American Institute of Mathematical Sciences

Dynamic analysis of an $SEIR$ epidemic model with a time lag in awareness allocated funds

 1 LMA, Department of Mathematics, FST, B.P. 416-Tanger Principale, Tanger, Morocco 2 Ibn Tofail University, Laboratory of PDE, Algebra and Spectral Geometry, Department of Mathematics, Faculty of Sciences, Kénitra, BP 133, Morocco 3 Department of Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China

* Corresponding author: Mohamed El Fatini

Received  July 2019 Revised  July 2020 Published  October 2020

The spread of infectious diseases is often accompanied by a rise in the awareness programs to educate the general public about the infection risk and suggest necessary preventive practices. In the present paper we propose to study the impact of awareness on the dynamics of the classical $SEIR$ by considering the budget allocation to warn people as a new dynamic variable. In the model formulation, it is assumed that the susceptible individuals contract the infection via a direct contact with infected individuals, and that the transmission rate is presented by a general decreasing function of the availability of funds. We further introduced a time delay in the growth rate of the budget allocation related to the number of reported infected cases. The existence and the stability criteria of the equilibrium states are obtained in terms of the basic reproduction number $\mathcal{R}_{a}$. It is shown that $\mathcal{R}_{a} \leq 1$ is a necessary and sufficient condition for the global stability of the disease-free equilibrium, and by application of the geometric approach based on the third additive compound matrix we derived sufficient conditions for the global stability of the positive equilibrium state in the absence of delay. Our analysis reveals that awareness programs have the ability to reduce the infection prevalence. However, delay in providing funds destabilizes the system and give rise to periodic oscillations through Hopf-bifurcation. The direction and the stability of the bifurcating periodic solutions are investigated by using the normal form theory and central manifold theorem. Numerical simulations and sensitive analysis are provided to illustrate the theoretical findings..

Citation: Riane Hajjami, Mustapha El Jarroudi, Aadil Lahrouz, Adel Settati, Mohamed EL Fatini, Kai Wang. Dynamic analysis of an $SEIR$ epidemic model with a time lag in awareness allocated funds. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020285
References:
 [1] G. O. Agaba, Y. N. Kyrychko and K. B. Blyuss, Time-delayed SIS epidemic model with population awareness,, Ecological Complexity, 31 (2017), 50-56.  doi: 10.1016/j.ecocom.2017.03.002.  Google Scholar [2] F. A. Basir, Dynamics of infectious diseases with media coverage and two time delay, Mathematical Models and Computer Simulations, 10 (2018), 770-783.  doi: 10.1134/S2070048219010071.  Google Scholar [3] B. Berrhazi, M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise, Chaos Solitons Fractals, 105 (2017), 60-68.  doi: 10.1016/j.chaos.2017.10.007.  Google Scholar [4] B. Berrhazi, M. El Fatini, A. Lahrouz, A. Settati and R. Taki, A stochastic SIRS epidemic model with a general awareness-induced incidence, Phys. A, 512 (2018), 968-980.  doi: 10.1016/j.physa.2018.08.150.  Google Scholar [5] T. Caraballo, M. El Fatini, R. Pettersson and R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3483-3501.  doi: 10.3934/dcdsb.2018250.  Google Scholar [6] J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar [7] J. Cui, Y. Sun and and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar [8] J.-A. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mountain Journal of Mathematics, 38 (2008), 1323-1334.  doi: 10.1216/RMJ-2008-38-5-1323.  Google Scholar [9] M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, Lévy noise perturbation for an epidemic model with impact of media coverage, Stochastics, 91 (2019), 998-1019.  doi: 10.1080/17442508.2019.1595622.  