The presented research aims to design a new prevention class (P) in the HIV nonlinear system, i.e., the HIPV model. Then numerical treatment of the newly formulated HIPV model is portrayed handled by using the strength of stochastic procedure based numerical computing schemes exploiting the artificial neural networks (ANNs) modeling legacy together with the optimization competence of the hybrid of global and local search schemes via genetic algorithms (GAs) and active-set approach (ASA), i.e., GA-ASA. The optimization performances through GA-ASA are accessed by presenting an error-based fitness function designed for all the classes of the HIPV model and its corresponding initial conditions represented with nonlinear systems of ODEs. To check the exactness of the proposed stochastic scheme, the comparison of the obtained results and Adams numerical results is performed. For the convergence measures, the learning curves are presented based on the different contact rate values. Moreover, the statistical performances through different operators indicate the stability and reliability of the proposed stochastic scheme to solve the novel designed HIPV model.
Humankind is facing many deathly infection viruses for many years. HIV is one infectious virus that exists almost in each continent of the world with a low or high rate. Most viruses do not have proper treatment and vaccinations like coronavirus, dengue virus, and HIV [
All these citations related to HIV model have their individual novelty, merits and advantages. But no one has designed the prevention class in the nonlinear HIV model. The aim of this current work is to design the HIPV model and investigate numerically by using the artificial neural networks together with the competence of the hybrid of global and local search schemes called genetic algorithm (GA) and active-set approach (ASA), i.e., GA-ASA. The stochastic schemes have been applied to explore many linear/nonlinear, singular/non-singular and biological, delayed, fractional, prediction and pantograph differential models. Few well-known applications related to these applications are Thomas-Fermi singular system [
Some novel prominent features of the current work are described as:
The design of prevention class in the HIV nonlinear mathematical system is presented using the injection drug, safety measures, avoid from pregnancy and contact rate. The designed HIPV nonlinear system is effectively solved by using the ANN along with the optimization of GA-ASA combinations. For the convergence of HIPV system, the learning curves through GA and GA-ASAS are presented using different values of the contact rate. The consistent overlapped outcomes through GA-ASA and the Adams numerical routines validate the exactness of the designed scheme. The endorsement of the presentation is trained for different statistical valuations to get the numerical solutions of HIPV system. The advantages of the designed scheme are simply performed for the nonlinear HIPV system, easy to understand, operated efficiently, constancy and inclusive applicability are the other significant influences.
The remaining parts of current research are described as: Section 2 designates the methodology and statistical presentations. Section 3 shows the result simulations. Section 4 indicates the final declarations and future research reports.
The designed structure of ANNs by using the optimization of GA-ASA is presented in two steps for solving the nonlinear mathematical HIPV model, given as:
For the ANNs parameters, the design of fitness function is presented. Essential settings are provided to optimize the fitness function using the hybrid combination of GA-ASA.
The mathematical formulations to solve each class of the nonlinear Human Immunodeficiency Prevention Virus (HIPV) are obtainable in this section. The proposed solutions of the nonlinear HIPV model are respectively represented as
In the above system,
The log-sigmoid
The optimization of an error based ‘fitness function’ is performed using the GA-ASA procedures is given as:
In this section, the optimal performance through GA-ASA for solving the mathematical HIPV nonlinear model is presented. The designed ANNs structure through GA-ASA for solving the nonlinear mathematical HIPV model is shown in
GA is a global search optimization procedure, which is implemented to solve the stiff, complicated and nonlinear systems. In this study, GA is implemented as an optimization procedure to solve the HIPV model. To find the best solutions of the network, GA works through the selection operator, crossover process, reproduction practice and mutation procedure. Recently, GA has been applied in extensive optimization practices like as, system of hospitalization expenditure [
ASA is known as a quick and rapid local search optimization scheme, which is broadly executed to solve both types of models based on constrained/unconstrained systems. ASA is implemented in numerous optimizations networks based complex models. In recent few years, ASA is implemented to execute the real-time optimal control [
The statistical operator performances based on the “variance account for (VAF),” “mean absolute deviation (MAD),” “Theil's inequality coefficient (TIC)” and “semi interquartile range (S.I.R)” along with the Global performances of these operators are mathematically presented to solve the HIVP nonlinear model in this section.
