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## 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:

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##### References:
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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