Google Scholar [10] S. Funk, E. Gilad, C. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, AIDS Research and Therapy, Proceedings of the National Academy of Sciences USA, 106 (2009), 6872–6877. doi: 10.1073/pnas.0810762106.  Google Scholar [11] S. Funk, M. Salathé and V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of the Royal Society Interface, 7 (2010), 1247-1256.  doi: 10.1098/rsif.2010.0142.  Google Scholar [12] D. Greenhalgh, S. Rana, S. Samanta, T. Sardar, S. Bhattacharya and J. Chattopadhyay, Awareness programs control infectious disease–multiple delay induced mathematical model, Applied Mathematics and Computation, 251 (2015), 539-563.  doi: 10.1016/j.amc.2014.11.091.  Google Scholar [13] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer Science and Business Media, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [14] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge-New York, 1981.   Google Scholar [15] H. -F. Huo, P. Yang and H. Xiang, Stability and bifurcation for an SEIS epidemic model with the impact of media, Physica A: Statistical Mechanics and Its Applications, 490 (2018), 702-720.  doi: 10.1016/j.physa.2017.08.139.  Google Scholar [16] M. Y. Li and J. S. Muldowney, Dynamics of differential equations on invariant manifolds, Journal of Differential Equations, 168 (2000), 295-320.  doi: 10.1006/jdeq.2000.3888.  Google Scholar [17] P. van den Driessche, M. Y. Li and J. S. Muldowney, Global stability of SEIRS models in epidemiology, Canadian Applied Mathematics Quarterly, 7 (1999), 409-425.   Google Scholar [18] M. Y. Li and L. Wang, Global stability in some SEIR epidemic models, In Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, Springer, New York, 126 (2002), 295–311. doi: 10.1007/978-1-4613-0065-6_17.  Google Scholar [19] M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM Journal on Mathematical Analysis, 27 (1996), 1070-1083.  doi: 10.1137/S0036141094266449.  Google Scholar [20] J. Liu and Y. Jia, Dynamics of an SEIRS epidemic model with media coverage, Basic Sciences Journal of Textile Universities, 29 (2016), 18-25.   Google Scholar [21] R. Liu, J. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Computational and Mathematical Methods in Medicine, 8 (2007), 153-164.  doi: 10.1080/17486700701425870.  Google Scholar [22] Y. Liu and J.-A. Cui, The impact of media coverage on the dynamics of infectious disease, International Journal of Biomathematics, 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.  Google Scholar [23] A. K. Misra, R. K. Rai and Y. Takeuchi, Modeling the effect of time delay in budget allocation to control an epidemic through awareness, International Journal of Biomathematics, 11(2018), 1850027, 1–20. doi: 10.1142/S1793524518500274.  Google Scholar [24] A. K. Misra, A. Sharma and J. Li, A mathematical model for control of vector borne diseases through media campaigns, Discrete and Continuous Dynamical Systems-B, 18 (2013), 1909-1927.  doi: 10.3934/dcdsb.2013.18.1909.  Google Scholar [25] A. K. Misra, A. Sharma and V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, Journal of Biological Systems, 19 (2011), 389-402.  doi: 10.1142/S0218339011004020.  Google Scholar [26] A. K. Misra, R. K. Rai and Y. Takeuchi, Modeling the effect of time delay in budget allocation to control an epidemic through awareness, International Journal of Biomathematics, 11 (2018), 1850027, 20pp. doi: 10.1142/S1793524518500274.  Google Scholar [27] K. A. Pawelek, A. Oeldorf-Hirsch and L. Rong, Modeling the impact of Twitter on influenza epidemics, Mathematical Biosciences and Engineering, 11 (2014), 1337-1356.  doi: 10.3934/mbe.2014.11.1337.  Google Scholar [28] M. S. Rahman and M. L. Rahman, Media and education play a tremendous role in mounting AIDS awareness among married couples in Bangladesh, AIDS Research and Therapy, 4 (2007), Article 10. doi: 10.1186/1742-6405-4-10.  Google Scholar [29] S. Samanta, Effects of awareness program and delay in the epidemic outbreak, Mathematical Methods in the Applied Sciences, 40 (2017), 1679-1695.  