Intelligent computing based supervised learning for solving nonlinear system of malaria endemic model

1.   Introduction

Malaria is a mosquito-borne disease that is spread by one-celled Plasmodium parasites ^[1]. Malaria's ability to continue increasing mortality and morbidity causes significant public health and economic challenges in developing countries ^[2]. About two billion people are constantly at risk of infection around the world ^[3], some areas of Africa have been severely affected, with children and women making up the majority of the victims. Malaria kills at least one millions of people per year in Sub-Saharan Africa, according to the World Health Organization (WHO) and it has the potential to become even worse as a result of climate change and the (HIV) ^[4,5]. Malaria spread in humans by bites from an infected anopheles Mosquito. Medical symptoms including discomfort, headaches, body aches, fever, nausea, sweats and vomiting appear a few days after the bites. After a blood meal, mosquitoes collect infection from an infected person. Malaria is a life-threatening disease that can be prevented and treated if caught early. Current methods of controlling the disease include pesticides, medicines, treated bed nets and treatments to prevent the disease. In certain areas, these measures resulted in significant reductions in morbidity and mortality ^[6]. However, if the disease is not managed appropriately and treated, symptoms may worsen. The most significant impediment to malaria elimination is the occurrence of recurring malaria, which can be categorized as relapse and re-infection ^[7,8]. On the other hand, the appearance of the relapse symptoms of recurrence of non-parasites in the liver, such as later the parasites have cleared their blood ^[9,10,11]. Reinfection, on the other hand, is not caused by medication failure; rather, it is the recurrence of malaria diseases reported by parasite infection from innovative infectious mosquito bites.

Various mathematical models (see, for example, ^{[12,13,14,15]}), which describe the dynamic transmission of malaria population has evolved since the early works of May and Anderson ^[16], Macdonald ^[17], and Ross ^[18]. Anguelov et al. ^[12] suggested a mathematical model that would use sterile insect technology to reduce the number of wild lady anospheles mosquitos. Ghosh ^[13] has developed a model of the dynamics of malaria with this assumption after the density of the mosquito population increased in terms of the attractiveness of the environment. The authors of ^[14] used the stability principle of differential equations to investigate malaria dynamic behavior with nonlinear infection forces. Niger and Gumel ^[19] developed a deterministic model to evaluate the impact of re-infection on the dynamics of malaria transmission. The impact of relapse on the slow and fast dynamics of a malaria disease model were studied by Li et al. ^[20]. The authors of ^[21] investigated a malaria model in which the recovered people not only arrival to the susceptible individuals class, but also back to the infectious individuals class (relapse). A number of many other mathematical malaria models of relapse have been discussed in the works ^[22,23,24]. However, we have implemented a complex model of six classes based on the current scenarios, Susceptible people ($S_1$), infectious people ($I_1$), pseudo recovered people ($R_1$), Recovered people prone to re-infection ($R_2$), mosquito Susceptible ($S_2$) and mosquito infection ($I_2$) classes, i.e. malaria model for numerical investigation ^[25]. Recently, the proposed malaria model is mathematically analysis through various methods some of them as ^{[26,27,28,29]}.

The power of artificial intelligence-based computing solvers has been misused, leading to the widespread application of technology and applied science to estimate and solve many problems. Recent research in bioinformatics, reactive transport systems, nano-fluidics, atomic physics, plasma physics, electricity energy, nanotechnology, Van-der-Pol oscillatory systems, entropy optimization system in the fluid flow system and functional mathematics has been reported to be of paramount importance (see ^{[30,31,32,33,34,35,36,37,38,39,40,41,42,43]} and references cited in it). The above outcomes are motivational affinities for the malaria model to investigate in the artificial intelligence base, numerical computing solver. According to our review of the literature, Levenberg-Marquardt artificial neural networks (LMANNs) have yet to be used to resolve initial value problems (IVBs) of non-linear ordinary differential equation (ODE) systems that describe malaria dynamics. The following is an emphasis on the revolutionary contributions of the work to be implemented in malaria models for Levenberg-Marquardt artificial neural networks (LMANNs).

● The effect of the malaria model, which is expressed by a non-linear system containing six ordinary differential equations corresponding to the initial value problems (IVPs), is investigated using the supervised learning infrastructure based on 2 layers structure of the Levenberg-Marquardt artificial neural networks (LMANNs).

● Using the Adams numerical method, the mean square error (MSE) analysis of computationally achieved results to compute merit function designed by LMANNs considering six class reference solutions based on malaria epidemic model performed efficiently by index. Moreover, the comparative study validates the outcomes attained through the designed solver.

