# A Novel Computational Instrument Based on a Universal Mixture Density Network with a Gaussian Mixture Model as a Backbone for Predicting COVID-19 Variants’ Distributions

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- The number of compartments or stages (M);
- (2)
- The forward and backward inner flows between the compartments;
- (3)
- The weighting parameters;
- (4)
- The number of numerical computation libraries.

^{th}compartment can be considered as the rate of individuals’ transition to the (n ± k)

^{th}compartment, where n, k = {1, 2, …, M} and n ≠ k. This cascading form constitutes a system of ordinary differential equations (SODE). Consequently, we can build simple to complex CMs depending on the number and details of the formal elements. To find an optimal solution for an SODE, one must optimize the whole system of equations, not the individual equations [1,2]. While it is possible to find an analytical solution for a simple (i.e., M = 3 and no backward flows) CM, a system’s optimal solution can typically be found using a numerical computation library, such as the Python Differential Evolution (DE) library [35]. The optimal solution and the framework are the foundational pillars of the targeted model of the COVID-19 infection rate. Such a model can be as simple as SIR [3,4] (susceptible, infected, and recovered), as detailed as SEIR [4,5,6] (susceptible, exposed, infected, and recovered), or as extensive as those in [1,2,7,8,9] with many backward flows.

- (1)
- regional density distribution;
- (2)
- age categories;
- (3)
- social distancing and masking;
- (4)
- vaccination;
- (5)
- climate data;
- (6)
- asymptomatic infection;
- (7)
- healthcare resources.

## 2. COVID-19 Input Data Modes

- One in which the time-dependent variable of the infection compartment’s population is Z(t), at which Z(t) $\in $ {S(t), P(t), E(t), I(t), M(t), H(t), Q(t), D(t), R(t)}. Table 1 exhibits the definitions of the mentioned time-dependent variables. For the rest of this communication, the time-independent notation for the compartment rate variables is dropped to facilitate reading, when necessary.
- The compartment parameter variable ψ is defined as ψ $\in $ {α, β, є, ϒ, δ, η, λ, μ, χ, σ, ρ, φ}. Table 2 exhibits the definitions of these parameters as the weights from the input compartment to the target compartment.

## 3. Mixture Density Network (MDN) with Multiple Outputs

#### 3.1. Neural Networks (NNs): An Introductory Overview

**x**

_{i}of the processing unit is multiplied by the connection weight w

_{kj}, which simulates the learning in an artificial neural network (ANN) by adjusting the strength or weight of the connection.

**x**) and output (

**y**) is then formulated as

- (1)
- one input layer that digests a given set of input variables
**x**≡ {x_{1}, x_{2}, …, x_{J}}, where J is the number of input features, - (2)
- one hidden layer with K neurons, and
- (3)
- one output layer with one neuron to produce an associated mapping
**y**.

**x**to a set of output variables

**y**≡ {y

_{1}, y

_{2}, …, y

_{n}}. In practice, such a network is trained to utilize a finite set of samples, which can be denoted by [{

**x**}

_{q}, {

**y**}

_{q}], where q = 1, 2, …, Q are the training sets under investigation. In other words, the principal aim of network training is to model the causal sources of event data. Hence, the best possible predictions for the y vector can be made when the trained network is presented with a new value of

**x**. The data source can be expressed statistically in terms of the probability density function (PDF) p(

**x**,

**y**) in a joint-input target space.

**y**. Nevertheless, these techniques are good for comparing and assessing the validity of a proposed NN design. For associated mapping scenarios, such as that of the COVID-19 infection rate that we consider in this study, it is suitable to decompose the joint probability density p(

**x**,

**y**) into the product of the conditional density of the target data p(

**y**|

**x**), which is conditioned on that of the input data p(

**x**) [41]: p(

**x**,

**y**) = p(

**y**|

**x**) p(

**x**), where the density p(

**x**) = ∫ p(

**x**,

**y**) dy plays a crucial role in confirming the predictions of the trained networks. Nevertheless, to predict the value of y corresponding to the input set {x

