# Basic reproduction ratios for periodic and time-delayed compartmental models with impulses

- 54 Downloads

## Abstract

Much work has focused on the basic reproduction ratio \({\mathcal {R}}_0\) for a variety of compartmental population models, but the theory of \({\mathcal {R}}_0\) remains unsolved for periodic and time-delayed impulsive models. In this paper, we develop the theory of \({\mathcal {R}}_0\) for a class of such impulsive models. We first introduce \({\mathcal {R}}_0\) and show that it is a threshold parameter for the stability of the zero solution of an associated linear system. Then we apply this theory to a time-delayed computer virus model with impulse treatment and obtain a threshold result on its global dynamics in terms of \({\mathcal {R}}_0\). Numerically, it is found that the basic reproduction ratio of the time-averaged delayed impulsive system may overestimate the spread risk of the virus.

## Keywords

Impulsive models Time delay Basic reproduction ratio Computer virus Threshold dynamics## Mathematics Subject Classification

34A37 92D30 37N25## Notes

### Acknowledgements

Zhenguo Bai would like to thank the China Scholarship Council (201606965021) for financial support during the period of his overseas study and to express his gratitude to the Department of Mathematics and Statistics, Memorial University of Newfoundland, for its kind hospitality. We are also grateful to two anonymous referees for careful reading and valuable comments which led to improvements of our original manuscript.

