# A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics

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## Abstract

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## 1. Introduction

- The solution of the mathematical expression for the nonlinear SITR model for novel COVID-19 dynamics is calculated viably by using the novel application of the intelligent neuro-evolution-based integrated computing paradigm, i.e., FF-ANN-GASQP.
- Closely matching of the results of the proposed FF-ANN-GASQP solver with the solutions of the reference state of the art numerical procedure of Adams methods for variants of the nonlinear SITR-based mathematical model established the value and worth.
- Authentication and verification of the performance through statistical assessments studies is proven on multiple implementations of FF-ANN-GASQP in terms of Theil’s inequality coefficient (TIC) as well as root mean square error (RMSE)-based indices.
- In addition to the precise and accurate solutions for the SITR-based mathematical model of the COVID-19 pandemic, other valuable perks are that it is easy to understand the concepts, and it also has smooth operation, exhaustive applicability, consistency, and extendibility.

## 2. Proposed Methodology

- To exploit the FF-ANN models, an error-based objective function is introduced.
- Optimize the objective function for system (1) using the hybrid GA-SQ programming approach.

#### 2.1. ANN Modeling

**W**and given as:

#### 2.2. Optimization Technique: Hybrid of GA with SQP

## 3. Performance Measures

## 4. Results and Discussion

#### Nonlinear SITR Model Based on COVID-19

^{−4}to 10

^{−5}for all the parameters of the nonlinear SITR model using five numbers of neurons.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Process workflow of proposed FF-ANN-GASQP structure for SITR model based on COVID-19. FF-ANN-GASQP: feed-forward artificial neural networks trained with global search genetic algorithms and speedy fine tuning by sequential quadratic programming.

**Figure 5.**Absolute error (AE) results for all the variables of the nonlinear SITR model based on COVID-19 using five neurons.

**Figure 6.**Convergence analysis based on the performance measures of Theil’s inequality coefficient (TIC) and root mean square error (RMSE) values for all the variables of the nonlinear SITR model.

Variable | Description |
---|---|

${S}_{1}(\chi )$ | Non-infected individuals |

${S}_{2}(\chi )$ | Non-infected older or major diseased people |

$I(\chi )$ | Rate of infected from COVID-19 |

$R(\chi )$ | Recovery rate from COVID-19 |

$T(\chi )$ | Treatment |

Parameter | Description |
---|---|

$\beta $ | Contact rate |

$B$ | Rate of natural birth |

$\delta $ | Reduce infection from treatment |

$\sigma $ | Fever, tiredness and dry cough rate |

$\mu $ | Recovery rate |

$\alpha $ | Death rate |

$\rho $ | Rate of infection from treatment |

$\psi $ | Healthy food rate |

$\epsilon $ | Sleep rate |

${A}_{j},\hspace{0.17em}j=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4,5$ | Initial conditions |

Start the GA process |

Inputs: The individuals with genes equally representing the decision values of FF-ANN as: |

$\mathit{W}=[{\mathit{W}}_{{S}_{1}},\hspace{0.17em}{\mathit{W}}_{{S}_{2}},{\mathit{W}}_{I},{\mathit{W}}_{T},{\mathit{W}}_{R}]$, where ${\mathit{W}}_{{S}_{1}}=[{\mathit{\varphi}}_{{S}_{1}},{\mathit{w}}_{{S}_{1}},{\mathit{b}}_{{S}_{1}}]$, ${\mathit{W}}_{{S}_{2}}=[{\mathit{\varphi}}_{{S}_{2}},{\mathit{w}}_{{S}_{2}},{\mathit{b}}_{{S}_{2}}]$, |

${\mathit{W}}_{I}=[{\mathit{\varphi}}_{I},{\mathit{w}}_{I},{\mathit{b}}_{I}]$, ${\mathit{W}}_{T}=[{\mathit{\varphi}}_{T},{\mathit{w}}_{T},{\mathit{b}}_{T}]$ and ${\mathit{W}}_{R}=[{\mathit{\varphi}}_{R},{\mathit{w}}_{R},{\mathit{b}}_{R}]$ as per the details provided |

in the system (3). |

Population: Number of chromosomes in a set define a population as: |

$\mathit{P}={[{\mathit{W}}_{1},\hspace{0.17em}\hspace{0.17em}{\mathit{W}}_{2},{\mathit{W}}_{3},\dots ,{\mathit{W}}_{n}]}^{t}$, for ith component |

${\mathit{W}}_{i}=[{\mathit{W}}_{{S}_{1},i},\hspace{0.17em}{\mathit{W}}_{{S}_{2},i},{\mathit{W}}_{I,i},{\mathit{W}}_{T,i},{\mathit{W}}_{R,i}]$ with |

${\mathit{W}}_{{S}_{1},i}=[{\mathit{\varphi}}_{{S}_{1},i},{\mathit{w}}_{{S}_{1},i},{\mathit{b}}_{{S}_{1},i}]$, ${\mathit{W}}_{{S}_{2},i}=[{\mathit{\varphi}}_{{S}_{2},i},{\mathit{w}}_{{S}_{2},i},{\mathit{b}}_{{S}_{2},i}]$, ${\mathit{W}}_{I,i}=[{\mathit{\varphi}}_{I,i},{\mathit{\omega}}_{I,i},{\mathit{b}}_{I,i}]$, |