_{j}} of feature x, we need to focus on the conditional density p(

**y**|

**x**) model rather than the average target value. This is a fundamental aspect on which we should keep our focus.

#### 3.2. Mixture Density Network (MDN): The Theory

_{n}(t)}, where n є {1, 2, …, N}, and N is the maximum number of variants in a COVID-19 infection wave. The GMM is computationally equipped to determine the {I

_{n}(t)} set from the corresponding infected population I(t), which is a principal input into the input layer (Figure 1).

_{n}(t)}. With different health controls across regions and the nature of the variants, type j of the variant may dominate the other types k, where j, k є {1, 2, 3, …, N} and k $\ne j$. Suppose that we construct an ML regression model (e.g., an NN) to estimate the population of an individual I

_{n}(t). In that case, we have two potential deficiencies:

_{n}(t) to propose the right health control actions.

_{q}} into the input layer of the network, determining weights in a subsequent layer, and eventually modelling an estimate of the output y. Then, once a possible backpropagation action has been commenced, the weights in the NN can be altered when necessary, and the best estimate of y is eventually provided as a prediction that yields a single value. This single value is inadequate for effective health planning because different types of variants require different diagnoses, treatments, and levels of health control measures.

_{jk}, u

_{jk}) until the minimum error is small enough to meet the target requirements of the necessary output. In summary, the distinguishing difference is that the NN in Figure 6 produces two values: one for μ and one for σ. These two founding attributes are basically what is required to, for example, determine the PDF of the Gaussian distribution here. Consequently, to predict µ and the associated σ of the expected Gaussian distribution, there should be two neurons in the last layer (as shown in Figure 6) as opposed to one, as shown in Figure 5.

_{n}, σ

_{n}} sets, each corresponding to one COVID-19 variant I

_{n}(t). These distributions should be nearly sufficient to describe the known information and published data I(t) on COVID-19. It is a known fact that COVID-19 variants have different levels of population dominance. In other words, the forecasted distributions of the variants will have different levels of contribution/weight (α

_{n}) for the synthesized I(t). To map this requirement to our new NN design, the NN output should include N of {μ

_{n}, σ

_{n}, α

_{n}} sets, with each set corresponding to one predicted COVID-19 variant. The new NN design can be expected to have the layout shown in Figure 7. This MDN with a GMM as a backbone can be used to predict the individual components I

_{n}(t) of the COVID-19 infection compartments. In conclusion, MDNs are built from two components—an NN and a mixture model [40], which is implemented in the form of a GMM object, as illustrated in Figure 8b. Hence, the output layer of an MDN should be equipped with a GMM to synthesize the hidden layer’s outputs to the relevant distribution components, as shown in Figure 7.

#### 3.3. Architecture, Design, and Implementation Notes

## 4. Results and Discussion

#### 4.1. Use Case Implementation Dictionary

_{n}(t) per country. This array of outputs should be enough to conclude the reliability and validity of the proposed MDN computational instrument. The results are divided into two categories: (1) each set of three I

_{n}(t) values and the corresponding synthesized and predicted COVID-19 infection rate I(t) and (2) the MDN loss performance curves.

- (1)
**Input Modes**: In Figure 1, the diagram shows the deployment of the two implementations depending on the type of COVID-19 input data for Canada and Saudi Arabia. The first input feed was raw COVID-19 data from the WHO [37]; hence, we named this source “WHO COVID-19 data”. The second input feed was the optimal solution of the SODE in Equations (1)–(9), which was derived in Section 2. The results are depicted in Figure 4 (for Canada) and Figure 5 (for Saudi Arabia). In this study, we named this input feed “PCom-SEIR” because it relied on a modification of the PCom-SEIR model.- (2)
**Data Ranges**: The two input feeds covered two spans: 300 and 500 days.- (3)
**The MDN implementation runs**: There were three implementation runs; each run involved the utilization of one activation function in the hidden layers. The activation function set was {relu, tanh, sigmoid}.