In the above network,
In this section, the details solution for solving the HIPV nonlinear model are provided. For the convergence measures of the prevention class, six different cases have been presented based on the contact rate. Moreover, the obtained results have been compared with the Adams numerical results to authenticate the correctness of the HIPV nonlinear model. The statistical performances are also provided by taking different measures to check the consistency and reliability of the proposed scheme. The updated mathematical form of the nonlinear HIPV system with suitable parameters is given as:
The formulation of the fitness function using the above system becomes as:
The optimization of the above fitness function is performed to solve the HIPV nonlinear system using the hybrid combination of GA-ASA for 100 executions by taking 120 variables. The best weight vectors show the proposed solutions of the HIPV nonlinear mathematical model presented as:
The fitness function shown in
The proposed numerical solutions are obtained using the above systems (16–19) for 0 to 1 input with the step size of 0.1 of the HIPV nonlinear mathematical model along with the best weights drawn in the
The graphical representations along with histograms and boxplots using the statistical actions to authenticate the convergence measures are given in
Min | Max | Med | Mean | S.I Range | ST.D | |
---|---|---|---|---|---|---|
0 | 1.6377E-08 | 4.6720E-04 | 4.5848E-06 | 1.8610E-05 | 5.3442E-06 | 5.4089E-05 |
0.1 | 5.9156E-08 | 4.7400E-02 | 1.2960E-05 | 1.0287E-03 | 1.1693E-05 | 6.6058E-03 |
0.2 | 1.9430E-08 | 6.0309E-02 | 2.7709E-05 | 1.3097E-03 | 2.2189E-05 | 8.4029E-03 |
0.3 | 4.7200E-07 | 8.1658E-02 | 3.0612E-05 | 1.7736E-03 | 3.1225E-05 | 1.1383E-02 |
0.4 | 2.3255E-07 | 1.0989E-01 | 2.9581E-05 | 2.3783E-03 | 4.7282E-05 | 1.5318E-02 |
0.5 | 1.0969E-07 | 1.4800E-01 | 3.8530E-05 | 3.1968E-03 | 4.8653E-05 | 2.0629E-02 |
0.6 | 9.9906E-07 | 1.9944E-01 | 6.2370E-05 | 4.3087E-03 | 6.4859E-05 | 2.7800E-02 |
0.7 | 7.9126E-07 | 2.6870E-01 | 8.6570E-05 | 5.8108E-03 | 9.0269E-05 | 3.7455E-02 |
0.8 | 6.5015E-07 | 3.6193E-01 | 9.9611E-05 | 7.8360E-03 | 1.2367E-04 | 5.0451E-02 |
0.9 | 1.3928E-06 | 4.8746E-01 | 1.3191E-04 | 1.0550E-02 | 1.6817E-04 | 6.7948E-02 |
1 | 2.5086E-06 | 6.5658E-01 | 2.0238E-04 | 1.4192E-02 | 2.1230E-04 | 9.1522E-02 |
Min | Max | Med | Mean | S.I Range | ST.D | |
---|---|---|---|---|---|---|
0 | 8.6503E-10 | 1.5785E-05 | 7.1864E-07 | 1.8610E-05 | 1.1335E-06 | 3.3787E-06 |
0.1 | 1.3550E-08 | 5.3561E-05 | 3.2246E-06 | 1.0287E-03 | 2.6408E-06 | 7.2566E-06 |
