understanding the population dynamics due to seasonal changes becomes very important for predicting and controlling disease transmission risks. In order to investigate the impact of time-dependent delays on disease control, we propose an SEIRS epidemic model with a periodic latent period. We introduce the basic reproduction ratio \(R_0\) for this model and establish a threshold type result on its global dynamics in terms of \(R_0\). More precisely, we show that the disease-free periodic solution is globally attractive if \(R_0<1\); while the system admits a positive periodic solution and the disease is uniformly persistent if \(R_0>1\). Numerical simulations are also carried out to illustrate the analytic results. In addition, we find that the use of the temporal average of the periodic delay may underestimate or overestimate the real value of \(R_0\).

## Keywords

Periodic SEIRS model Time-dependent latent period Basic reproduction ratio Periodic solution Uniform persistence## Mathematics Subject Classification

34K13 37N25 92D30## Notes

### Acknowledgements

This work was supported in part by the China Scholarship Council (201506460020) and the Natural Science and Engineering Research Council of Canada. We are grateful to two referees for their valuable comments and suggestions which led to an improvement of our original manuscript.

## References

- Altizer S, Dobson A, Hosseini P, Hudson P, Pascual M, Rohani P (2006) Seasonality and the dynamics of infectious diseases. Ecol Lett 9(4):467–484CrossRefGoogle Scholar
- Anderson RM, May RM (1979) Population biology of infectious diseases I. Nature 280:361–367CrossRefGoogle Scholar
- Aron JL, Schwartz IB (1984) Seasonality and period-doubling bifurcations in an epidemic model. J Theor Biol 110(4):665–679MathSciNetCrossRefGoogle Scholar
- Beck-Johnson LM, Nelson WA, Paaijmans KP, Read AF, Thomas MB, Bjørnstad ON (2013) The effect of temperature on Anopheles mosquito population dynamics and the potential for malaria transmission. PLoS ONE 8(11):e79276CrossRefGoogle Scholar
- Cooke KL, van den Driessche P (1996) Analysis of an SEIRS epidemic model with two delays. J Math Biol 35:240–260MathSciNetCrossRefGoogle Scholar
- Dowell SF (2001) Seasonal variation in host susceptibility and cycles of certain infectious diseases. Emerg Infect Dis 7(3):369–374CrossRefGoogle Scholar
- Dowell SF, Whitney CG, Wright C, Rose CE Jr, Schuchat A (2003) Seasonal patterns of invasive pneumococcal disease. Emerg Infect Dis 9:573–579CrossRefGoogle Scholar
- Fares A (2011) Seasonality of tuberculosis. J Glob Infect Dis 3(1):46–55CrossRefGoogle Scholar
- Fisman DN (2007) Seasonality of infectious diseases. Annu Rev Public Health 28:127–143CrossRefGoogle Scholar
- Gao LQ, Mena-Lorca J, Hethcote HW (1995) Four SEI endemic models with periodicity and separatrices. Math Biosci 128(1–2):157–184MathSciNetCrossRefGoogle Scholar
- Grassly NC, Fraser C (2006) Seasonal infectious disease epidemiology. Proc Biol Sci 273(1600):2541–2550CrossRefGoogle Scholar
- Greenman J, Kamo M, Boots M (2004) External forcing of ecological and epidemiological systems: a resonance approach. Physica D Nonlinear Phenom 190(1–2):136–151CrossRefGoogle Scholar
- Groberg WJ, McCoy RH, Pilcher KS, Fryer JL (1978) Relation of water temperature to infections of Coho Salmon (
*Oncorhynchus kisutch*), Chinook Salmon (*O. tshawytscha*) and Steelhead Trout (*Salmo gairdneri*) with*Aeromonas salmonicida*and*A. hydrophila*. J Fish Res Board Can 35(1):1–7CrossRefGoogle Scholar - Hale JK, Verduyn Lunel SM (1993) Introduction to functional differential equations. Springer, New YorkCrossRefGoogle Scholar
- Hethcote H (1976) Qualitative analyses of communicable disease models. Math Biosci 28:335–356MathSciNetCrossRefGoogle Scholar
- Huang G, Takeuchi Y, Ma W, Wei D (2010) Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bull Math Biol 72(5):1192–1207MathSciNetCrossRefGoogle Scholar
- Jiao J, Chen L, Cai S (2008) An SEIRS epidemic model with two delays and pulse vaccination. J Syst Sci Complex 21:217–225MathSciNetCrossRefGoogle Scholar
- Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics II—the problem of endemicity. Proc R Soc A 138:55–83CrossRefGoogle Scholar
- Kermack WO, McKendrick AG (1991) Contributions to the mathematical theory of epidemics I. Bull Math Biol 53:33–55Google Scholar