- (1)
- Diagrams of the COVID-19 variants’ distributions {I
_{n}(t)} and their predicted I(t) values as a synthesized Gaussian distribution; - (2)
- Diagrams of the MDN’s loss performance, which includes
- (a)
- loss vs. epochs;
- (b)
- val_loss vs. epochs.

#### 4.2. Implementation Configurations and Environments

#### 4.3. The MDN’s Predictions of COVID-19 Variants

_{n}, σ

_{n}, α

_{n}}. The outputs from these dense sub-layers were passed to a dense concatenation layer, which coordinated with the GMM object (Figure 7 and Figure 8b) to compute the individual COVID-19 variant profiles. In this manner, the objective of the MDN of predicting three Gaussian components, with each component corresponding to a COVID-19 variant, was fulfilled. The results are shown in Figures 9 and 11 for Canada and in Figures 10 and 12 for Saudi Arabia.

_{n}(t) values as Gaussian distributions with different sets of distribution parameters {µ, σ, α}. As illustrated in these figures, different values of µ and σ imply that each variant candidate in the set {I

_{n}(t)} has a different extent of time (lengths in days) and different start/end days, with the different variants’ peaks mostly taking place on different days (the unit of time of observation). Such information is vital to health authorities when planning health control measures and preparing for the impact of the spread of severe infections. The profiles of the components predicted by the MDN corresponding to COVID-19 variant I

_{n}(t) values are different from those reported in Figure 2 and Figure 3 of [1]. In the referred study, the COVID-19 variant profiles are not fully Gaussian in shape; all variants practically share the same start and end dates but have different peaks.

#### 4.4. The Impacts of Epistemic Uncertainty and Aleatoric Uncertainty on Component Predictions

**Y**=

**L**(

**I**, θ)

**L**is the learned function that maps the input

**I**to the output

**Y**using the parameter θ. Here, the epistemic uncertainty (EU), which describes what the model does not know, is derived from θ and the inherent aleatoric uncertainty (AU), which is part of the data-generating process of

**I**. A high EU was found in part of the input feature space of the COVID-19 data published in [37], sporadically populated with data samples. In such an m-dimensional space, many parameters might explain the given data points, which gives rise to uncertainty. Based on this line of thinking, we decided to deploy two executions (see Figure 1) corresponding to the two data mode use cases. The first run took the output of the PCom-SEIR engine as an input (Figure 9 for Canada and Figure 10 for Saudi Arabia).

- (1)
- The fitting level of the PCom-SEIR data feed for both data ranges and both countries was 100% optimal, regardless of the activation function.
- (2)
- The fitting level of the WHO data for both data ranges and both countries was between 33% and 66%. The 33% optimal fitting occurred when we used the sigmoid activation function.
- (3)
- By observing the performance in Figure 9B(c) and Figure 11B(c) for Canada and in Figure 10B(c) and Figure 12B(c) for Saudi Arabia, one can notice that the sigmoid activation function produced 100% optimal fitting for both input data feeds. Relu and Tanh had a 100% optimal fit for the PCom-SEIR input data feed and nearly zero for the WHO input data feed. This meant that the algorithm, alongside the statistical profile of the input data, played a part in governing the model fitting for the same set of batch sizes and epochs.