doi: 10.1002/mma.4089.  Google Scholar [30] G. P. Sahu and J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, Journal of Mathematical Analysis and Applications, 421 (2015), 1651-1672.  doi: 10.1016/j.jmaa.2014.08.019.  Google Scholar [31] A. Sharma and A. K. Misra, Modeling the impact of awareness created by media campaigns on vaccination coverage in a variable population, Journal of Biological Systems, 22 (2014), 249-270.  doi: 10.1142/S0218339014400051.  Google Scholar [32] P. Song and Y. Xiao, Global hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease, Journal of Mathematical Biology, 76 (2018), 1249-1267.  doi: 10.1007/s00285-017-1173-y.  Google Scholar [33] J. M. Tchuenche, N. Dube, C. P. Bhunu, R. J. Smith and C. T. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11 (2011), S5. doi: 10.1186/1471-2458-11-S1-S5.  Google Scholar [34] Y. Wang, J. Cao, J. Zhen, H. Zhang and G.-Q. Sun, Impact of media coverage on epidemic spreading in complex networks, Physica A: Statistical Mechanics and its Applications, 392 (2013), 5824-5835.  doi: 10.1016/j.physa.2013.07.067.  Google Scholar [35] W. Wang and S. Ruan, Simulating the SARS outbreak in Beijing with limited data, Journal of Theoretical Biology, 227 (2004), 369-379.  doi: 10.1016/j.jtbi.2003.11.014.  Google Scholar [36] F. Yang and S. Ruan, A generalization of the Butler-McGehee lemma and its applications in persistence theory, Differential and Integral Equations, 9 (1996), 1321-1330.   Google Scholar

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References:
 [1] G. O. Agaba, Y. N. Kyrychko and K. B. Blyuss, Time-delayed SIS epidemic model with population awareness,, Ecological Complexity, 31 (2017), 50-56.  doi: 10.1016/j.ecocom.2017.03.002.  Google Scholar [2] F. A. Basir, Dynamics of infectious diseases with media coverage and two time delay, Mathematical Models and Computer Simulations, 10 (2018), 770-783.  doi: 10.1134/S2070048219010071.  Google Scholar [3] B. Berrhazi, M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, A stochastic SIRS epidemic model incorporating media coverage and driven by Lévy noise, Chaos Solitons Fractals, 105 (2017), 60-68.  doi: 10.1016/j.chaos.2017.10.007.  Google Scholar [4] B. Berrhazi, M. El Fatini, A. Lahrouz, A. Settati and R. Taki, A stochastic SIRS epidemic model with a general awareness-induced incidence, Phys. A, 512 (2018), 968-980.  doi: 10.1016/j.physa.2018.08.150.  Google Scholar [5] T. Caraballo, M. El Fatini, R. Pettersson and R. Taki, A stochastic SIRI epidemic model with relapse and media coverage, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3483-3501.  doi: 10.3934/dcdsb.2018250.  Google Scholar [6] J. Cui, Y. Sun and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar [7] J. Cui, Y. Sun and and H. Zhu, The impact of media on the control of infectious diseases, Journal of Dynamics and Differential Equations, 20 (2008), 31-53.  doi: 10.1007/s10884-007-9075-0.  Google Scholar [8] J.-A. Cui, X. Tao and H. Zhu, An SIS infection model incorporating media coverage, Rocky Mountain Journal of Mathematics, 38 (2008), 1323-1334.  doi: 10.1216/RMJ-2008-38-5-1323.  Google Scholar [9] M. El Fatini, A. Laaribi, R. Pettersson and R. Taki, Lévy noise perturbation for an epidemic model with impact of media coverage, Stochastics, 91 (2019), 998-1019.  doi: 10.1080/17442508.2019.1595622.  Google Scholar [10] S. Funk, E. Gilad, C. Watkins and V. A. A. Jansen, The spread of awareness and its impact on epidemic outbreaks, AIDS Research and Therapy, Proceedings of the National Academy of Sciences USA, 106 (2009), 6872–6877. doi: 10.1073/pnas.0810762106.  Google Scholar [11] S. Funk, M. Salathé and V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of the Royal Society Interface, 7 (2010), 1247-1256.  doi: 10.1098/rsif.2010.0142.  