● Levenberg-Marquardt back-propagation is used to carry through validation, testing, and training in order to obtain decision variables for artificial neural networks (ANNs) for each epoch index increment. To meet this goal, we have considered a model of six complex classes based on the Susceptible people (${S}_1$), the infections people (${I}_1$), the Pseudo recovered people (${R}_1$), the recovered people prone to re-infection ($R_2$), the susceptible mosquitoes ($S_2$) and the infections mosquitoes ($I_2$).

● Accurate performance, convergence and reliability of (LMANNs) to resolve the malaria model with reference dataset for variation of two different parameters are endorsed by correlation, regression curve and histograms with error analysis and comparative study.

Section 2 provides a brief description of mathematical models of the malaria consisting system of nonlinear differential equations, LMANNs methodology is given in Sect. 3, the numerical analysis and discussion are provided in Sect. 4 for various cases of malaria dynamics, we conclude the research article in the last section.

2.   Modelling of mathematical for malaria models

The following 4 classes of the human population are considered for the development of transmission dynamics with the rehabilitation and re-infection of the malaria model, namely, the $S_1(t)$ denoted susceptible people class at time $t$, $I_1(t)$ is an infectious people class, $R_1(t)$ denotes a pseudo-recovered people with the possibility of infection reactivation (relapse), and $R_2(t)$ is a recovered people with the possibility of re-infection. Hence, the total number of human population $N_1(t)$ is calculated as follows: $N_1(t) = S_1 (t) + I_1 (t) + R_1 (t) + R_2 (t)$, on the other hand, the total number of mosquito population $N_2(t)$ is divided into $2$ groups: susceptible class $S_2(t)$ and infectious class $I_2(t)$, $N_2(t) = S_2(t) + I_2(t)$ The dynamics of malaria transmission, including re-infection and relapse, are defined by the system of non-linear initial value problem for deterministic mathematical model below;

Where, in the nomenclature table, definitions of each parameter on the malaria model (2.1)–(2.7) are given. For the dynamics of malaria, a graphical representation of the malaria model is shown in Figure 1 to more evidently decode the information.

Consider the malaria model equations as a dynamical system

The Eqs (2.8)–(2.14) mathematical explanation of the dynamical system of the malaria model.

3.   Methodology

This section covers the most important aspects of our proposed mathematical modelling of performance matrix. Three steps have been used to execute the mathematical modelling: In step one, the malaria model is evaluated by changing two different parameters, which is pointed to as the input reference dataset point for FFNN, step two, LMBNNs model layer structure formulation and training of LMBNNs models is considered. LMBNNs is executed in step three with Levenberg-Marquardt Solver. A graphical summary of the study presented is shown in Figure 2. It presents the Adams Predictive Accuracy Method of the system (2.8)–(2.14). To increase the level of accuracy of the results with the information provided by the predicted results, we first used the predictive solution and then the whole numerical approach using the Adams method configuration. The predictor-corrector method (2.8)–(2.14) of the equations may be given as (3.1)–(3.7).

In the case of the first equation of set (3.2)–(3.6), the relation of the predictor 2-step formula is given:

While, in the case of the first equation of set (3.2)–(3.6) the 2-step formula of the corrector relationship is written as:

Accordingly, in set (3.2)–(3.6) the formulas are formulated for the predictive and accuracy method for the remaining equations. The FFNN data-set can be configured with the Adams numerical method summarized in Eqs (2.8)–(2.14) to solve the malaria model. However, in the study presented, we have developed an FFNN data-set using the mathematical NDSolve routine with the algorithm Adams' for each scenario of malaria model. To solve each scenario of the malaria model, the layer structure of FFNN models with log sigmoid activation function and 40 numbers of neurons in the hidden layer are used. The architecture designed by FFNN is shown in Figure 3. The Levenberg-Marquardt Method (LMM) back-propagation is used to train FFNN, which entails explaining the merit function of the error basis for FFNN-LMM. The merit function of mean square error (MSE) is built on metrics and objective optimization the function is executed with LMM for each case.

The following is a measure of the performance of a mathematical note by AE, merit figure, i.e. MSE and regression coefficient:

Here, $S_q$, $\hat{S}_q$ and $\overline{S}_q$ stand for reference, $qth$ estimates the input solution and the means, while the $m$ represents the total number of grids in the input. The $R$ unit value, i.e., the parameter required for successful modeling is the square root of $R^2$, and the absolute and mean square error must be equal to zero for successful modeling scenarios.

4.   Numerical results and discussion

Simulated studies of numbers with the necessary explanations are presented here for the first order non-linear ODE (2.1)–(2.7) system in which the malaria model by using the proposed LMANNs method. The epidemic model is represented. The values of the several parameters of the malaria model are discussed in Table 1.