#### 4.5. The MDN Model’s Accuracy

#### 4.6. General Remarks

## 5. Conclusions

- (1)
- The new MDN-based computational instrument was proven to be valid and credible through the examination of twelve use cases covering COVID-19 data sources, date ranges (i.e., data scope), and several MDN implementation configurations. The MDN’s outputs for these use cases clearly indicated that it produced an interpretable model for the growth of the COVID-19 infection rate [45].
- (2)
- Our approach can provide vital information to health authorities when planning health control measures and preparing for a severe spread of infections. This can be argued because our results indicated that the MDN produced COVID-19 variants’ In(t) values as Gaussian distributions with different sets of distribution parameters {µ, σ, α}. Different values of µ and σ imply that each variant candidate in the {In(t)} set has a different extent of time (length of days), different start/end days, and different variant peaks mostly taking place on different days (which is the unit of time of observation). Accordingly, the information above is vital to health authorities when planning health control measures and preparing for the impact of the severe spread of infections. On the other hand, in [1], the COVID-19 variant profiles had the same extent of time and practically the same peaks, as evidenced when comparing the prediction results for the COVID-19 variants’ In(t) values in this study.
- (3)
- Another indicator of the universality and practicality of our proposed MDN as a tool for predicting COVID-19 variants was demonstrated. It was shown that the relu, tanh, and sigmoid activation functions essentially had comparable sets of COVID-19 variant distributions, each with different parameter sets {µ, σ, α} that were compatible with the reality of the emergence and spread of COVID-19 variants.
- (4)
- (5)
- The MDN with the sigmoid activation function produced a multipeaked I(t) value by producing one of the components as a late-peaking variant component (Figure 13).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Al-Hadeethi, Y.; El Ramley, I.F.; Mohammed, H.; Barasheed, A.Z. A New Polymorphic Comprehensive Model for COVID-19 Transition Cycle Dynamics with Extended Feed Streams to Symptomatic and Asymptomatic Infections. Mathematics
**2023**, 11, 1119. [Google Scholar] [CrossRef] - Al-Hadeethi, Y.; Ramley, I.F.E.; Sayyed, M.I. Convolution model for COVID-19 rate predictions and health effort levels computation for Saudi Arabia, France, and Canada. Sci. Rep.
**2021**, 11, 22664. [Google Scholar] [CrossRef] [PubMed] - Anastassopoulou, C.; Russo, L.; Tsakris, A.; Siettos, C. Data-based analysis, modelling and forecasting of the COVID-19 outbreak. PLoS ONE
**2020**, 15, e0230405. [Google Scholar] [CrossRef] [PubMed] - Bakhta, A.; Boiveau, T.; Maday, Y.; Mula, O. Epidemiological forecasting with model reduction of compartmental models. Application to the COVID-19 pandemic. Biology
**2020**, 10, 22. [Google Scholar] [CrossRef] - Chang, Y.-C.; Liu, C.-T. A Stochastic Multi-Strain SIR Model with Two-Dose Vaccination Rate. Mathematics
**2022**, 10, 1804. [Google Scholar] [CrossRef] - Liu, X.; Ding, Y. Stability and numerical simulations of a new SVIR model with two delays on COVID-19 booster vaccination. Mathematics
**2022**, 10, 1772. [Google Scholar] [CrossRef] - Putra, S.; Mutamar, Z.K. Estimation of parameters in the SIR epidemic model using particle swarm optimisation. Am. J. Math. Comput. Model.