## References

- Bacaër N, Ait Dads EH (2012) On the biological interpretation of a definition for the parameter \(R_0\) in periodic population models. J Math Biol 65:601–621MathSciNetCrossRefGoogle Scholar
- Bacaër N, Guernaoui S (2006) The epidemic threshold of vector-borne diseases with seasonality. J Math Biol 53:421–436MathSciNetCrossRefGoogle Scholar
- Bainov D, Simeonov PS (1993) Impulsive differential equations: periodic solutions and applications. Longman Harlow, New YorkzbMATHGoogle Scholar
- Ballinger G, Liu X (2000) Existence, uniqueness and boundedness results for impulsive delay differential equations. Appl Anal 74:71–93MathSciNetCrossRefGoogle Scholar
- Billings L, Spears WM, Schwartz IB (2002) A unified prediction of computer virus spread in connected networks. Phys Lett A 297:261–266MathSciNetCrossRefGoogle Scholar
- Burlando L (1991) Monotonicity of spectral radius for positive operators on ordered Banach spaces. Arch Math 56:49–57MathSciNetCrossRefGoogle Scholar
- Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. Wiley series in mathematical and computational biology. Wiley, West SussexGoogle Scholar
- Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio \(R_0\) in the models for infectious disease in heterogeneous populations. J Math Biol 28:365–382MathSciNetCrossRefGoogle Scholar
- Du Y (2006) Order structure and topological methods in nonlinear partial differential equations. Maximum principles and applications, vol 1. World Scientific, New JerseyzbMATHGoogle Scholar
- Faria T, Oliveira JJ (2016) On stability for impulsive delay differential equations and application to a periodic Lasota–Wazewska model. Discrete Contin Dyn Syst Ser B 21:2451–2472MathSciNetCrossRefGoogle Scholar
- Faria T, Oliveira JJ (2019) Existence of positive periodic solutions for scalar delay differential equations with and without impulses. J Dyn Differ Equ 31:1223–1245MathSciNetCrossRefGoogle Scholar
- Gourley SA, Liu R, Wu J (2007) Eradicating vector-borne diseases via age-structured culling. J Math Biol 54:309–335MathSciNetCrossRefGoogle Scholar
- Heffernan JM, Smith RJ, Wahl LM (2005) Perspectives on the basic reproductive ratio. J R Soc Interface 2:281–293CrossRefGoogle Scholar
- Hethcote HW (2000) The mathematics of infectious diseases. SIAM Rev 42:599–653MathSciNetCrossRefGoogle Scholar
- Inaba H (2012) On a new perspective of the basic reproduction number in heterogeneous environments. J Math Biol 65:309–348MathSciNetCrossRefGoogle Scholar
- Liang X, Zhao X-Q (2007) Asymptotic speeds of spread and traveling waves formonotone semiflows with applications. Commun Pure Appl Math 60:1–40CrossRefGoogle Scholar
- Liang X, Zhang L, Zhao X-Q (2019) Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J Dyn Differ Equ 31:1247–1278MathSciNetCrossRefGoogle Scholar
- Liu X, Takeuchi Y (2007) Periodicity and global dynamics of an impulsive delay Lasota–Wazewska model. J Math Anal Appl 327:326–341MathSciNetCrossRefGoogle Scholar
- Mitchell C, Kribs C (2017) A comparison of methods for calculating the basic reproductive number for periodic epidemic systems. Bull Math Biol 79:1846–1869MathSciNetCrossRefGoogle Scholar
- Ren J, Yang X, Yang L-X, Xu Y, Yang F (2012a) A delayed computer virus propagation model and its dynamics. Chaos Solitons Fractals 45:74–79MathSciNetCrossRefGoogle Scholar
- Ren J, Yang X, Zhu Q, Yang L, Zhang C (2012b) A novel computer virus model and its dynamics. Nonlinear Anal Real World Appl 13:376–384MathSciNetCrossRefGoogle Scholar
- Shen J, Li J (2009) Existence and global attractivity of positive periodic solutions for impulsive predator–prey model with dispersion and time delays. Nonlinear Anal Real World Appl 10:227–243MathSciNetCrossRefGoogle Scholar
- Tang S, Liang J, Tan Y, Cheke R (2013) Threshold conditions for integrated pest management models with pesticides that have residual effects. J Math Biol 66:1–35MathSciNetCrossRefGoogle Scholar
- Terry AJ (2010) Impulsive culling of a structured population on two patches. J Math Biol 61:843–875MathSciNetCrossRefGoogle Scholar
- Terry AJ, Gourley SA (2010) Perverse consequences of infrequently culling a pest. Bull Math Biol 72:1666–1695MathSciNetCrossRefGoogle Scholar
- Thieme HR (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J Appl Math 70:188–211MathSciNetCrossRefGoogle Scholar
- van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48MathSciNetCrossRefGoogle Scholar
- Wang W, Zhao X-Q (2008) Threshold dynamics for compartmental epidemic models in periodic environments. J Dyn Differ Equ 20:699–717MathSciNetCrossRefGoogle Scholar
- Xu D, Zhao X-Q (2005) Dynamics in a periodic competitive model with stage structure. J Math Anal Appl 311:417–438MathSciNetCrossRefGoogle Scholar
- Yang Y, Xiao Y (2012) Threshold dynamics for compartmental epidemic models with impulses. Nonlinear Anal Real World Appl 13:224–234MathSciNetCrossRefGoogle Scholar
- Yang L, Yang X (2014) The pulse treatment of computer viruses: a modeling study. Nonlinear Dyn 76:1379–1393MathSciNetCrossRefGoogle Scholar
- Yang Z, Huang C, Zou X (2018) Effect of impulsive controls in a model system for age-structured population over a patchy environment. J Math Biol 76:1387–1419MathSciNetCrossRefGoogle Scholar
- Yuan H, Chen G (2008) Network virus-epidemic model with the point-to-group information propagation. Appl Math Comput 206:357–367MathSciNetzbMATHGoogle Scholar
- Zhao X-Q (2017) Basic reproduction ratios for periodic compartmental models with time delay. J Dyn Differ Equ 29:67–82MathSciNetCrossRefGoogle Scholar
- Zou CC, Gong W, Towsley D, Gao L (2005) The monitoring and early detection of internet worms. IEEE/ACM Trans Netw 13:961–974CrossRefGoogle Scholar