 Research
 Open access
 Published:
Bioinspired algorithm integrated with sequential quadratic programming to analyze the dynamics of hepatitis B virus
BeniSuef University Journal of Basic and Applied Sciences volumeÂ 13, ArticleÂ number:Â 71 (2024)
Abstract
Background
There are a variety of lethal infectious diseases that are seriously affecting people's lives worldwide, particularly in developing countries. Hepatitis B, a fatal liver disease, is a contagious disease spreading globally. In this paper, a new hybrid approach of feed forward neural networks is considered to investigate aspects of the SEACTR (susceptible, exposed, acutely infected, chronically infected, treated, and recovered) transmission model of hepatitis B virus disease (HBVD). The combination of genetic algorithms and sequential quadratic programming, namely CGASQP, is applied, where genetic algorithm (GA) is used as the main optimization algorithm and sequential quadratic programming (SQP) is used as a fastsearching algorithm to finetune the outcomes obtained by GA. Considering the nature of HBVD, the whole population is divided into six compartments. An activation function based on mean square errors (MSEs) is constructed for the best performance of CGASQP using proposed model.
Results
The solution's confidence is boosted through comparative analysis with reference to the Adam numerical approach. The results revealed that approximated results of CGASQP overlapped the reference approach up to 3â€“9 decimal places. The convergence, resilience, and stability characteristics are explored through mean absolute deviation (MAD), Theilâ€™s coefficient (TIC), and root mean square error (RMSE), as well as minimum, semiinterquartile range, and median values with respect to time for the nonlinear proposed model. Most of these values lie around 10^{âˆ’10}â€“10^{âˆ’4} for all classes of the model.
Conclusion
The results are extremely encouraging and indicate that the CGASQP framework is very effective and highly feasible for implementation. In addition to excellent reliability and level of precision, the developed CGASQP technique also stands out for its simplicity, wider applicability, and flexibility.
1 Background
In the era of technology, irrespective of matter whether a challenge is simple or difficult, intelligent and optimal problemsolving techniques are significant in every field. Researchers and programmers are working to improve the productivity and intelligence of software and machines. To provide effective and ideal searching algorithm solutions, artificial intelligence (AI) is utilized in this context. One of the most used and highly developed heuristic search techniques in AI is the genetic algorithm. A genetic algorithm (GA) is a search metaheuristic method that is drawn from the evolution of biological organisms [1]. GA used inheritance, mutation, selection, and some other procedures to identify the best solution for a given issue. GA utilizes parallel populationbased algorithm with stochastic selection of numerous individual solutions, probabilistic crossover, and stochastic mutation which distinguish it from other algorithms. Some of these components are present in many other search techniques, but only GA has this specific arrangement [2]. Genetic algorithms are used in a variety of fields, including engineering, economics, biology, biotechnology, computer science, mathematics, chemistry, manufacturing, medicine, and other fields [3,4,5,6,7]. It has both benefits and drawbacks. The global optimal region can be found using genetic algorithms swiftly, but the precise local optimum in a region takes a considerable time to find. To improve the search for the global optima, a genetic algorithm and local search approach are combined. A local search can enhance the efficiency of a genetic algorithm, provided that both of these components work together to achieve the optimization objective. In hybrid optimization, there is an opportunity to give best outcomes from both (global and local search) [8].
It is usual to integrate many search strategies in order to improve the inadequacies of the individual algorithms and produce a more resilient optimization technique. Utilizing local optimization algorithms, new methods have been proposed in recent years to address the GAâ€™s poor accuracy. These methods combine local search techniques, which are effective in locating local optima and global search strategies. These have been effectively applied to address a wide range of issues [9,10,11] and experimental research. These are widely known as hybrid algorithms. The findings [12, 13] demonstrated that hybrid techniques are more effective and frequently discover superior solutions. Elsiedy et al. [14] investigated integrated GA with local search sequential quadratic programming (SQP) using an objective function to optimize the Polymer spur gear system. The designed technique is characterized by high accuracy and reliability, ease of implementation, simple intellectual approaches, broad applicability, and adaptability. Butt et al. [15] inspected the nanopolymeric Casson nanoliquid using the effect of thermal radiation and Joule dissipation by employing metaheuristic CGASQP. It is revealed that the ability of a GA to search globally and the capability of SQP to search locally both effectively guarantee the global optimum retardants and the accompanying parameter values.