**2019**, 4, 83–93. [Google Scholar] [CrossRef] - Margenov, S.; Popivanov, N.; Ugrinova, I.; Hristov, T. Mathematical Modeling and Short-Term Forecasting of the COVID-19 Epidemic in Bulgaria: SEIRS Model with Vaccination. Mathematics
**2022**, 10, 2570. [Google Scholar] [CrossRef] - Mamis, K.; Farazmand, M. Stochastic compartmental models of the COVID-19 pandemic must have temporally correlated uncertainties. Proc. R. Soc. A
**2023**, 479, 20220568. [Google Scholar] [CrossRef] - Mbuvha, R.; Marwala, T. On data-driven management of the COVID-19 outbreak in South Africa. medRxiv
**2020**. [Google Scholar] [CrossRef] - Gatto, A.; Accarino, G.; Aloisi, V.; Immorlano, F.; Donato, F.; Aloisio, G. Limits of Compartmental Models and New Opportunities for Machine Learning: A Case Study to Forecast the Second Wave of COVID-19 Hospitalizations in Lombardy, Italy. Informatics
**2021**, 8, 57. [Google Scholar] [CrossRef] - Wondyfraw, T.A.; Sama, S.T. Stochastic model of the transmission dynamics of COVID-19 pandemic. Adv. Differ. Equ.
**2021**, 2021, 457. [Google Scholar] - Hoertel, N.; Blachier, M.; Blanco, C.; Olfson, M.; Massetti, M.; Rico, M.S.; Limosin, F.; Leleu, H. A stochastic agent-based model of the SARS-CoV-2 epidemic in France. Nat. Med.
**2020**, 26, 1417–1421. [Google Scholar] [CrossRef] [PubMed] - Yan, L.; Zhang, H.T.; Xiao, Y.; Wang, M.; Guo, Y.; Sun, C.; Tang, X.; Jing, L.; Li, S.; Zhang, M.; et al. Prediction of criticality in patients with severe COVID-19 infection using three clinical features: A machine learning-based prognostic model with clinical data in Wuhan. medRxiv 2020. [CrossRef]
- Frausto-Solís, J.; Hernández-González, L.J.; González-Barbosa, J.J.; Sánchez-Hernández, J.P.; Román-Rangel, E. Convolutional Neural Network–Component Transformation (CNN–CT) for Confirmed COVID-19 Cases. Math. Comput. Appl.
**2021**, 26, 29. [Google Scholar] [CrossRef] - Alanazi, S.A.; Kamruzzaman, M.M.; Alruwaili, M.; Alshammari, N.; Alqahtani, S.A.; Karime, A. Measuring and Preventing COVID-19 Using the SIR Model and Machine Learning in Smart Health Care. J. Healthc. Eng.
**2020**, 2020, 8857346. [Google Scholar] [CrossRef] [PubMed] - Ahmad, Z.; Almaspoor, Z.; Khan, F.; El-Morshedy, M. On predictive modeling using a new flexible Weibull distribution and machine learning approach: Analysing the COVID-19 data. Mathematics
**2022**, 10, 1792. [Google Scholar] [CrossRef] - Yadav, S.K.; Akhter, Y. Statistical Modeling for the Prediction of Infectious Disease Dissemination with Special Reference to COVID-19 Spread. Front. Public Health
**2021**. [Google Scholar] [CrossRef] - Zain, Z.M.; Alturki, N.M. COVID-19 pandemic forecasting using CNN-LSTM: A hybrid approach. J. Control Sci. Eng.
**2021**, 2021, 8785636. [Google Scholar] [CrossRef] - Wang, L.; Lin, Z.Q.; Wong, A. Covid-net: A tailored deep convolutional neural network design for detection of COVID-19 cases from chest x-ray images. Sci. Rep.
**2020**, 10, 19549. [Google Scholar] [CrossRef] - Zisad, S.N.; Hossain, M.S.; Hossain, M.S.; Andersson, K. An Integrated Neural Network and SEIR Model to Predict COVID-19. Algorithms
**2021**, 14, 94. [Google Scholar] [CrossRef] - Wieczorek, M.; Siłka, J.; Wo’zniak, M. Neural network powered COVID-19 spread Forecasting model. Chaos Solitons Fractals
**2020**, 140, 110203. [Google Scholar] [CrossRef] - Schiassi, E.; de Florio, M.; D’Ambrosio, A.; Mortari, D.; Furfaro, R. Physics-informed neural networks and functional interpolation for data-driven parameters discovery of epidemiological compartmental models. Mathematics
**2021**, 9, 2069. [Google Scholar] [CrossRef] - Hussein, H.I.; Mohammed, A.O.; Hassan, M.M.; Mstafa, R.J. Lightweight deep CNN-based models for early detection of COVID-19 patients from chest X-ray images. Expert Syst. Appl.
**2023**, 223, 119900. [Google Scholar] [CrossRef] [PubMed] - Tamang, S.K.; Singh, P.D.; Datta, B. Forecasting of COVID-19 cases based on prediction using artificial neural network curve fitting technique. Glob. J. Environ. Sci. Manag.
**2020**, 6, 53–64. [Google Scholar] - Huang, C.J.; Chen, Y.H.; Ma, Y.; Kuo, P.H. Multiple-input deep convolutional neural network model for covid-19 forecasting in china. medRxiv
**2020**. [Google Scholar] [CrossRef] - Gomez-Cravioto, D.A.; Diaz-Ramos, R.E.; Cantu-Ortiz, F.J.; Ceballos, H.G. Data Analysis and Forecasting of the COVID-19 Spread: A Comparison of Recurrent Neural Networks and Time Series Models. Cogn. Comput.