0.2 | 7.9559E-09 | 6.8027E-05 | 3.9649E-06 | 1.3097E-03 | 3.7177E-06 | 9.8190E-06 |
0.3 | 1.9216E-08 | 7.5179E-05 | 6.5913E-06 | 1.7736E-03 | 4.5963E-06 | 1.2129E-05 |
0.4 | 4.9970E-08 | 5.6836E-05 | 6.7106E-06 | 2.3783E-03 | 6.0846E-06 | 1.2446E-05 |
0.5 | 5.9831E-09 | 4.7231E-05 | 8.2174E-06 | 3.1968E-03 | 6.3420E-06 | 1.1193E-05 |
0.6 | 1.0263E-08 | 4.6508E-05 | 7.4482E-06 | 4.3087E-03 | 5.2442E-06 | 9.5789E-06 |
0.7 | 3.9926E-08 | 7.1441E-05 | 4.1949E-06 | 5.8108E-03 | 3.3778E-06 | 1.0191E-05 |
0.8 | 3.6312E-08 | 1.0434E-04 | 4.0627E-06 | 7.8360E-03 | 3.0320E-06 | 1.3172E-05 |
0.9 | 2.2420E-10 | 1.2800E-04 | 5.8047E-06 | 1.0550E-02 | 4.3344E-06 | 1.5324E-05 |
1 | 1.8850E-08 | 1.2486E-04 | 3.5504E-06 | 1.4192E-02 | 3.2511E-06 | 1.4563E-05 |
The statistical presentations taking different gages like Minimum (Min), S.I Range, standard deviation (ST. D), Maximum (Max), Mean and Median (Med) are used to check the validation of the HIPV nonlinear model given in
Min | Max | Median | Mean | S.I Range | ST.D | |
---|---|---|---|---|---|---|
0 | 2.5989E-10 | 1.8505E-05 | 4.6175E-07 | 1.8610E-05 | 7.8245E-07 | 2.9527E-06 |
0.1 | 1.4111E-07 | 5.3657E-05 | 2.3457E-06 | 1.0287E-03 | 3.0229E-06 | 8.0955E-06 |
0.2 | 5.7269E-08 | 7.7215E-05 | 3.6965E-06 | 1.3097E-03 | 3.6843E-06 | 1.1442E-05 |
0.3 | 2.8241E-07 | 9.0006E-05 | 4.5533E-06 | 1.7736E-03 | 4.2207E-06 | 1.2120E-05 |
0.4 | 1.0675E-07 | 6.5357E-05 | 4.2574E-06 | 2.3783E-03 | 5.3379E-06 | 1.1839E-05 |
0.5 | 3.8237E-08 | 6.1991E-05 | 5.7020E-06 | 3.1968E-03 | 6.3486E-06 | 1.1551E-05 |
0.6 | 1.0820E-08 | 7.2882E-05 | 6.6137E-06 | 4.3087E-03 | 5.3160E-06 | 1.1922E-05 |
0.7 | 8.3392E-08 | 6.9889E-05 | 5.0312E-06 | 5.8108E-03 | 5.0082E-06 | 1.2358E-05 |
0.8 | 1.9028E-08 | 7.5191E-05 | 3.7824E-06 | 7.8360E-03 | 4.0082E-06 | 1.2316E-05 |
0.9 | 6.4691E-08 | 6.6702E-05 | 4.8486E-06 | 1.0550E-02 | 3.7728E-06 | 1.1000E-05 |
1 | 1.0121E-07 | 5.5042E-05 | 3.4958E-06 | 1.4192E-02 | 3.2566E-06 | 8.2567E-06 |
Min | Max | Med | Mean | S.I Range | ST.D | |
---|---|---|---|---|---|---|
0 | 1.0775E-09 | 8.3983E-05 | 1.0944E-06 | 3.5984E-06 | 1.4599E-06 | 9.2831E-06 |
0.1 | 1.9899E-07 | 2.8312E-04 | 8.1502E-06 | 1.4794E-05 | 5.8441E-06 | 2.9998E-05 |
0.2 | 6.0195E-07 | 1.8907E-04 | 1.1277E-05 | 1.8719E-05 | 1.0430E-05 | 2.3941E-05 |
0.3 | 1.2154E-07 | 1.5281E-04 | 7.8086E-06 | 1.4799E-05 | 8.3632E-06 | 2.0144E-05 |
0.4 | 8.5862E-09 | 1.1888E-04 | 5.3149E-06 | 1.0123E-05 | 4.7844E-06 | 1.6262E-05 |
0.5 | 2.2421E-07 | 9.4371E-05 | 7.4011E-06 | 1.1462E-05 | 6.6073E-06 | 1.3407E-05 |
0.6 | 1.7104E-07 | 7.3380E-05 | 8.6283E-06 | 1.4250E-05 | 8.2804E-06 | 1.4298E-05 |