GAs typically incorporate hybridization with effective local search strategies for quick parameter optimization of problemspecific variables. The SQP algorithmbased efficient constrained optimization solvers are used for swift local convergence of the variables. Based on the previous investigation, the work deals with local optimal search, named sequential quadratic programming (SQP). The reader is referred to reported research [16, 17] if they are interested in learning more about the subject words, theory, underlying ideas, mathematical context, importance, implementations, and applications of SQP.
Hepatitis B is a contagious illness primarily affecting the human liver and is brought on by the hepatitis B virus (HBV). Hepatitis B is most commonly caused by liver malignancy, liver disease, and organ scarring. Hepatitis B can spread horizontally and vertically from an infected person to a susceptible person. If the virus spreads from one person to another by contaminated blood, tattoos, sexual contact, wounds, needle sticks, water, food, etc., this type of transmission is referred to as horizontal. Hepatitis virus transmission from mother to newborn child occurs vertically [18]. More than 257 million individuals have a chronic liver disease worldwide, and 0.88 million people have died from hepatitis B infection, according to World Health Organization (WHO) 2015 report [19].
The good comprehensive mathematical models provide us with a thorough grasp of fast transmission and assist us in properly assessing the efficacy of control measures in the discipline of mathematical epidemiology. Mathematical models based on the characteristics of the hepatitis B virus have been constructed by many researchers from all over the world [20,21,22,23]. Liu et al. [24] created a nonlinear epidemic after researching the HBV spreads during various stages of infection. After that, a fractional Atangana Baleanu (AB) derivative model of the hepatitis B virus is developed incorporating vaccine effects. Gul et al. [25] examined a new mathematical model to explore the dynamic simulation of HBV disease in asymptomatic carriers through fractional derivatives. Gao et al. [26] conferred the HBV model through the CaputoFabrizio fractional derivative with immunological delay, while also paying attention to the suggested model's dimensional stability. Din et al. [27] proposed a thorough investigation of a stochastic delayed system that rules the HBV transmission mechanism by taking into account white noises and vaccinationsâ€™ impacts. Fayyaz et al. [28] exemplified the HBV model numerically with nonstandard finite differenceÂ method having positivity property, absolute stability, dynamically consistent, and convergence with continuous system. Li and Din [29] constructed an epidemic HBV model with nonlinear incidence rate after evaluating the transmissibility related to various HBV infection phases. Additionally, by utilizing the AB derivative operator with vaccination impacts, the proposed model was simulated. Zada et al. [30] invented nonlinear HBV model to cope the hepatitis B from society by taking three controlling parameters awareness, vaccination, and treatment. They suggested that we must work together with the health department to put our control strategies into action, and the government must take appropriate action in pandemic areas to identify susceptible, infected, and latent populations. Separating and identifying acute and chronic infections in a community is also useful.
Wodajo and Mekonnen [31] developed the aspects of nonlinear deterministic hepatitis B virus disease (HBVD) model considering intervention of vaccination and treatment. They considered the two ways of HBV infection transmission: horizontal and vertical transmission. The prevalence of chronic infection, the death rate from HBV illness, and the exposure rate are also taken into account. The research helped to determine which class of infectious agents is more important in the spread of HBVD and the intervention of vaccination with treatment is the most effective way to stop the spread of the illness. Vertical transmission is controlled by providing treatment to those who are chronically infected, whereas horizontal transmission is reduced by administering vaccines to those who are susceptible, particularly for the newborn populations. Therefore, the treatment and vaccine have various effects on HBV control. Considering nature of HBVD, the whole population \(N\left( t \right)\) is divided into six compartments: susceptible S, exposed E, acutely infected A, chronically infected C, treated T,, and recovered R classes. Following is the nonlinear dynamic HBVD system (1) used in the study:
where \(I_{j} ,j = 1, \ldots ,6\) are initial conditions of the HBVD system. Details of the parameters for the system (1) are described in TableÂ 1.
In this paper, the aspects of SEACTR transmission model of hepatitis B virus disease (HBVD) are investigated by a new hybrid approach of feed forward neural networks (FFNNs). The combination of genetic algorithm and sequential quadratic programming, i.e., CGASQP, is applied, where GA is used as a main optimization algorithm and SQP as a fastsearching algorithm to finetune the outcomes obtained by GA. After reviewing abundant literature, it is proven that no one has considered the HBVD model employing neural networkbased CGASQP procedure. The competency of these neural networkbased methodologies motivates authors to work in miscellaneous fields such as mixed convective secondorder slip flow [32, 33], Jeffrey nanofluid [34], double diffusive stream [35], carbon nanotube model [36], Darcyâ€“Forchheimer nanofluid [37], skin sores illness model [38], smoking nonlinear model [39], Covid19 model [40], controlling of mega cities air pollution [41], cantilever piezoelectric mass beam model [42], and Stuxnet virus dynamics [40].
The salient features of this study optimizing CGASQP for the dynamics of hepatitis B virus disease (HBVD) model are:

To attain significantly less effort required to produce precise results, a methodology that is simple to implement and extendable to analogous systems, continuous outcomes within the input training span, reduced sensitivity to computational round off errors, and a viable alternative to solve HBV model.

To solve the nonlinear HBV system in the frame of feed forward network, a log sigmoid function based on mean square error is developed.

To optimize the solution, GA is used as a main optimization algorithm and SQP is a fastsearching algorithm used to finetune the outcomes trained by GA.

To verify the dependability, durability, and efficiency, the outcomes of CGASQP with Adam method results are compared and their absolute error is calculated.

To explore the convergence, resilience, and stability characteristics, MAD, TIC, and RMSE as well as mean, semiinterquartile range, and median values are also found for nonlinear HBVD model.
The remaining structure of proposed study is designed as: the design methodology of CGASQP is explained in Sect.Â 2.0, the performance metrics are described in Sect.Â 3.0, the results of numerical experiments and a comparison study are analyzed in Sect.Â 4.0, and the paper is concluded in Sect.Â 5.0.
2 Designed methodology
The proposed methodology consists of two phases:

A.
Initially, the neural network modeling of ordinary differential equations (ODEs) system (1) is established in form of MSEbased activation function.

B.
Later on, learning process is done for many runs of CGASQP optimization.
2.1 Neural networkbased formulation
The solution of all classes (susceptible S, exposed E, acutely infected A, chronically infected C, treated T, and recovered R classes) of HBVD model is simulated with rth derivative in form of FFNNsbased ODEs (1):
Here \(\alpha ,\lambda\) and \(\gamma\) are the unidentified vectors of \(W\) defined as:
\(W = \left[ {W_{S} ,W_{E} ,W_{A} ,W_{C} ,W_{T} ,W_{R} } \right]\) for \(W_{S} = \left[ {\alpha_{S} ,\lambda_{S} ,\gamma_{S} } \right],\) \(W_{E} = \left[ {\alpha_{E} ,\lambda_{E} ,\gamma_{E} } \right],\) \(W_{A} = \left[ {\alpha_{A} ,\lambda_{A} ,\gamma_{A} } \right],\) \(W_{C} = \left[ {\alpha_{C} ,\lambda_{C} ,\gamma_{C} } \right],W_{T} = \left[ {\alpha_{T} ,\lambda_{T} ,\gamma_{T} } \right],W_{R} = \left[ {\alpha_{R} ,\lambda_{R} ,\gamma_{R} } \right],\) where
In system (2), the log sigmoid function \(\chi \left( t \right) = \left( {1 + e^{{\left( {  t} \right)}} } \right)^{  1}\) is developed as an activation function with its derivative generally as:
The sum of MSEs as an objective function for HBVD system (1) is described as:
where \(t_{i} = ih,\,\)\(\hat{S}_{i} = \hat{S}\left( {t_{i} } \right),\,\hat{E}_{i} = \hat{E}\left( {t_{i} } \right),\,\)\(\hat{A}_{i} = \hat{A}\left( {t_{i} } \right),\,\hat{C}_{i} = \hat{C}\left( {t_{i} } \right),\,\)\(\hat{T}_{i} = \hat{T}\left( {t_{i} } \right),\,\hat{R}_{i} = \hat{R}\left( {t_{i} } \right),\,\) and \(hQ = 1.\) The MSEs objective functions defined for all classes of HBVD are \(\hat{e}_{1} ,\hat{e}_{2} ,\hat{e}_{3} ,\hat{e}_{4} ,\hat{e}_{5} ,\hat{e}_{6}\) and \(\hat{e}_{7}\) representing the initial conditions (ICs) of system (1). So, for the finest achieved weights, the difference between output and exact solutions is approximately equal to zero as, \(\left[ {\hat{S}\left( t \right) \to S\left( t \right),} \right.\)\(\hat{E}\left( t \right) \to E\left( t \right),\) \(\hat{A}\left( t \right) \to A\left( t \right),\) \(\hat{C}\left( t \right) \to C\left( t \right),\) \(\hat{T}\left( t \right) \to T\left( t \right),\)\(\left. {\hat{R}\left( t \right) \to R\left( t \right)} \right],\) where \(E_{{{\text{HBVD}}}} \to 0\).
2.2 Hybrid GASQP learning process
A learning procedure of hybrid combination of genetic algorithm and sequential quadratic programming, known as CGASQP, is applied to optimize the objective function (9), in this section.
The GA needs a fitness statistic or activation function since it simulates the idea of solution evolution by stochastically creating generations of solution populations. The GA is a parallel populationbased evolutionary search algorithm that uses mechanisms that imitate biological evolution to systematically select, combine, and modify parameters to solve optimization and modeling issues. Natural selection is simulated by GAs, which means that only those species that can adapt to changes in their environment will live, reproduce, and pass on to the next generation.