Figure 2 explain the complete process workflow diagram of the offered LMANNs. In the neural network toolbox in the Matlab setting, the offered LMANNs are implemented via 'ntstool' (neural time series tool), while Levenberg-Marquardt back-propagation is used to train neural network weights. The reference data-set of malaria model is created for 100 days as step-size inputs of 1.0 through the Adams numerical approach solutions by using the built-in Mathematica environment NDSolve function for a numerical non-linear system of ordinary differential equations results for every case of the malaria model. The values for $S_1, \, I_1, \, R_1, \, R_2, \, S_2$ and $I_2$ classes for 101 input data-set points that are randomly dispersed to yield a set for validation, testing, and training with $5\%, \, 5\%,$ and $90\%$ respectively. Two layered structure LMANNs the computational model based on neural networks with Levenberg-Marquardt back production with 40 hidden layers is contracted for the results of the malaria model that shown in Figure 3.

In scenario 1 we discussed 3 different cases of changing the values of $\tau = 0.06, \, 0.30$ and $0.85$ by keeping all other parameter are fixed to examine the behavior of Susceptible people class $(S_1)$, infectious people class $(I_1)$, pseudo recovered people class $(R_1)$, Recovered people prone to re-infection class $(R_2)$, mosquito Susceptible class $(S_2)$ and mosquito infection class $(I_2)$. The effects of changing the human re-infection parameter $(t)$ and constant rate of relapse $(\eta)$ on the dynamical behavior of recovered human and infectious population are shown in sub-Figure 4a-f and 5a-f. As shown in Figure 4b and 4d, increasing the re-infection parameter $(\tau)$ increases the population of infection people $I_1$, which reduces the human recovered population R2 over time. Similar, in scenario 2 with three different cases by changing the values of $\eta = 2.8\times10^{-03}, \, 2.8\times10^{-02}$ and $2.8\times10^{-01}$ with other parameter are constant to examine the behavior of the Susceptible people class $(S_1)$, infectious people class $(I_1)$, pseudo recovered people class $(R_1)$, Recovered people prone to re-infection class $(R_2)$, mosquito Susceptible class $(S_2)$ and mosquito infection class $(I_2)$. It is seen in Figure 5c, increasing the constant rate of relapse $(\eta)$ decrease the population pseudo recovered people.

Figures 4–15 describe comparative outcomes by graphical results of various scenarios and cases. Therefore, for the Susceptible people class $(S_1)$, infectious people class $(I_1)$, pseudo recovered people class $(R_1)$, Recovered people prone to re-infection class $(R_2)$, mosquito Susceptible class $(S_2)$ and mosquito infection class $(I_2)$ the graphical and numerical outcomes of LMANNs are defined to explain the performance corresponding to $100$ days for case $1-3$ with both scenarios.

Numerical results are represented in sub-Figure 4a-f and 5a-f. For all cases of each malaria model class, the numerical outcomes obtained by the offered technique are presented in Tables 3–8. The graphical compression is expressed by performance analysis of computational scenarios in the sub-Figure 6a, 6c, 6e, 7a, 7c and 7e. Performance consist of on mean square error (MSE) which is the difference between simulation and observation, the less value of MSE indicates the best performance.

It is noted that Figure 6c of the malaria model describes better performance with scenario 2 as compared to other cases of both scenario because the mean square error (MES) of case 2 is the smallest best validation performance (BVP $= 7.817\times 10^{-07}$ at epoch 1000). Figures 6a, 6c, 6e, 7a, 7c and 7e display the performance of all remaining cases with best validation performance $3.6665\times 10^{-06}, \, 5.5209\times 10^{-06}, \, 9.1707\times 10^{-07}, 3.5802\times 10^{-06}, \, 7.5891\times 10^{-06}$ along with 764,554, 1000,635 and 440 respectively. MSE has specific values for training, testing, and validating of both scenarios in every case of malaria model are shown in Table 2 respectively. Figures 6b, 6, d, 6f, 7b, 7d and 7f show the auto-correlation of both scenarios with three different cases. In the graph we define the correlation through the area. For each input point, the error dynamics are additionally estimated using an error histogram and the results of the graphically are presented in sub-Figure 8a-f of the malaria model.

The reference zero-line error bin has an error of around $[7.8\times 10^{-05}, \, -2.4\times 10^{-04}, \, 5.4\times 10^{-04}, \, -1.9 \times 10^{-04}, \, -4.3 \times 10^{-04}, \,$ and $\, -1.5 \times 10^{-03}$] for various cases, respectively. Specifies the value of the maximum result of the proposed method on the zero lines. In sub-Figures 9a-d and 10a-b describes the analysis of regression plots for validation, testing, and training of the malaria model. Regression value (R = 1) indicates that during computation, there is a very close correlation between output and target values.

