Deterministic and ANN-based modelling of measles transmission using real epidemiological data

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Deterministic and ANN-based modelling of measles transmission using real epidemiological dataDownload PDF Download PDF ArticleOpen accessPublished: 16 September 2026Kamil Shah1,Changqing Du1,Ali Akgül2,6,7,8,9,10,Farad Sameer Alshammari3,Sanaa Ahmed Bajri4 &…Hamiden Abd El-Wahed Khalifa5 Scientific Reports volume 16, Article number: 28798 (2026) Cite this articleSave articleView saved researchAbstractThis study aims to develop a nonlinear SVLIQR epidemic model to investigate measles transmission dynamics by incorporating vaccination, latency, quarantine, and treatment effects. The qualitative properties of the model, including positivity and boundedness of solutions, are established. The basic reproduction number is derived using the next-generation matrix method, and the existence of disease-free and endemic equilibrium points is examined. The local stability of the disease-free equilibrium is analysed using the Routh–Hurwitz criterion, while global stability is studied through the Castillo-Chávez approach. In addition, model parameters are estimated using the least squares method based on reported measles cases in China from 2005 to 2017. To improve the numerical approximation and capture the nonlinear dynamics of the system, an ANN-based solver coupled with the ode45 scheme is implemented. The results show that the model fits the reported data well, and the ANN framework provides highly accurate and convergent approximations under different epidemiological parameter settings. Numerical simulations further indicate that higher vaccination and quarantine rates significantly reduce measles transmission. Overall, the proposed framework provides a useful and reliable tool for understanding measles dynamics and assessing effective intervention strategies.IntroductionMeasles is an infectious viral disease, which is highly contagious and a member of the family Paramyxoviridae and is a constant threat to human health all over the world. A short period of incubation, estimated at 10–12 days characterises the disease with clinical manifestations that include conjunctive coryze, cough, fever and generalised rash1. It is one of the most contagious infectious diseases known as it is transmitted mostly by respiratory droplets and direct contact with infected environments. Despite the fact that there is an effective and cheap vaccine, measles outbreaks are still witnessed in most parts of the world, particularly in middle-income nations. WHO reports show that the changes in normal vaccination services have caused increasingly more unvaccinated children all over the world, the first major setback in vaccination coverage since 20092. This has been coupled with a sharp rise in the number of reported measles cases in various countries and clearly evidence of the continued weakness of the public health3.Measles outbreaks are not only triggered by biological factors but are highly sensitive to social, economic and logistical factors. These challenges underscore the criticality of knowledge concerning the interaction between measures of public health and disease transmission processes, like vaccination and quarantine. Mathematical modelling has proved to be a beneficial and useful instrument in the study of dynamics of infectious diseases and assessment of public health response4,5,6,7,8,9,10,11. Mathematical models have the benefit of enabling a rigorous study of thresholds, equilibrium points and stability properties to be studied by formulating epidemiological processes as nonlinear differential equations. Such methods have been effectively used to a wide variety of infectious disease modelling, including HIV/AIDS, COVID-19, cholera, TB, and so on12,13,14,15. Combining the real epidemiological data with mathematical modelling offers a powerful framework for understanding the complex dynamics of the disease and for supporting the evidence-based public health decision-making. Integration of real epidemiological data, with mathematical modelling, provides a robust structure towards explaining the convoluted nature of the disease, and in aiding in the evidence-based decision-making of a population. The importance of vulnerable children in the measles outbreaks also became pointed out by Wallinga et al.16. as well as the fact that with a high vaccination rate, the epidemics can be prevented. Rehman et al.17 and James et al.18 used delayed SIQR model to show that improvement of any control measure is most effective (such as hospitalisation and vaccination and quarantine) compared to single intervention measures. Fakhruddin et al.19 reasoned on the deterministic model and highlighted accessibility to the treatment, and Kudus et al.20 reasoned on the effects of the program of a doublet dose of the vaccine as a way of alleviating the outbreaks. Many mathematical models have been suggested to decompose