**2021**. [Google Scholar] [CrossRef] - Feng, C.; Wang, L.; Chen, X.; Zhai, Y.; Zhu, F.; Chen, H.; Wang, Y.; Su, X.; Huang, S.; Tian, L.; et al. A Novel triage tool of artificial intelligence-assisted diagnosis aid system for suspected COVID-19 pneumonia in fever clinics. Ann Transl Med.
**2021**, 9, 201. [Google Scholar] [CrossRef] - Jin, C.; Chen, W.; Cao, Y.; Xu, Z.; Tan, Z.; Zhang, X.; Deng, L.; Zheng, C.; Zhou, J.; Shi, H.; et al. Development and evaluation of an artificial intelligence system for COVID-19 diagnosis. Nat. Commun.
**2020**, 11, 5088. [Google Scholar] [CrossRef] - Xie, J.; Hungerford, D.; Chen, H.; Abrams, S.T.; Li, S.; Wang, G.; Wang, Y.; Kang, H.; Bonnett, L.; Zheng, R.; et al. Development and external validation of a prognostic multivariable model on admission for hospitalised patients with COVID-19. medRxiv 2020. [CrossRef]
- Wynants, L.; van Calster, B.; Collins, G.S.; Riley, R.D.; Heinze, G.; Schuit, E.; Bonten, M.M.J.; Dahly, D.L.; Damen, J.A.; Debray, T.P.A.; et al. Prediction models for diagnosis and prognosis of COVID-19: Systematic review and critical appraisal. BMJ
**2020**, 369. [Google Scholar] [CrossRef] - Rahimi, I.; Chen, F.; Gandomi, A.H. A review on COVID-19 forecasting models. Neural Comput. Appl.
**2023**, 35, 23671–23681. [Google Scholar] [CrossRef] [PubMed] - Naudé, W. Artificial intelligence vs COVID-19: Limitations, constraints and pitfalls. AI Soc.
**2020**, 35, 761–765. [Google Scholar] [CrossRef] [PubMed] - Britton, T. Stochastic epidemic models: A survey. Math. Biosci.
**2010**, 225, 24–35. [Google Scholar] [CrossRef] [PubMed] - Storn, R.; Price, K. Differential evolution–a simple and efficient heuristic for global optimisation over continuous spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Technical Report. Available online: https://publications.aston.ac.uk/ (accessed on 14 April 2024).
- WHO Data. Available online: https://covid19.who.int/WHO-COVID-19-global-data.csv (accessed on 14 April 2024).
- Python Optimization (scipy. Optimise). Available online: https://docs.scipy.org/doc/scipy/tutorial/optimize.html (accessed on 10 October 2022).
- Lerch, F.; Ultsch, A.; Lötsch, J. Distribution Optimization: An evolutionary algorithm to separate Gaussian mixtures. Sci. Rep.
**2020**, 10, 648. [Google Scholar] [CrossRef] [PubMed] - Covões, T.F.; Hruschka, E.R.; Ghosh, J. Evolving gaussian mixture models with splitting and merging mutation operators. Evol. Comput.
**2016**, 24, 293–317. [Google Scholar] [CrossRef] [PubMed] - Li, M.; Dushoff, J.; Bolker, B.M. Fitting mechanistic epidemic models to data: A comparison of simple Markov chain Monte Carlo approaches. Stat. Methods Med. Res.
**2018**, 27, 1956–1967. [Google Scholar] [CrossRef] [PubMed] - Rafique, D.; Velasco, L. Machine learning for network automation: Overview, architecture, and applications [Invited Tutorial]. J. Opt. Commun. Netw.
**2018**, 10, D126–D143. [Google Scholar] [CrossRef] - Saleh, B. Photoelectron Statistics: With Applications to Spectroscopy and Optical Communication; Springer: Cham, Switzerland, 2013; Volume 6. [Google Scholar]
- Hüllermeier, E.; Waegeman, W. Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Mach. Learn.
**2021**, 110, 457–506. [Google Scholar] [CrossRef] - Montesinos López, O.A.; Montesinos López, A.; Crossa, J. Multivariate Statistical Machine Learning Methods for Genomic Prediction; Springer Nature: Berlin, Germany, 2022; Volume 691. [Google Scholar]
- Rahmani, R.; Yusof, R. A new simple, fast and efficient algorithm for global optimization over continuous search-space problems: Radial movement optimization. Appl. Math. Comput.
**2014**, 248, 287–300. [Google Scholar] [CrossRef]