Figure 11a-f display the training state of both scenarios with three different case. The training state show the better convergence rate. The value of both Mu and gradient corresponding to epoch show either the convergence is slow or fast. Furthermore, for all three cases of the malaria model through LMANNs, the convergence parameter achieved in terms of execution time, MSE, executed epochs, back-propagation, Mu step-size, and performance is tabulated in Tables 2 and the time of all cases describes the complexity of the suggested process. The back-propagation of step-size Mu and gradient values are about $[10^{-05}, \, 10^{-05}, \, 10^{-05}, \, 10^{-04}, \, 10^{-05}$ and $10^{-05}$] and [$1.58 \times 10^{-01}, \, 1.63 \times 10^{-02}, \, 5.74 \times 10^{-03}, \, 1.31 \times 10^{-03}, \, 1.13 \times 10^{-01}$ and $5.26\times 10^{-02}$] as shown in sub Figure 11a–f for case 1–3 with both scenario, respectively witch shows the convergence for all scenarios along all case of the proposed model through the designed solver is good.

For every case of the malaria model, the results define the convergent performance and accuracy of the proposed system. The small Mu value can be observed to lead the better convergent outcomes. The analysis of fitness plots for the malaria model is shown in sub-Figures 12a-d and 13a-b, the error defined is the difference between target and output for testing, training, and validation at every scenario of input data-set point of malaria model.

The results obtained through LMANNs correspond to the reference (ref) of the Adams numerical solution in every case for all six groups of the malaria model, so absolute errors (AEs) are calculated to access the accuracy gauges. The absolute error (AEs) of all classes are shown in sub-Figure 14a–f and 15a–f for $S_1$, $I_1$, $R_1$, $R_2$, $S_2$ and $I_2$ respectively, for scenarios 1-2 tabular in Tables 9–14. The range of absolute error for Susceptible people class $(S_1)$ is $10^{-03}$ to $10^{-04}$ of case 1 and 3 for scenario 1 and case 1 for scenario 2, $10^{-03}$ to $10^{-05}$ of case 2 for both scenario and $10^{-03}$ to $10^{-04}$, $10^{-06}$ of case 3 for scenario 2 respectively. The range absolute error for infection people class ($I_1$) is $10^{-03}$ to $10^{-04}$ of case 1 2 and 3 for both scenario and $10^{-03}$ to $10^{-05}$ of case 2 for scenario 2 respectively. The range of absolute error for pseudo recovered people class ($R_1$), recovered people prone to re-infection class ($R_2$), mosquito susceptible class ($S_2$) and mosquito infection class ($I_2$) are $10^{-03}$ to $10^{-04}$, $10^{-04}$ to $10^{-05}$ and $10^{-03}$ to $10^{-06}$ of case 1 for both scenario, $10^{-04}$, $10^{-03}$ to $10^{-05}$, $10^{-03}$ to $10^{-04}$, $10^{-04}$ to $10^{-05}$ of case 2 for both scenario, $10^{-03}$ to $10^{-04}$, $10^{-03}$ to $10^{-05}$ of case 3 for both scenarios respectively. The consistency of the suggested technique is shown by these ranges of absolute error for all groups of each case with both scenario, which is up to 10 decimal places.

5.   Conclusions

Through neural networks with Levenberg-Marquardt back-propagation, the integrated computing intelligent platform is presented to obtain the solution of the malaria mathematical model representing the spread of malaria that is constructed based on a real dataset. The input dataset for the malaria disease has been developed using the Adams numerical solver for various groups. The $5\%$, $5\%$, and $90\%$ of reference data-sets are used as validation, testing, and training for LMANNs. The following key findings of the malaria model can be observed based on the above numerical analysis and investigation.

● ODEs representing the radioactive spread of the malaria disease are analyzed with the support of LMANN.

● Comparing the offered results with the numerical outcomes obtained by Adams method up-to 11 decimal point shows the consistency and accuracy of the suggested LMANNs.

● The feature of the suggested approach is further validated by numerically and graphically explanation based on error histogram, regression dynamics, mean square error and Convergence plots.

● The dynamics of malaria model are greatly influenced by the variation of parameters of interest.

● The efficiency of the computational process improves due to complexity mean square error (MSE), time series, regression, and histogram.

The designed solver LMANN for the analysis of structures representing the fluid flow systems ^{[45,46,47,48,49,50]} could be implemented in the future.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University (KKU) for funding this research project Number (R.G.P.2/248/43).

Conflict of interest

The authors declare that there are no conflicts of interest.