A mathematical investigation of an "SVEIR" epidemic model for the measles transmission
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A generalized "SVEIR" epidemic model with general nonlinear incidence rate has been proposed as a candidate model for measles virus dynamics. The basic reproduction number $ \mathcal{R} $, an important epidemiologic index, was calculated using the next generation matrix method. The existence and uniqueness of the steady states, namely, disease-free equilibrium ($ \mathcal{E}_0 $) and endemic equilibrium ($ \mathcal{E}_1 $) was studied. Therefore, the local and global stability analysis are carried out. It is proved that $ \mathcal{E}_0 $ is locally asymptotically stable once $ \mathcal{R} $ is less than. However, if $ \mathcal{R} > 1 $ then $ \mathcal{E}_0 $ is unstable. We proved also that $ \mathcal{E}_1 $ is locally asymptotically stable once $ \mathcal{R} > 1 $. The global stability of both equilibrium $ \mathcal{E}_0 $ and $ \mathcal{E}_1 $ is discussed where we proved that $ \mathcal{E}_0 $ is globally asymptotically stable once $ \mathcal{R}\leq 1 $, and $ \mathcal{E}_1 $ is globally asymptotically stable once $ \mathcal{R} > 1 $. The sensitivity analysis of the basic reproduction number $ \mathcal{R} $ with respect to the model parameters is carried out. In a second step, a vaccination strategy related to this model will be considered to optimise the infected and exposed individuals. We formulated a nonlinear optimal control problem and the existence, uniqueness and the characterisation of the optimal solution was discussed. An algorithm inspired from the Gauss-Seidel method was used to resolve the optimal control problem. Some numerical tests was given confirming the obtained theoretical results.
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Keywords:
- "SVEIR" epidemic model,
- nonlinear incidence rate,
- lyapunov function,
- LaSalle's invariance principle,
- sensitivity aanalysis,
- optimal control
Citation: Miled El Hajji, Amer Hassan Albargi. A mathematical investigation of an 'SVEIR' epidemic model for the measles transmission[J]. Mathematical Biosciences and Engineering, 2022, 19(3): 2853-2875. doi: 10.3934/mbe.2022131
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Abstract
A generalized "SVEIR" epidemic model with general nonlinear incidence rate has been proposed as a candidate model for measles virus dynamics. The basic reproduction number $ \mathcal{R} $, an important epidemiologic index, was calculated using the next generation matrix method. The existence and uniqueness of the steady states, namely, disease-free equilibrium ($ \mathcal{E}_0 $) and endemic equilibrium ($ \mathcal{E}_1 $) was studied. Therefore, the local and global stability analysis are carried out. It is proved that $ \mathcal{E}_0 $ is locally asymptotically stable once $ \mathcal{R} $ is less than. However, if $ \mathcal{R} > 1 $ then $ \mathcal{E}_0 $ is unstable. We proved also that $ \mathcal{E}_1 $ is locally asymptotically stable once $ \mathcal{R} > 1 $. The global stability of both equilibrium $ \mathcal{E}_0 $ and $ \mathcal{E}_1 $ is discussed where we proved that $ \mathcal{E}_0 $ is globally asymptotically stable once $ \mathcal{R}\leq 1 $, and $ \mathcal{E}_1 $ is globally asymptotically stable once $ \mathcal{R} > 1 $. The sensitivity analysis of the basic reproduction number $ \mathcal{R} $ with respect to the model parameters is carried out. In a second step, a vaccination strategy related to this model will be considered to optimise the infected and exposed individuals. We formulated a nonlinear optimal control problem and the existence, uniqueness and the characterisation of the optimal solution was discussed. An algorithm inspired from the Gauss-Seidel method was used to resolve the optimal control problem. Some numerical tests was given confirming the obtained theoretical results.
References
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- Figure 1. "SVEIR" epidemic compartmental model
- Figure 2. $ \Lambda = 0.5, \epsilon = 1, \gamma = 3, \mathcal{R} = 0.94 < 1 $ (left) and $ \Lambda = 0.5, \epsilon = 0.5, \gamma = 4, \mathcal{R} = 0.59 < 1 $ (right)
- Figure 3. $ \Lambda = 1, \epsilon = 0.5, \gamma = 0.3, \mathcal{R} = 10.82 > 1 $ (left) and $ \Lambda = 5, \epsilon = 0.5, \gamma = 0.5, \mathcal{R} = 28.94 > 1 $ (right)
- Figure 4. Behaviours for $ \alpha = 1, \beta = 10 $ (left) and $ \alpha = 10, \beta = 1 $(right)