**Figure 3.**(Canada) COVID-19 data reported in [37] and the output of the PCom-SEIR model.

**Figure 4.**(Saudi Arabia) COVID-19 data reported in [37] and the output of the PCom-SEIR model.

**Figure 5.**A typical feedforward NN with three feature input layers, one hidden layer, and a single output layer.

**Figure 7.**A NN with three output sets (sub-layers) of the mean (µ), variance (σ), and contribution (α), synthesizing three Gaussian distributions of I

_{n}(t).

**Figure 13.**The MDN’s multipeaked predictions. (

**A**) Hypothetical COVID-19 data and the corresponding PCom-SEIR solution. (

**B**) MDN’s prediction of COVID-19 variant components I

_{n}(t) and the overall infection rate I(t).

Variable (Z(t)) | Description | |
---|---|---|

1 | S(t) | The population of the susceptible compartment. |

2 | P(t) | The population of the protected compartment. |

3 | E(t) | The population of the exposed compartment. |

4 | I(t) | The infection population of the symptomatic infection compartment. |

5 | M(t) | The population with asymptomatic infection. |

6 | Q(t) | The population of the quarantined compartment. |

7 | H(t) | The population of the hospitalised compartment. |

8 | D(t) | The population of the dead compartment. |

Parameter (ψ) | Input Parameter | ||
---|---|---|---|

From Compartment | To Compartment | ||

1 | α | Susceptible (S) | Protected (P) |

2 | Β | Exposed (E) | |

3 | Φ | Protected (P) | |

4 | Φ_{e} | Protected P_{e} = P * Φ component_{e} | Symptomatic infection (I) |

5 | Φ_{m} | Protected P_{m} = P * Φ_{m} component | Asymptomatic infection (M) |

6 | ϒ | Exposed (E) | Symptomatic infection (I) |

7 | є | Asymptomatic infection (M) | |

8 | I | Symptomatic infection (I) | Symptomatic infection |

9 | ϒ | Exposed component | Symptomatic infection (I) |

10 | η | Quarantine (Q) | |

11 | λ | Death (D) | |

12 | Τ | Asymptomatic (M) | Recovered (R) |

13 | Χ | Hospitalisation (H) | Recovered (R) |

14 | Ρ | Death (D) | |

15 | Μ | Quarantine (Q) | Recovered (R) |

16 | Σ | Death (D) | |

17 | Φ | Hospitalisation (H) | |

18 | r | Recovered component | Symptomatic infection |

MDN–GMM Network Data | ||||
---|---|---|---|---|

300 Data Range | 500 Data Range | 300 Data Range | 500 Data Range | |

Input Data Size | 319 | 571 | 283 | 535 |

Training Data Size | 212 | 380 | 188 | 356 |

Testing Data Size | 107 | 191 | 95 | 179 |

Output Data Size | 107 | 191 | 95 | 179 |

Neurons | 10 | 10 | 10 | 10 |

Batch Size | 64 | 64 | 64 | 64 |