the pattern of the expansion of the measles epidemic due to the influences of different epidemiological states of affairs21,22,23,24,25,26,27,28,29,30,31,32. Uncommon immune-mediated complications have been documented post-COVID-19 vaccination like VKH disease and Sweet syndrome33,34. Early detection is the emphasis of HLH post-COVID-19 inoculation and fast screening techniques, like the LIP score35,36.ANNs offer a multi-purpose computational model of complex programs like image interpretation, language processing and the like. They can be trained using supervised, unsupervised or reinforcement learning frameworks and hence the adaptable use of ANNs to various fields. Although ANNs are applicable in a wide range of fields, they have certain limitations inherent to their nature, including the complexity of the calculations, the need to have large volumes of training data, and overfitting. Regardless of these constraints, neural networks can be a versatile and useful tool in all forms of mathematical modelling37,38,39,40,41,42,43,44,45.In spite of the fact that some mathematical models have been designed to examine the transmission of measles, a number of the existing studies conducted are based on focusing primarily on classical epidemiological dynamics with no concomitant incorporation of vaccination, latency, quarantine, treatment, real epidemiological data, and ANN-based computational analysis into a single framework. Some earlier works examined particular control strategies such as vaccination, hospitalization, or quarantine, while others investigated deterministic epidemic behaviour; however, limited attention has been given to combining analytical epidemiological modelling with data-driven parameter estimation and ANN-based numerical approximation for measles transmission. Therefore, there remains a need for a comprehensive framework that not only captures the nonlinear transmission dynamics of measles under multiple intervention mechanisms, but also improves predictive accuracy using real reported data and artificial neural networks. This study addresses this gap by developing a nonlinear SVLIQR measles model incorporating vaccination, latency, quarantine, and treatment, estimating model parameters from reported measles cases in China, and employing an ANN-based numerical scheme to analysed the model dynamics and control effects more accurately.This study aims to develop a nonlinear SVLIQR model to investiga\(\gamma\)te the dynamics of the measles in the population while explicitly accounting for vaccination latency, quarantine and treatment processes. The key properties of the measles model, including boundedness, positivity, the existence of equilibrium states, the basic reproduction number, and the stability analysis of the measles-free equilibrium, is investigated using analytical techniques, providing insight into conditions for the measles elimination. The least squares method is employed to estimate the parameter values using the measles surveillance cases from 2005 to 2017, providing a realistic representation of disease dynamics. Furthermore, the neural network-based numerical framework is implemented to obtain more accurate and realistic dynamics of the measles, as well as the impact of different parameters on the dynamics of measles.Formulation of the modelThe detailed transmission dynamics of the measles in the population according to their disease status have been shown in Fig. 1.Fig. 1Full size imageShows the flow sheet for the measles transmission.Figure 1 illustrate that the total population is divided into six compartments and is denoted by \(\:N\left(t\right)\). Susceptible individuals who are vulnerable to contracting the disease is denoted by \(\:S\left(t\right)\), the vaccination compartment means individuals who receive the vaccine to acquire resistance against measles are denoted by \(\:V\left(t\right)\), the exposed compartment is denoted by \(\:L\left(t\right)\), infected individuals compartment is denoted by \(\:I\left(t\right)\), the isolated or quarantine individuals compartment is denoted by \(\:Q\left(t\right)\) and recovered individuals compartment is denoted by \(\:R\left(t\right)\).Table 1 The description of parameters, its values and sources.Full size tableBased on the above detail description of the measles transmission dynamics in the population the mathematical model for the measles can be constructed as follow:$$\left\{ \begin{gathered} \frac{{dS}}{{dt}}=\Lambda N - \gamma S - \frac{{\omega SI}}{N} - \beta S+\alpha V, \hfill \\ \frac{{dV}}{{dt}}=\beta S - \gamma V - \alpha V - \frac{{\phi VI}}{N} - \tau V, \hfill \\ \frac{{dL}}{{dt}}=\frac{{\phi VI}}{N}+\frac{{\omega SI}}{N} - \gamma L - \kappa L, \hfill \\ \frac{{dI}}{{dt}}=\kappa L - \left( {\nu +\gamma +\lambda +\varepsilon } \right)I, \hfill \\ \frac{{dQ}}{{dt}}=\lambda I - \left( {\sigma +\gamma +\xi } \right)Q, \hfill \\ \frac{{dR}}{{dt}}=\tau V\ominus +\varepsilon I+\xi Q - \gamma R. \hfill \\ \end{gathered} \right.