- Kot M (2001) Elements of mathematical ecology. Cambridge University Press, New YorkCrossRefGoogle Scholar
- Liang X, Zhao X-Q (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun Pure Appl Math 60:1–40MathSciNetCrossRefGoogle Scholar
- Liang X, Zhang L, Zhao X-Q (2017) Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J Dyn Differ Equ. https://doi.org/10.1007/s10884-017-9601-7
- Liu L, Zhao X-Q, Zhou Y (2010) A Tuberculosis model with seasonality. Bull Math Biol 72(4):931–952MathSciNetCrossRefGoogle Scholar
- London WP, Yorke JA (1973) Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates. Am J Epidemiol 98(6):453–468CrossRefGoogle Scholar
- Lou Y, Zhao X-Q (2017) A theoretical approach to understanding population dynamics with seasonal developmental durations. J Nonlinear Sci 27(2):573–603MathSciNetCrossRefGoogle Scholar
- Lovell DJ, Hunter T, Powers SJ, Parker SR, van den Bosch F (2004) Effect of temperature on latent period of septoria leaf blotch on winter wheat under outdoor conditions. Plant Pathol 53:170–181CrossRefGoogle Scholar
- Ma W, Song M, Takeuchi Y (2004) Global stability of an SIR epidemic model with time delay. Appl Math Lett 17(10):1141–1145MathSciNetCrossRefGoogle Scholar
- Mateus JP, Silva CM (2017) Existence of periodic solutions of a periodic SEIRS model with general incidence. Nonlinear Anal Real World Appl 34:379–402MathSciNetCrossRefGoogle Scholar
- Meyer FP, Warren JW, Carey TG (1983) A guide to integrated fish health management in the Great Lakes basin. Great Lakes Fishery Commission, Ann Arbor, MI. Special Publication, 83–2: 272pGoogle Scholar
- Nakata Y, Kuniya T (2010) Global dynamics of a class of SEIRS epidemic models in a periodic environment. J Math Anal Appl 363(1):230–237MathSciNetCrossRefGoogle Scholar
- Nisbet RM, Gurney WSC (1983) The systematic formulation of population models for insects with dynamically varying instar duration. Theor Popul Biol 23(1):114–135MathSciNetCrossRefGoogle Scholar
- Omori R, Adams B (2011) Disrupting seasonality to control disease outbreaks: the case of koi herpes virus. J Theor Biol 271:159–165MathSciNetCrossRefGoogle Scholar
- Purse BV, Mellor PS, Rogers DJ, Samuel AR, Mertens PP, Baylis M (2005) Climate change and the recent emergence of bluetongue in Europe. Nat Rev Microbiol 3(2):171–181CrossRefGoogle Scholar
- Qi L, Cui J (2013) The stability of an SEIRS model with nonlinear incidence, vertical transmission and time delay. Appl Math Comput 221:360–366MathSciNetzbMATHGoogle Scholar
- Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. American Mathematical Society, ProvidencezbMATHGoogle Scholar
- Snieszko SF (1974) The effects of environmental stress on outbreaks of infectious diseases of fishes. J Fish Biol 6(2):197–208CrossRefGoogle Scholar
- Sultan B, Labadi K, Guégan JF, Janicot S (2005) Climate drives the meningitis epidemics onset in West Africa. PLoS Med 2:43–49CrossRefGoogle Scholar
- Towers S, Vogt-Geisse K, Zheng Y, Feng Z (2011) Antiviral treatment for pandemic influenza: assessing potential repercussions using a seasonally forced SIR model. J Theor Biol 289:259–268MathSciNetCrossRefGoogle Scholar
- Walter W (1997) On strongly monotone flows. Ann Pol Math 66:269–274MathSciNetCrossRefGoogle Scholar
- Wang W (2002) Global behavior of an SEIRS epidemic model with time delays. Appl Math Lett 15:423–428MathSciNetCrossRefGoogle Scholar
- Wang X, Zhao X-Q (2017a) Dynamics of a time-delayed Lyme disease model with seasonality. SIAM J Appl Dyn Syst 16(2):853–881MathSciNetCrossRefGoogle Scholar
- Wang X, Zhao X-Q (2017b) A malaria transmission model with temperature-dependent incubation period. Bull Math Biol 79(5):1155–1182MathSciNetCrossRefGoogle Scholar
- Vaidya NK, Wahl LM (2015) Avian influenza dynamics under periodic environmental conditions. SIAM J Appl Math 75(2):443–467MathSciNetCrossRefGoogle Scholar
- Zhang T, Teng Z (2007) On a nonautonomous SEIRS model in epidemiology. Bull Math Biol 69(8):2537–2559MathSciNetCrossRefGoogle Scholar
- Zhao X-Q (2017a) Basic reproduction ratios for periodic compartmental models with time delay. J Dyn Differ Equ 29(1):67–82MathSciNetCrossRefGoogle Scholar
- Zhao X-Q (2017b) Dynamical systems in population biology, 2nd edn. Springer, New YorkCrossRefGoogle Scholar