(A) Canada | (B) Saudi Arabia |

Country | Canada | Saudi Arabia | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Data Range | 300–500 Days | 300–500 Days | ||||||||||

Data Input Mode | PCom-SEIR | WHO Data | PCom-SEIR | WHO Data | ||||||||

Activation Function | relu | tanh | sigmoid | relu | tanh | sigmoid | relu | tanh | sigmoid | relu | tanh | sigmoid |

Variant’s Distribution | Figure 9A | Figure 9A | Figure 9A | Figure 11A | Figure 11A | Figure 11A | Figure 10A | Figure 10A | Figure 10A | Figure 12A | Figure 12A | Figure 12A |

Loss Performance | Figure 9B(a) | Figure 9B(b) | Figure 9B(c) | Figure 11B(a) | Figure 11B(b) | Figure 11B(c) | Figure 10B(a) | Figure 10B(b) | Figure 10B(c) | Figure 12B(a) | Figure 12B(b) | Figure 12B(c) |

Network Parameters | Table 3A | Table 3B |

Country | Canada | Saudi Arabia | |||||||
---|---|---|---|---|---|---|---|---|---|

Input Mode | PCom-SEIR | WHO | PCom-SEIR | WHO | |||||

Data Range | 300 | 500 | 300 | 500 | 300 | 500 | 300 | 500 | |

relu | loss | ||||||||

Accuracy | 97.49 | 98.25 | 97.49 | 76.88 | 96.47 | 98.50 | 96.11 | 99.07 | |

tanh | loss | ||||||||

Accuracy | 97.49 | 99.30 | 97.81 | 77.93 | 97.88 | 98.88 | 98.23 | 98.88 | |

sigmoid | loss | ||||||||

Accuracy | 97.18 | 99.12 | 96.87 | 77.76 | 96.11 | 98.13 | 97.17 | 99.44 | |

Fitting Level % | 100% | 100% | 33% | 33% | 100% | 100% | 33% | 66% | |

Loss Legend | Optimal fit | Underfitting | Overfitting |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Al-Hadeethi, Y.; El Ramley, I.F.; Mohammed, H.; Bedaiwi, N.M.; Barasheed, A.Z.
A Novel Computational Instrument Based on a Universal Mixture Density Network with a Gaussian Mixture Model as a Backbone for Predicting COVID-19 Variants’ Distributions. *Mathematics* **2024**, *12*, 1254.
https://doi.org/10.3390/math12081254

**AMA Style**

Al-Hadeethi Y, El Ramley IF, Mohammed H, Bedaiwi NM, Barasheed AZ.
A Novel Computational Instrument Based on a Universal Mixture Density Network with a Gaussian Mixture Model as a Backbone for Predicting COVID-19 Variants’ Distributions. *Mathematics*. 2024; 12(8):1254.
https://doi.org/10.3390/math12081254

**Chicago/Turabian Style**

Al-Hadeethi, Yas, Intesar F. El Ramley, Hiba Mohammed, Nada M. Bedaiwi, and Abeer Z. Barasheed.
2024. "A Novel Computational Instrument Based on a Universal Mixture Density Network with a Gaussian Mixture Model as a Backbone for Predicting COVID-19 Variants’ Distributions" *Mathematics* 12, no. 8: 1254.
https://doi.org/10.3390/math12081254