Each generation comprises a group of individuals, and each individual indicates a potential solution or a position in the search area. Each individual is depicted by a string of characters, integers, floats, or bits. This string resembles a chromosome. Starting with a population of chromosomes that is created at random, the genetic algorithm begins. Then, based on the fitness of each chromosome, it performs a selection procedure and recombination. The next generation is created by combining offspring chromosomes with parent genetic elements. The iteration of this method continues until a halting requirement is met [43]. This approach can be used to find solutions to problems with high levels of complexity, which frequently involve massive, nonlinear, and discrete features.
Hybridization of GA with local search optimization boosts the performance of GA. The GA with local search SQP is designed to optimize the solution of HBVD model. This approach utilized the resilience and global optimization strength of GA. The outcome of the GA's global optimization is used as the starting point for the SQP algorithm, and the challenge of obstacle avoidance in the challenging terrain of the target landing location is then solved by obtaining the local optimal solution. The recently reported applications include inverse heat transfer problem [44], locationallocation modeling for active healthcare facility placement [45], power network optimization [46], eyecorneal model [47], Ebola virus system [48], hybrid fluid [49], and human dermal region system [50]. FigureÂ 1 depicts the procedural workflow of CGASQP for HBVD model. Additionally, the pseudocode of the algorithm (CGASQP) is presented in TableÂ 2. This selection of parameters requires precautions, experience, and considerations because small changes could cause premature convergence.
2.3 Performance tools
Three different performance measuresâ€”mean absolute deviation (MAD), Theilâ€™s coefficient (TIC), and root mean square error (RMSE)â€”are used in this section to evaluate the performance of designed CGASQP for transmission model of hepatitis B virus disease (HBVD).
2.4 Results
The results with numerical experimentations are explained in this section to investigate the solution of SEACTR transmission model of hepatitis B virus disease (HBVD). The solution credibility is increased through comparison with the reference Adam numerical approach (ANA). The nonlinear HBVD model's convergence, robustness, and stability properties are studied through MAD, TIC, and RMSE, and minimum, semiinterquartile range, and median values are also calculated.
2.5 Numeric modeling for HBVD system
The nonlinear ODEs for HBVD system (1) after employing the appropriate values of TableÂ 1 are expressed as follows:
The objective function (4) is constructed for system (15) as:
2.6 Graphical interpretations
The optimization of objective function (16) is performed for HBVD model using FFNNs with 5 neurons and 100 trials employing CGASQP approach. The optimized finest weights achieved by CGASQP for interval [0, 1] and 5 neurons are depicted in Fig.Â 2. Moreover, the approximated solution for these weights is also given by using system (3a):
The outcomes of Adam numerical approach (ANA) with proposed CGASQP are compared in Fig.Â 3 for all classes of HBVD model. To check the matching level of reference ANA and approximated solution attained by CGASQP, absolute errors (AEs) are also presented graphically in Fig.Â 4. The results reveal that approximated results of CGASQP overlap the reference ANA up to 3â€“9 decimal places. The AE for susceptible class \(S\left( t \right)\) is \({10}^{7}\) to \({10}^{5}\), for exposed class \(E\left( t \right)\) is \({10}^{9}\) to \({10}^{5}\), for acutely infected class \(A\left( t \right)\) is \({10}^{7}\) to \({10}^{5}\), for chronically infected class \(C\left( t \right)\) is \({10}^{6}\) to \({10}^{3}\), for treated class \(T\left( t \right)\) is \({10}^{6}\) to \({10}^{4}\) and for recovered class \(R\left( t \right)\) is \({10}^{8}\) to \({10}^{5}\), which endorsed the reliability of CGASQP.
