Bayesian Noise Modelling for State Estimation of the Spread of COVID-19 in Saudi Arabia with Extended Kalman Filters
Abstract
:1. Introduction
2. Kalman Filtering and the Pandemic Model
Multivariate Distributions
3. Methodology for Model Parameters and Uncertainty Estimation
4. Results and Discussions
4.1. Bayesian Inference Results and Error Distributions
4.2. Assumptions on the Error Distributions
- The distribution with a full covariance matrix between I and D;
- The distribution with no correlation between I and D, i.e., diagonal covariance matrix ;
- The distribution with a full covariance matrix, , between I and D;
- The distribution with no correlation between I and D, i.e., diagonal covariance matrix, .
4.3. Sampling from the Posterior of the Unknown Noise Parameters Using Nested Sampling
4.4. Bayesian Model Comparison for Selecting the Best Noise Distribution
4.5. State Estimation Using Extended Kalman Filters with the Estimated Noise Distributions
5. Conclusions
5.1. Main Contributions and Limitations
5.2. Future Work
- This work can be further extended with other, more complex nonlinear epidemiological compartmental models under the EKF framework with more experimental evidence.
- Another possible future direction is to include more model comparisons within different skewness distribution families using similar approaches such as the class of skewness distributions presented in [52]. However, the challenge with this class is dealing with the extra parameters as well as prior range selection, which should be carried out carefully as there is a lack of interpretability in the literature which needs further investigation.
- Unknown microbes or microorganisms that arrive from space, which are called panspermia, as discussed in [54,55]; however, there is a lack of evidence for this hypothesis and microbial data collection in space is limited due to the high cost, the complexity of the experimental setup and the high level of risk involved. As there are no clear findings of microorganisms coming from asteroids, comets or spacecraft, the evidence of possible infection is rather low. COVID-19 data can help in improving modelling of unknown pandemics from outer space.
- The outbreak of experimental microbes in scientific laboratories, due to accidents or poor management practices, can result in infections such as Brucella abortus, which can cause foetal death in pregnant women. Moreover, the origin of COVID-19 is still unknown, and whether it emerged through natural spillover, trans-species migration or a laboratory accident is still uncertain. However, a laboratory accident cannot be excluded as a potential risk for similar or even larger future pandemics, as discussed in [56,57].
- Future pandemics may also be caused by anthropogenic roots such as political conflicts or wars between countries, continents and specific genotypes, which can be modelled and controlled using the COVID-19 data as a test scenario [58,59]. This may also include manipulating the birth rate and other constants between the model compartments related to the health system’s infrastructure, such as the hospital capacity, quarantine period and reinfection rate in a country or region.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
KF | Kalman filter |
EKF | Extended Kalman filter |
MVN | Multivariate normal distribution |
MSN | Multivariate skew normal distribution |
Probability density function | |
CDF | Cumulative density function |
KDE | Kernel density estimate |
MAP | Maximum a posteriori estimate |
RMSE | Root Mean Squared Error |
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Model | |||||
---|---|---|---|---|---|
with correlation | - | - | |||
without correlation | - | - | - | ||
with correlation | |||||
without correlation | - |
Model | log | ||
---|---|---|---|
Model-1: with correlation | 60 | −263.215 ± 0.329 | 715 |
Model-2: without correlation | 40 | −266.940 ± 0.403 | 463 |
Model-3: with correlation | 100 | −263.546 ± 0.265 | 1239 |
Model-4: without correlation | 80 | −263.548 ± 0.297 | 989 |
EKF | Covariance Matrices | Infected RMSE | Death RMSE |
---|---|---|---|
EKF1 with correlation | = 500 × , = | 67.1907 | 26.9871 |
EKF2 without correlation | = 500 × , = diag[82.017, 989.20] | 61.2482 | 26.6587 |
EKF3 with correlation | = 500 × , = | 66.5684 | 26.9649 |
EKF4 without correlation | = 500 × , = diag[93.17, 994.56] | 66.8042 | 26.9032 |
EKF5 | = 500 × , = diag[100, 1000], Ref. [1] | 70.1258 | 27.0861 |
Parameter | Parameter Value | ||
---|---|---|---|
Description | Mean | Standard Deviation | |
infection rate | 3 | 4.4 | |
reinfection rate | 0.0028 | 1.4 | |
incubation period | 0.0353 | 0.00361 | |
q | quarantine rate | 0.9593 | 0.0033 |
quarantine period | 0.5939 | 0.0521 | |
d | death rate | 4 | 2.7 |
recovery rate | 0.9586 | 0.0031 |
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Alyami, L.; Panda, D.K.; Das, S. Bayesian Noise Modelling for State Estimation of the Spread of COVID-19 in Saudi Arabia with Extended Kalman Filters. Sensors 2023, 23, 4734. https://doi.org/10.3390/s23104734
Alyami L, Panda DK, Das S. Bayesian Noise Modelling for State Estimation of the Spread of COVID-19 in Saudi Arabia with Extended Kalman Filters. Sensors. 2023; 23(10):4734. https://doi.org/10.3390/s23104734
Chicago/Turabian StyleAlyami, Lamia, Deepak Kumar Panda, and Saptarshi Das. 2023. "Bayesian Noise Modelling for State Estimation of the Spread of COVID-19 in Saudi Arabia with Extended Kalman Filters" Sensors 23, no. 10: 4734. https://doi.org/10.3390/s23104734