Google Scholar [12] D. Greenhalgh, S. Rana, S. Samanta, T. Sardar, S. Bhattacharya and J. Chattopadhyay, Awareness programs control infectious disease–multiple delay induced mathematical model, Applied Mathematics and Computation, 251 (2015), 539-563.  doi: 10.1016/j.amc.2014.11.091.  Google Scholar [13] J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer Science and Business Media, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar [14] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge-New York, 1981.   Google Scholar [15] H. -F. Huo, P. Yang and H. Xiang, Stability and bifurcation for an SEIS epidemic model with the impact of media, Physica A: Statistical Mechanics and Its Applications, 490 (2018), 702-720.  doi: 10.1016/j.physa.2017.08.139.  Google Scholar [16] M. Y. Li and J. S. Muldowney, Dynamics of differential equations on invariant manifolds, Journal of Differential Equations, 168 (2000), 295-320.  doi: 10.1006/jdeq.2000.3888.  Google Scholar [17] P. van den Driessche, M. Y. Li and J. S. Muldowney, Global stability of SEIRS models in epidemiology, Canadian Applied Mathematics Quarterly, 7 (1999), 409-425.   Google Scholar [18] M. Y. Li and L. Wang, Global stability in some SEIR epidemic models, In Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, Springer, New York, 126 (2002), 295–311. doi: 10.1007/978-1-4613-0065-6_17.  Google Scholar [19] M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM Journal on Mathematical Analysis, 27 (1996), 1070-1083.  doi: 10.1137/S0036141094266449.  Google Scholar [20] J. Liu and Y. Jia, Dynamics of an SEIRS epidemic model with media coverage, Basic Sciences Journal of Textile Universities, 29 (2016), 18-25.   Google Scholar [21] R. Liu, J. Wu and H. Zhu, Media/psychological impact on multiple outbreaks of emerging infectious diseases, Computational and Mathematical Methods in Medicine, 8 (2007), 153-164.  doi: 10.1080/17486700701425870.  Google Scholar [22] Y. Liu and J.-A. Cui, The impact of media coverage on the dynamics of infectious disease, International Journal of Biomathematics, 1 (2008), 65-74.  doi: 10.1142/S1793524508000023.  Google Scholar [23] A. K. Misra, R. K. Rai and Y. Takeuchi, Modeling the effect of time delay in budget allocation to control an epidemic through awareness, International Journal of Biomathematics, 11(2018), 1850027, 1–20. doi: 10.1142/S1793524518500274.  Google Scholar [24] A. K. Misra, A. Sharma and J. Li, A mathematical model for control of vector borne diseases through media campaigns, Discrete and Continuous Dynamical Systems-B, 18 (2013), 1909-1927.  doi: 10.3934/dcdsb.2013.18.1909.  Google Scholar [25] A. K. Misra, A. Sharma and V. Singh, Effect of awareness programs in controlling the prevalence of an epidemic with time delay, Journal of Biological Systems, 19 (2011), 389-402.  doi: 10.1142/S0218339011004020.  Google Scholar [26] A. K. Misra, R. K. Rai and Y. Takeuchi, Modeling the effect of time delay in budget allocation to control an epidemic through awareness, International Journal of Biomathematics, 11 (2018), 1850027, 20pp. doi: 10.1142/S1793524518500274.  Google Scholar [27] K. A. Pawelek, A. Oeldorf-Hirsch and L. Rong, Modeling the impact of Twitter on influenza epidemics, Mathematical Biosciences and Engineering, 11 (2014), 1337-1356.  doi: 10.3934/mbe.2014.11.1337.  Google Scholar [28] M. S. Rahman and M. L. Rahman, Media and education play a tremendous role in mounting AIDS awareness among married couples in Bangladesh, AIDS Research and Therapy, 4 (2007), Article 10. doi: 10.1186/1742-6405-4-10.  Google Scholar [29] S. Samanta, Effects of awareness program and delay in the epidemic outbreak, Mathematical Methods in the Applied Sciences, 40 (2017), 1679-1695.  doi: 10.1002/mma.4089.  Google Scholar [30] G. P. Sahu and J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, Journal of Mathematical Analysis and Applications, 421 (2015), 1651-1672.  doi: 10.1016/j.jmaa.2014.08.019.  Google Scholar [31] A. Sharma and A. K. Misra, Modeling the impact of awareness created by media campaigns on vaccination coverage in a variable population, Journal of Biological Systems, 22 (2014), 249-270.  