${\mathit{W}}_{T,i}=[{\mathit{\varphi}}_{T,i},{\mathit{w}}_{T,i},{\mathit{b}}_{T,i}]$, and ${\mathit{W}}_{R,i}=[{\mathit{\varphi}}_{R,i},{\mathit{w}}_{R,i},{\mathit{b}}_{R,i}]$ |

Output: The best global decision variables/trained weights of the ANN-GASQ programming scheme denoted as W_{GA-Best} |

Initialization: Generate chromosome W and P with pseudo random numbers. |

Initialization is performed for {GA} and {gaoptimset} routines with |

suitable declarations and settings. |

Fitness evaluated: Calculate the fitness and its parts shown in Equations (5)–(10) for each {W} in P. |

Termination: Terminate the procedure, when any requirements meets |

• {Achieved Fitness = 10^{−20}},{Generations = 60} |

• {Tolerances: {TolFun= 10^{−20} and TolCon =10^{−21}}, |

• {StallGenLimit=100},{Population size = 300} |

• Others values: default. |

When the above conditions meet, go to storage |

Ranking: Rank is proficient for every ‘W’ of ‘P’ indicates the attained |

fitness. |

Reproduction: |

• {Selection: selectionuniform} |

• {Mutations: mutationadaptfeasible}. |

• {Crossover: crossoverheuristic}. |

• {Elitism: Transmit 5% individuals in P} |

Go to fitness assessment step. |

Storage: Store the W, i.e., the weight vector, fitness assessment,_{GA-Best} |

generations, time and count of functions for the present run of |

GAs. |

End of GA |

SQP Process Start |

Inputs: Start point is W_{GA-Best} |

Output: GASQP best weights are denoted as W_{GASQ} |

Initialize: Set the limited constraints, iterations and other values of |

optimset. |

Terminate: The SQ programming process terminates when one the criteria |

meets |

{Generations = 900}, {Fitness = 10^{−18}}, |

(TolFun = TolCon = TolX = 10^{−22}) and {MaxFunEvals = 285000}. |

While (Terminate) |

Fitness Evaluate: Compute fitness value of every W of P by using |

system (4) to (10). |

Adjustments: Fine-tune {fmincon} with SQ programming scheme to tune |

W and adjust again the fitness value by using systems (4) to (10). |

Accumulate Store fitness, time, W, function counts and generations for multiple trials of SQ programming._{GASQ} |

End of the SQ programming scheme |

Data Generations |

Repeat the procedure 30 times based on the GASQ programming to get a |

massive dataset of the optimization variables of ANNs for numerical |

solutions of the SITR model based on COVID-19 |

Symbol | Parameter Description | Assigned Value |
---|---|---|

$\beta $ | Contact rate | 0.3 |

$B$ | Natural birth rate | 0.3 |

$\delta $ | Reduce infection from the treatment | 0.3 |

$\sigma $ | Fever, tiredness, and dry cough rate | 0.005 |

$\mu $ | Recovery rate | 0.1 |

$\alpha $ | Death rate | 0.25 |

$\rho $ | Rate of infection from the treatment | 0.3 |

$\psi $ | Healthy food rate | 0.2 |

$\epsilon $ | Sleep rate | 0.1 |

Scenarios | Variable Parameter | Case I | Case II | Case III | Case IV |
---|---|---|---|---|---|

1 | Contact Rate | $\beta $ = 0.25 | $\beta $ = 0.30 | $\beta $ = 0.35 | $\beta $ = 0.40 |

2 | Recovery Rate | $\mu $ = 0.08 | $\mu $ = 0.10 | $\mu $ = 0.12 | $\mu $ = 0.14 |

3 | Death Rate | $\alpha $ = 0.20 | $\alpha $ = 0.25 | $\alpha $ = 0.30 | $\alpha $ = 0.35 |

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## Share and Cite

**MDPI and ACS Style**

Umar, M.; Sabir, Z.; Raja, M.A.Z.; Shoaib, M.; Gupta, M.; Sánchez, Y.G.
A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics. *Symmetry* **2020**, *12*, 1628.
https://doi.org/10.3390/sym12101628

**AMA Style**

Umar M, Sabir Z, Raja MAZ, Shoaib M, Gupta M, Sánchez YG.
A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics. *Symmetry*. 2020; 12(10):1628.
https://doi.org/10.3390/sym12101628

**Chicago/Turabian Style**

Umar, Muhammad, Zulqurnain Sabir, Muhammad Asif Zahoor Raja, Muhammad Shoaib, Manoj Gupta, and Yolanda Guerrero Sánchez.
2020. "A Stochastic Intelligent Computing with Neuro-Evolution Heuristics for Nonlinear SITR System of Novel COVID-19 Dynamics" *Symmetry* 12, no. 10: 1628.
https://doi.org/10.3390/sym12101628