| Citation: |
Hui Wu, Yafei Zhao, Chen Zhang, Jianhong Wu, Jie Lou. STRUCTURAL AND PRACTICAL IDENTIFIABILITY ANALYSES ON THE TRANSMISSION DYNAMICS OF COVID-19 IN THE UNITED STATES[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1475-1495. doi: 10.11948/20210300
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STRUCTURAL AND PRACTICAL IDENTIFIABILITY ANALYSES ON THE TRANSMISSION DYNAMICS OF COVID-19 IN THE UNITED STATES
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Abstract
We formulate an epidemic model to capture essential epidemiology of COVID-19 and major public health interventions. We start with a system of differential equations involving six compartments, and we use the Goodman and Weare affine invariant ensemble Markov Chain Monte Carlo algorithm (GWMCMC) to identify a simplified version of the full model that consists of only four compartments. We examine well-posedness of the relevant parameter estimation problem for the given observations using the U.S. epidemic data; study the reliability of model selection; analyze the structural identifiability of the selected model; and conduct a practical identifiability analysis on the selected model using the GWMCMC algorithm. Our study shows that the selected model is structurally identifiable for the confirmed cases, and for small measurement errors, key parameters such as the transmission rate are practically identifiable. We also analyze the stability of the selected model and prove the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium by constructing appropriate Lyapunov functions. Our numerical experiments show that the U.S. will undergo damped transit oscillations towards the endemicity.
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Keywords:
- Dynamic model /
- COVID-19 /
- model selection /
- identifiability analysis
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References
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Hui Wu, Yafei Zhao, Chen Zhang, Jianhong Wu, Jie Lou. STRUCTURAL AND PRACTICAL IDENTIFIABILITY ANALYSES ON THE TRANSMISSION DYNAMICS OF COVID-19 IN THE UNITED STATES[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1475-1495. doi: 10.11948/20210300
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Figure 1. The compartment diagram of the full-model.
$ \lambda $ is the constant recruitment rate;$ \beta $ is the transmission rate of$ I $ compartment;$ \eta_{i} , i=E, A, Q $ are the multiple of the transmission rate of$ E $ ,$ A $ and$ Q $ compartment relative to$ I $ compartment respectively;$ c $ is the proportion of asymptomatic infection;$ \gamma_{i}, i=A, I, Q $ represent the recovery rate of$ A $ ,$ I $ and$ Q $ compartment respectively;$ \delta_{i}, i=I, Q $ represent the disease-induced death rate of$ I $ and$ Q $ compartment respectively;$ \frac{1}{\alpha} $ is the incubation period;$ q_{i}, i=A, I $ represent the detection rate of$ A $ and$ I $ compartment respectively;$ \xi $ is the rate of the recovered patients return to the susceptible population due to the weakening or disappearance of antibodies;$ \omega $ is the natural mortality rate which is not shown in the figure. - Figure 2. Fitting results. (a) Fitting of Model 4 to the data of COVID-19 in the United States from February 29 to March 14, 2020. (b) It's 95% confidence interval.
- Figure 3. The parameter estimates of the Model 4 for 1000 synthetic data generated by Gaussian noise. True parameters are indicated by red stars.
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Figure 4. The frequency distribution histograms of
$ {\cal R}_{0} $ from 1000 simulations for measurement error levels$ \sigma_{0}=10\% $ and$ \sigma_{0}=20\% $ . - Figure 5. Fitting results. (a)Piecewise fit of the model (4.1) to the data of COVID-19 in the United States from March 15 to May 10, 2021. (b)95% confidence interval.
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Figure 6. Prediction in different situations.(a) The prediction of total cases with
$ \kappa=11.24, \epsilon=0.12 $ and$ t<t^{*} $ . (b) The prediction of the number of confirmed cases with different values of$ \kappa_1 $ and$ \epsilon_1 $ .