doi: 10.1142/S0218339014400051.  Google Scholar [32] P. Song and Y. Xiao, Global hopf bifurcation of a delayed equation describing the lag effect of media impact on the spread of infectious disease, Journal of Mathematical Biology, 76 (2018), 1249-1267.  doi: 10.1007/s00285-017-1173-y.  Google Scholar [33] J. M. Tchuenche, N. Dube, C. P. Bhunu, R. J. Smith and C. T. Bauch, The impact of media coverage on the transmission dynamics of human influenza, BMC Public Health, 11 (2011), S5. doi: 10.1186/1471-2458-11-S1-S5.  Google Scholar [34] Y. Wang, J. Cao, J. Zhen, H. Zhang and G.-Q. Sun, Impact of media coverage on epidemic spreading in complex networks, Physica A: Statistical Mechanics and its Applications, 392 (2013), 5824-5835.  doi: 10.1016/j.physa.2013.07.067.  Google Scholar [35] W. Wang and S. Ruan, Simulating the SARS outbreak in Beijing with limited data, Journal of Theoretical Biology, 227 (2004), 369-379.  doi: 10.1016/j.jtbi.2003.11.014.  Google Scholar [36] F. Yang and S. Ruan, A generalization of the Butler-McGehee lemma and its applications in persistence theory, Differential and Integral Equations, 9 (1996), 1321-1330.   Google Scholar
Tendency of the basic reproductive number with $a_0$ and $\beta_1$ for $\beta(M) = \beta_0 -\dfrac{\beta_1 M}{p+ M}$
Comparison of infected individuals with respect to time in absence and presence of budget allocation, for $\beta(M) = \beta_0 -\dfrac{\beta_1 M}{p+ M}$ (for $\tau = 0$)
variation of I with respect to time for different $a_1$, $p$, $\beta_1,$ for $\beta(M) = \beta_0 -\dfrac{\beta_1 M}{p+ M}$
Variation of $S(t)$, $E(t)$, $I(t)$, $M(t)$ and the phase portrait in $M-I-E$ and $M-I-S$ spaces for $\tau = 69 < \tau_{0} = 79.996$ and $\beta(M) = \beta_0 -\dfrac{\beta_1 M}{p+ M},$ which shows that the system (3) is stable around the endemic equilibrium
Variation of $S(t)$, $E(t)$, $I(t)$, $M(t)$ and the phase portrait in $M-E-I$ and $M-S-I$ spaces for $\tau = 86 > \tau_{0}$ and $\beta(M) = \beta_0 -\dfrac{\beta_1 M}{p+ M},$ which depicts that the system (3) exhibit a stable periodic behavior around the endemic equilibrium
Appearance of a stable limit cycle in $M-E-I$ space for $\tau=86> \tau_{0}= 79.996$ and $\beta(M)= \beta_0 -\dfrac{\beta_1 M}{p+ M}.$
Comparison of infected individuals with respect to time t in absence and presence of budget allocation, for $\beta(M) = \beta_0e^{-\delta M}$ and $\tau = 0.$
variation of I with respect to time for different $\delta$, $M_0$, $a_1$ in the absence of delay $(\tau = 0)$ with $\beta(M) = \beta_0e^{-\delta M}.$
Variation of $S(t)$, $E(t)$, $I(t)$, $M(t)$ and the phase portrait in $I-S-M$ and $M-E-I$ spaces for $\tau = 39 < \tau_{0} = 79.996$ and $\beta(M) = \beta_0e^{-\delta M}$, which shows that the system (3) is stable around the endemic equilibrium
Variation of $S(t)$, $E(t)$, $I(t)$, $M(t)$ and the phase portrait in $M-E-I$ and $I-S-M$ spaces for $\tau = 55 > \tau_{0}$ and $\beta(M) = \beta_0e^{-\delta M}$, which depicts that the system (3) exhibit a periodic behavior around the endemic equilibrium
Appearance of a stable limit cycle in M-E-I space for $\tau=86> \tau_{0}= 79.996$ and $\beta(M)= \beta_0e^{-\delta M}$
Table of the parameter values used in the numerical simulations for $\beta(M)= \beta_0 -\dfrac{\beta_1 M}{p+ M}$
 Parameters $\mu$ $\beta_{0}$ $\beta_1$ $p$ $\alpha$ $\gamma$ $M_0$ $a_1$ $a_2$ Values 0.014 0.16 0.12 2.8 0.83 0.0601 1 2.38 0.012
 Parameters $\mu$ $\beta_{0}$ $\beta_1$ $p$ $\alpha$ $\gamma$ $M_0$ $a_1$ $a_2$ Values 0.014 0.16 0.12 2.8 0.83 0.0601 1 2.38 0.012
Table of the parameter values used in the numerical simulations for $\beta(M)=\beta_{0} e^{-\delta M}$
 Parameters $\mu$ $\beta_{0}$ $\delta$ $\alpha$ $\gamma$ $M_0$ $a_1$ $a_2=0.04$ Values 0.15 0.29 0.3 0.12 0.0108 0.4 0.89 0.04
 Parameters $\mu$ $\beta_{0}$ $\delta$ $\alpha$ $\gamma$ $M_0$ $a_1$ $a_2=0.04$ Values 0.15 0.29 0.3 0.12 0.0108 0.4 0.89 0.04
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