0.7 | 7.2227E-08 | 9.0064E-05 | 8.9980E-06 | 1.3166E-05 | 5.3049E-06 | 1.4642E-05 |
0.8 | 2.8017E-08 | 7.7007E-05 | 7.7262E-06 | 1.1395E-05 | 5.3109E-06 | 1.3465E-05 |
0.9 | 1.2282E-07 | 1.0137E-04 | 9.4099E-06 | 1.4776E-05 | 7.0165E-06 | 1.7939E-05 |
1 | 1.9025E-07 | 1.0178E-04 | 7.0364E-06 | 1.2025E-05 | 6.3172E-06 | 1.5699E-05 |
The performance of the global “G-MAD,” “G-EVAF” and “G-TIC” operatives for hundred executions based on the proposed scheme are shown in
Class | ‘G-MAD’ | ‘G-TIC’ | ‘G-EVAF’ | |||
---|---|---|---|---|---|---|
Min | S.I Range | Min | S.I Range | Min | S.I Range | |
6.3352E-05 | 7.0866E-05 | 3.9481E-09 | 4.4186E-09 | 8.8507E-08 | 3.0806E-07 | |
5.8032E-06 | 3.3711E-06 | 3.2011E-10 | 1.8574E-10 | 2.3833E-07 | 3.2563E-07 | |
4.7674E-06 | 3.5962E-06 | 2.7718E-10 | 2.2256E-10 | 1.1764E-06 | 2.5215E-06 | |
8.7662E-06 | 5.2399E-06 | 4.9977E-10 | 2.8481E-10 | 3.8918E-08 | 5.5154E-08 |
The presented work is related to introducing a new prevention class in the nonlinear HIV system named HIPV model. HIV is a virus with no treatment, so prevention is one of the best options to control or spread this type of dangerous virus. The introduced prevention class uses the four subclasses named injection drug, safety measures, avoid from pregnancy and contact rate. This designed HIPV nonlinear model is solved using the ANNs and optimization is performed using the hybrid procedure of GA-ASA. For the convergence of the HIPV model, six different contact rate values have been analyzed through the process of GA-ASA. The reliable accuracy through convergence is obtained using the stochastic GA-ASA process for solving the nonlinear HIPV mathematical model. For the correctness of the designed HIPV model and the proposed stochastic scheme, the proposed results through the GA-ASA optimization process overlapped with the Adams numerical solutions. An activation log-sigmoid function is applied along with 120 variables. For the precision and accuracy of the proposed numerical approach, the statistical based performances of TIC, MAD and E-VAF for 100 executions using 120 variables have been provided. For the TIC, MAD and E-VAF convergence, most of the runs have achieved a very high accuracy level for solving each class of the nonlinear HIPV model. Moreover, the valuations through statistics based Min, ST.D, Mean, S.I range, Max, Median further validate the value of the designed ANN along with the GA-ASA. The global presentations with high ranks of these statistical operators have also been performed for solving the nonlinear HIPV system.
In the future, the designed ANN along with GA-ASA is proficient to solve the prediction model [