Control and elimination in an SEIR model for the disease dynamics of COVID-19 with vaccination
doi: 10.3934/math.2023411
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Abstract: COVID-19 has become a serious pandemic affecting many countries around the world since it was discovered in 2019. In this research, we present a compartmental model in ordinary differential equations for COVID-19 with vaccination, inflow of infected and a generalized contact rate. Existence of a unique global positive solution of the model is proved, followed by stability analysis of the equilibrium points. A control problem is presented, with vaccination as well as reduction of the contact rate by way of education, law enforcement or lockdown. In the last section, we use numerical simulations with data applicable to South Africa, for supporting our theoretical results. The model and application illustrate the interesting manner in which a diseased population can be perturbed from within itself.
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Key words:
- basic reproduction number /
- generalized contact rate /
- infected immigrants /
- awareness /
- alertness
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$ P_0 $ The size of the population at the disease-free state $ A_0 $ Rate of birth of newborns, assumed to be susceptibles; Note that $ A_0 > 0 $ $ A_1 $ Rate of inflow into the $ E $-class; $ A_1\geq 0 $ $ \theta(I) $ Contact rate, a function of $ I $ $ \delta_1 $ and $ \delta_2 $ Disease-induced mortality rate, for $ E $ and $ I $ classes, respectively $ z $ Vaccination rate $ \alpha $ Transfer rate of from $ E $-class to $ I $-class $ \omega_1 $ Transfer rate of from $ E $-class to $ R $-class $ \omega_2 $ Transfer rate of from $ I $-class to $ R $-class $ \mu $ The average mortality rate by natural causes. 
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Table 1. Parameter values
Parameters Numerical values Source $ P_0 $ 69281690 cf. [33] $ \mu $ $ 8.43\times 10^{-5} $ [33] $ A_0 $ $ \mu P_0 $ (standard) $ A_1 $ 0 nominal $ \beta $ $ 0.39/P_0 $ fitted $ b $ $ 7\times 10^{-5} $ fitted $ \alpha $ 1/9 [32] $ z $ 0 see Remark 6.2.1 $ \delta_1 $ $ 0.05\delta_2 $ estimate $ \delta_2 $ $ 0.0015 $ [12] $ \omega_1 $ $ 0.05\omega_2 $ estimate $ \omega_2 $ 0.006 [28]
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