Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse

  • Abdelhai Elazzouzi Department of MPI, University Sidi Mouhamed Ben Abdellah, FP Taza, LSI Laboratory, Morocco
  • Abdesslem Lamrani Alaoui Department of Mathematics, University Moulay Ismaıl, FST Errachidia, M2I Laboratory, MAMCS Group, Morocco
  • Mouhcine Tilioua
  • Delfim F. M. Torres Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
Keywords: Global stability, Nonlinear incidence function, distributed delay, Lyapunov functionals, Relapse.


We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number R0 for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when R0 < 1. Moreover,we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when R0 > 1.


E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal. 47 (2001), no. 6, 4107–4115.

M. Budinˇcevi´c, A comparison theorem of differential equations, Novi Sad J. Math. 40 (2010), no. 1, 55–56.

Y. Enatsu, E. Messina, Y. Nakata, Y. Muroya, E. Russo and A.Vecchio, Global dynamics of a delayed SIRS epidemic model with a wide class of nonlinear incidence rates, J. Appl. Math. Comput. 39 (2012), no. 1-2, 15–34.

Y. Enatsu, Y. Nakata and Y. Muroya, Global stability of SIRS epidemic models with a class of nonlinear incidence rates and distributed delays, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 3, 851–865.

P. Georgescu and H. Zhang, A Lyapunov functional for a SIRI model with nonlinear incidence of infection and relapse, Appl. Math.Comput.

(2013), no. 16, 8496–8507.

J. K. Hale, Ordinary differential equations, Wiley-Interscience, New York, 1969.

J. Hale, Theory of functional differential equations, second edition, Springer-Verlag, New York, 1977.

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,Bull. Math. Biol. 68 (2006), no. 3, 615–626.

A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng. 1 (2004), no. 1, 57–60.

Y. Kuang, Delay differential equations with applications in population dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.

J. P. LaSalle, The stability of dynamical systems, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1976.

J. LaSalle and S. Lefschetz, Stability by Liapunov’s direct method, with applications, Mathematics in Science and Engineering, Vol. 4, Academic Press, New York, 1961.

C.-H. Li, C.-C. Tsai and S.-Y. Yang, Analysis of the permanence of an SIR epidemic model with logistic process and distributed time delay, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), no. 9, 3696–3707.

Q. Liu, D. Jiang, T. Hayat and B. Ahmad, Stationary distribution and extinction of a stochastic SIRI epidemic model with relapse,Stoch. Anal. Appl. 36 (2018), no. 1, 138–151.

J. P. Mateus, P. Rebelo, S. Rosa, C. M. Silva and D. F. M. Torres, Optimal control of non-autonomous SEIRS models with vaccination and treatment, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 6, 1179-1199. arXiv:1706.06843

H. N. Moreira and Y. Wang, Global stability in an S → I → R → I model, SIAM Rev. 39 (1997), no. 3, 496–502.

B. G. S. A. Pradeep, W. Ma and W. Wang, Stability and Hopf bifurcation analysis of an SEIR model with nonlinear incidence rate and relapse, J. Stat. Manag. Syst. 20 (2017), no. 3, 483–497.

A. Rachah and D. F. M. Torres, Analysis, simulation and optimal control of a SEIR model for Ebola virus with demographic effects, Commun. Fac. Sci. Univ. Ank. S´er. A1 Math. Stat. 67 (2018), no. 1, 179–197. arXiv:1705.01079

M. Sekiguchi, E. Ishiwata and Y. Nakata, Dynamics of an ultra-discrete SIR epidemic model with time delay, Math. Biosci. Eng. 15 (2018), no. 3, 653–666.

Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times,Nonlinear Anal. 42 (2000), no. 6, Ser. A: Theory Methods, 931–947.

D. Tudor, A deterministic model for herpes infections in human and animal populations, SIAM Rev. 32 (1990), no. 1, 136–139.

P. van den Driessche, L. Wang and X. Zou, Modeling diseases with latency and relapse, Math. Biosci. Eng. 4 (2007), no. 2, 205–219.

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48.

C. Vargas-De-Le´on, On the global stability of infectious diseases models with relapse, Abstraction & Application 9 (2013), 50–61.

R. Xu, Global dynamics of a delayed epidemic model with latency and relapse, Nonlinear Anal. Model. Control 18 (2013), no. 2,250–263.

H. Zhang, J. Xia and P. Georgescu, Multigroup deterministic and stochastic SEIRI epidemic models with nonlinear incidence rates and distributed delays: a stability analysis, Math. Methods Appl. Sci. 40 (2017), no. 18, 6254–6275.

How to Cite
Elazzouzi, A., Alaoui, A. L., Tilioua, M., & Torres, D. F. M. (2019). Analysis of a SIRI Epidemic Model with Distributed Delay and Relapse. Statistics, Optimization & Information Computing, 7(3), 545-557.
Research Articles