In Figs. 5, 6, and 7 correspondingly, the graphical representations of performance measuresâ€”MAD, TIC, and RMSEâ€”are illustrated for transmission model of hepatitis B virus disease (HBVD). The histogram graphs and Whisker and Box (WB) plots for MAD, TIC, and RMSE for strengthening the optimization of CGASQP are depicted in Figs. 8, 9, and 10, respectively. The WB plot presents numerical data about important characteristics of the distribution, precisely displaying the median, extreme values, and dispersion around the median. The MAD measures for all classes of HBVD model are given in Fig.Â 5 as: for susceptible \(S\left( t \right)\) and exposed \(E\left( t \right)\) classes, MAD values are about \({10}^{6}\) to \({10}^{4}\) in Fig.Â 5(i), in Fig.Â 5(ii) MAD for acutely \(A\left( t \right)\) and chronically \(C\left( t \right)\) infected classes are about \({10}^{6}\) to \({10}^{3}\), while for treated \(T\left( t \right)\) and recovered \(R\left( t \right)\) classes are between \({10}^{6}\) to \({10}^{4}\) as presented in Fig.Â 5(iii).
The TIC measures for all classes of HBVD model are specified in Fig.Â 6 as: for susceptible \(S\left( t \right)\) and exposed classes \(E\left( t \right)\), TIC values are around \({10}^{11}\) to \({10}^{8}\) in Fig.Â 6(i), in Fig.Â 6(ii) MAD for acutely \(A\left( t \right)\) and chronically \(C\left( t \right)\) infected classes are near \({10}^{11}\) to \({10}^{7}\), while for treated \(T\left( t \right)\) and recovered \(R\left( t \right)\) classes are between \({10}^{10}\) to \({10}^{8}\) as presented in Fig.Â 6(iii).
The RMSE measures for all classes of HBVD model are given in Fig.Â 7 as: for susceptible \(S\left( t \right)\) and exposed classes \(E\left( t \right)\), RMSE values are about \({10}^{6}\) to \({10}^{4}\) in Fig.Â 7(i), in Fig.Â 7(ii) MAD for acutely \(A\left( t \right)\) and chronically \(C\left( t \right)\) infected classes are almost \({10}^{6}\) to \({10}^{3}\), while for treated \(T\left( t \right)\) and recovered \(R\left( t \right)\) classes are between \({10}^{6}\) to \({10}^{4}\) as presented in Fig.Â 7(iii).
In Fig. 8(iâ€“vi), most of the values of WB plots for MAD measures lie around \({10}^{6}\) to \({10}^{4}\), for TIC measures are ranges in \({10}^{11}\) to \({10}^{8}\) in Fig. 9(i)â€“(vi), while for RMSE the WB plots values are \({10}^{6}\) to \({10}^{4}\) as shown in Fig. 10(i)(vi). Similarly, the histograms of errors lie between \({10}^{5}{10}^{4}\) for MAD and RMSE as presented in Figs. 8 and 10 (vii)(xii), whereas for TIC maximum values are \({10}^{9},{10}^{10}, {10}^{9}, {10}^{8}, {10}^{9}\) and \({10}^{9}\), respectively, for all classes of HBVD model as seen in Fig. 9(vii)(xii). All these graphical results are extremely encouraging and indicate that the CGASQP framework is very effective and highly feasible for implementation.
2.7 Statistical discussion
The proposed CGASQP strategy's reliability and convergence are further examined by utilizing statistical analysis for the minimum (Min), median (Med), and semiinterquartile range (SIQR) values in form of absolute errors in the interval [0,1] with step size 0.1 and the outcomes are presented in TableÂ 3, 4 and 5 for all classes of HBVD model. Most of the minimum values lie around \({10}^{9}{10}^{6}\) for susceptible class \(S\left( t \right)\), \({10}^{9}{10}^{7}\) for exposed \(E\left( t \right)\) and acutely infected \(A\left( t \right)\) classes, \({10}^{9}{10}^{4}\) for chronically infected \(C\left( t \right)\) and treated \(T\left( t \right)\) classes while \({10}^{10}{10}^{6}\) for recovered class \(R\left( t \right)\), respectively. The median statistics are \({10}^{7}{10}^{5}\) for susceptible class \(S\left( t \right)\), \({10}^{7}{10}^{6}\) for exposed \(E\left( t \right)\) and acutely infected \(A\left( t \right)\) classes, \({10}^{6}{10}^{4}\) for chronically infected class \(C\left( t \right)\) \({10}^{7}{10}^{4}\) for treated class \(T\left( t \right)\) and \({10}^{7}{10}^{5}\) for recovered class \(R\left( t \right)\), respectively. The many SIQR values lie around \({10}^{7}{10}^{6}\) for all classes of HBVD model. All the values are small, precise, and reliable and fulfill the consistency measure of CGASQP approach.
3 Concluding remarks
A metaheuristics algorithm is applied to investigate the aspects of the SEACTR transmission model of hepatitis B virus disease (HBVD) by utilizing the hybrid approach of FFNNs. The GA is the main optimization algorithm, and SQP is a fastsearching algorithm used to finetune the outcomes obtained by GA. The best performance of CGASQP is achieved by constructing an activation/objective function based on MSE for nonlinear transmission HBVD model. The main findings of this high potential study are as follows:

The outcomes of ANA with proposed CGASQP are compared. To check the matching level of reference ANA and approximated solution attained by CGASQP, absolute errors are also presented. The results reveal that approximated results of CGASQP overlap the reference ANA up to 3â€“9 decimal places.

The convergence, resilience, and stability characteristics are explored through mean absolute deviation, Theilâ€™s coefficient, and root mean square error, which endorsed the validity of CGASQP. The results are extremely encouraging and indicate that the CGASQP framework is very effective and highly feasible for implementation.

The proposed CGASQP strategy's reliability and convergence are further examined by utilizing statistical analysis for the minimum, median, and SIQR values in the form of absolute errors. Most of these values lie around \({10}^{10}{10}^{4}\) for all classes of HBVD model. All the values are small, precise, and reliable and fulfill the consistency measure of CGASQP approach.

In addition to excellent reliability and precision, the developed CGASQP technique stands out for its simplicity, wider applicability, and flexibility.
In the future era, hybrid heuristic and swarm optimized algorithms offer great potential for solving different nonlinear problems involving Covid19 models [51, 52], predatorâ€“prey model [53], chaotic systems [54, 55], Drude model [56], computer virusesâ€™ model [57], cancer immune model [58], and coronavirus disease model [59].
Even with these encouraging results, there are still areas for further research because of some study limitations. To get more insights and broaden its usefulness, incorporating the neural networkbased CGASQP process with other models and systems can be investigated. Examining the method in various contexts and situations can also improve and enhance its efficacy, leading to more thorough knowledge and complete solutions in the field.
Availability of data and materials
Not applicable.
Abbreviations
 SEACTR:

Susceptible, exposed, acutely infected, chronically infected, treated, and recovered classes
 HBV:

Hepatitis B virus
 HBVD:

Hepatitis B virus disease
 FFNNs:

Feed forward neural networks
 GA/GAs:

Genetic algorithm/genetic algorithms
 SQP:

Sequential quadratic programming
 CGASQP:

Combination of genetic algorithms and sequential quadratic programming
 ODEs:

Ordinary differential equations
 MSEs:

Mean square errors
 ANA:

Adam numerical approach
 AE/AEs:

Absolute error/absolute errors
 WB:

Whisker and Box
 TIC:

Theilâ€™s coefficient
 MAD:

Mean absolute deviation
 RMSE:

Root mean square error
 SIQR:

Semiinterquartile range
 Min:

Minimum
 Med:

Median
 SIQR:

Semiinterquartile range
 \(S\) :

Susceptible class
 \(E\) :

Exposed class
 \(A\) :

Acutely infected class
 C :

Chronically infected class
 T :

Treated class
 R :

Recovered class
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This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2024/R/1445).
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MS and KSN contributed to conceptualization; MS, RT, MAZR, and KSN performed writing original draft; MS, RT, and KSN contributed to software; MAZR contributed to methodology; MS and RT performed investigation. All the authors participated in writing, investigating, and validating the results of the paper.
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Shoaib, M., Tabassum, R., Raja, M.A.Z. et al. Bioinspired algorithm integrated with sequential quadratic programming to analyze the dynamics of hepatitis B virus. BeniSuef Univ J Basic Appl Sci 13, 71 (2024). https://doi.org/10.1186/s43088024005256
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DOI: https://doi.org/10.1186/s43088024005256