# Threshold dynamics of an infective disease model with a fixed latent period and non-local infections

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## Abstract

In this paper, we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction–diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local *scalar* reaction diffusion equation or equivalently by the basic reproduction number \({\mathcal{R}_0}\) for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and \({\mathcal{R}_0}\). In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute \({\mathcal{R}_0}\).

## Keywords

Infectious disease model Reaction–diffusion equation Non-local infection Delay Principal eigenvalue Basic reproduction number## Mathematics Subject Classification (2000)

35K57 37N25 92D30## Preview

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## References

- Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio in the models for infectious disease in heterogeneous population. J Math Biol 28: 365–382MathSciNetzbMATHCrossRefGoogle Scholar
- Engel KJ, Nagel R (2000) One-parameter semigroups for linear evolution equations. Springer, BerlinzbMATHGoogle Scholar
- Friedman A (1964) Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
- Hale JK (1988) Asymptotic behavior of dissipative systems. American Mathematical Society, ProvidencezbMATHGoogle Scholar
- Hirsch MW (1984) The dynamical systems approach to differential equations. Bull Am Math Soc 11: 1–64zbMATHCrossRefGoogle Scholar
- Hsu SB, Jiang J, Wang FB (2010) On a system of reaction-diffusion equations arising from competition with internal storage in an unstirred chemostat. J Differ Equ 248: 2470–2496MathSciNetzbMATHCrossRefGoogle Scholar
- Kerscher W, Nagel R (1984) Asymptotic behavior of one-parameter semigroups of positive operators. Acta Applicandae Math 2: 297–309MathSciNetzbMATHCrossRefGoogle Scholar
- Krkosek M, Lewis MA (2010) An
*R*_{0}theory for source-sink dynamics with application to Dreissena competition. Theor Ecol 3: 25–43CrossRefGoogle Scholar - Li J, Zou X (2009a) Generalization of the Kermack-McKendrick SIR model to a patchy environment for a disease with latency. Math Model Nat Phenom 4(2): 92–118MathSciNetzbMATHGoogle Scholar
- Li J, Zou X (2009b) Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain. Bull Math Biol 71: 2048–2079MathSciNetzbMATHCrossRefGoogle Scholar
- Li J, Zou X (2010) An epidemic model with non-local infections on a patchy environment. J Math Biol 60: 645–686MathSciNetzbMATHCrossRefGoogle Scholar
- Lou Y, Zhao X-Q (2011) A reaction-diffusion malaria model with incubation period in the vector population. J Math Biol 62: 543–568MathSciNetzbMATHCrossRefGoogle Scholar
- Magal P, Zhao X-Q (2005) Global attractors and steady states for uniformly persistent dynamical systems. SIAM J Math Anal 37: 251–275MathSciNetzbMATHCrossRefGoogle Scholar
- Martin R Jr, Smith HL (1990) Abstract functional differential equations and reaction-diffusion systems. Trans AMS 321: 1–44MathSciNetzbMATHGoogle Scholar
- Martin R Jr, Smith HL (1991) Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence. J Reine Angew Math 413: 1–35MathSciNetzbMATHGoogle Scholar
- Metz JAJ, Diekmann O (1986) The dynamics of physiologically structured populations. In: Metz JAJ, Diekmann O (eds) Springer, New YorkGoogle Scholar
- Pao CV (1992) Nonlinear Parabolic and Elliptic Equations. Plenum, New YorkzbMATHGoogle Scholar
- Pazy A (1983) Semigroups of linear operators and application to partial differential equations. Springer, BerlinCrossRefGoogle Scholar
- Protter MH, Weinberger HF (1984) Maximum principles in differential equations. Springer, BerlinzbMATHCrossRefGoogle Scholar
- Smith HL (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Math. Surveys Monogr 41, American Mathematical Society, ProvidenceGoogle Scholar
- Smith HL, Zhao X-Q (2001) Robust persistence for semidynamical systems. Nonlinear Anal 47: 6169–6179MathSciNetzbMATHCrossRefGoogle Scholar
- Thieme HR (1992) Convergence results and a Poincare-Bendixson trichotomy for asymptotically autonomous differential equations. J Math Biol 30: 755–763MathSciNetzbMATHCrossRefGoogle Scholar
- Thieme HR (2009) Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J Appl Math 70: 188–211MathSciNetzbMATHCrossRefGoogle Scholar
- Thieme HR, Zhao X-Q (2001) A non-local delayed and diffusive predator-prey model. Nonlinear Anal 2: 145–160MathSciNetzbMATHCrossRefGoogle Scholar
- van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.. Math Biosci 180: 29–48MathSciNetzbMATHCrossRefGoogle Scholar
- Wang W, Zhao X-Q (2011) A nonlocal and time-delayed reaction-diffusion model of dengue transmission. SIAM J Appl Math 71: 147–168MathSciNetzbMATHCrossRefGoogle Scholar
- Wu J (1996) Theory and applications of partial functional differential equations. Applied mathematical science, vol 119. Springer, BerlinCrossRefGoogle Scholar