$$(1)Initial condition$$S\left( 0 \right) \geqslant 0,V\left( 0 \right) \geqslant 0,L\left( 0 \right) \geqslant 0,I \geqslant 0,Q\left( 0 \right) \geqslant 0,R\left( 0 \right) \geqslant 0$$(2)Positivity of solutionTheorem # 1: The solution set \(S\left( t \right),V\left( t \right),L\left( t \right),I(t),Q\left( t \right),R\left( t \right)\) of the measles model (1) remains nonnegative for all \(t>0\), if \(S\left( 0 \right),V\left( 0 \right),L\left( 0 \right),I(0),Q\left( 0 \right),R\left( 0 \right) \geqslant 0\).$$\left\{ \begin{gathered} {\left. {\frac{{dS}}{{dt}}} \right|_{S=0}}=\Lambda N+\alpha V, \hfill \\ {\left. {\frac{{dV}}{{dt}}} \right|_{V=0}}=\beta S, \hfill \\ {\left. {\frac{{dL}}{{dt}}} \right|_{L=0}}=\frac{{\phi VI}}{N}+\frac{{\omega SI}}{N}, \hfill \\ {\left. {\frac{{dI}}{{dt}}} \right|_{I=0}}=\kappa L, \hfill \\ {\left. {\frac{{dQ}}{{dt}}} \right|_{Q=0}}=\lambda I, \hfill \\ {\left. {\frac{{dR}}{{dt}}} \right|_{R=0}}=\tau V+\varepsilon I+\xi Q. \hfill \\ \end{gathered} \right.$$(3)Measles free equilibrium pointsMeasles-free equilibrium points refers to the state when there is no infected individual by the measles exist in the population to calculate the measles free equilibrium point put right side of measles model (1) equal to zero and also consider \(L=I=Q=0\)$$\left\{ \begin{gathered} \Lambda N - \gamma S - \frac{{\omega SI}}{N} - \beta S+\alpha V=0, \hfill \\ \beta S - \gamma V - \alpha V - \frac{{\phi VI}}{N} - \tau V=0, \hfill \\ \frac{{\phi VI}}{N}+\frac{{\omega SI}}{N} - \gamma L - \kappa L=0, \hfill \\ \kappa L - \left( {\nu +\gamma +\lambda +\varepsilon } \right)I=0, \hfill \\ \lambda I - \left( {\sigma +\gamma +\xi } \right)Q=0, \hfill \\ \tau V\ominus +\varepsilon I+\xi Q - \gamma R=0. \hfill \\ \end{gathered} \right.$$(3)After some manipulation, we get the following Measles free equilibrium points:$$MFE=\left\{ {\frac{{\Lambda \left( {\alpha +\gamma +\tau } \right)}}{{\alpha \gamma +\left( {\beta +\gamma } \right)\left( {\gamma +\tau } \right)}},\frac{{\beta \Lambda }}{{\alpha \gamma +\left( {\beta +\gamma } \right)\left( {\gamma +\tau } \right)}},0,0,0,\frac{{\beta \Lambda \tau }}{{\gamma \left( {\alpha \gamma +\left( {\beta +\gamma } \right)\left( {\gamma +\tau } \right)} \right)}}} \right\}$$(4)The endemic equilibrium point refers to the state when the measles disease exists in the population. To calculate the endemic equilibrium point, put the right side of Eq. (1) is equal to zero, and we get the following points:$$\left\{ \begin{gathered} {S^*}=\frac{{Np\Lambda \left( {N\left( {\alpha +\gamma +\tau } \right)+I\phi } \right)}}{{N\left( {N\gamma \left( {\alpha +\beta +\gamma } \right)+N\left( {\beta +\gamma } \right)\tau +I\left( {\beta +\gamma } \right)\phi } \right)+I\left( {N\left( {\alpha +\gamma +\tau } \right)+I\phi } \right)\omega }}, \hfill \\ {V^*}=\frac{{{N^2}p\beta \Lambda }}{{N\left( {N\gamma \left( {\alpha +\beta +\gamma } \right)+N\left( {\beta +\gamma } \right)\tau +I\left( {\beta +\gamma } \right)\phi } \right)+I\left( {N\left( {\alpha +\gamma +\tau } \right)+I\phi } \right)\omega }}, \hfill \\ {L^*}=\frac{{Ip\Lambda \left( {N\beta \phi +N\left( {\alpha +\gamma +\tau } \right)\omega +I\phi \omega } \right)}}{{\left( {\gamma +\kappa } \right)\left( {N\left( {N\gamma \left( {\alpha +\beta +\gamma } \right)+N\left( {\beta +\gamma } \right)\tau +I\left( {\beta +\gamma } \right)\phi } \right)+I\left( {N\left( {\alpha +\gamma +\tau } \right)+I\phi } \right)\omega } \right)}}, \hfill \\ {L^*}=\frac{{I\lambda }}{{\gamma +\xi +\sigma }}, \hfill \\ {R^*}=\frac{1}{\gamma }\left( {I\left( {\epsilon +\frac{{\lambda \xi }}{{\gamma +\xi +\sigma }}} \right)+\frac{{{N^2}p\beta \Lambda \tau }}{{N\left( {N\gamma \left( {\alpha +\beta +\gamma } \right)+N\left( {\beta +\gamma } \right)\tau +I\left( {\beta +\gamma } \right)\phi } \right)+I\left( {N\left( {\alpha +\gamma +\tau } \right)+I\phi } \right)\omega }}} \right) \hfill \\ \end{gathered} \right.$$(5)Reproduction numberThe reproduction number is mainly tell us about the disease mechanisms of disease transmission and control. If the reproduction is greater than 1 it indicate that the disease will spread in the population and if the reproduction number is less than 1 it indicates that the disease will die out from the population. We used the next generation matrix method to derive the reproduction number of the measles model (1)50. For this purpose consider only the infectious class in the measles model (1) is given below:$$\left\{ \begin{gathered} \frac{{dL}}{{dt}}=\frac{{\phi VI}}{N}+\frac{{\omega SI}}{N} - \gamma L - \kappa L, \hfill \\ \frac{{dI}}{{dt}}=\kappa L - \left( {\nu +\gamma +\lambda +\varepsilon } \right)I, \hfill \\ \frac{{dQ}}{{dt}}=\lambda I - \left( {\sigma +\gamma +\xi } \right)Q. \hfill \\ \end{gathered} \right.$$(6)The Jacobean of P and Q is given below:\(P=\left( {\begin{array}{*{20}{c}} 0{\frac{{V\phi }}{{R+S+V}}+\frac{{S\omega }}{{R+S+V}}}0 \\ 000 \\ 000 \end{array}} \right), W=\left( {\begin{array}{*{20}{c}} {\gamma +\kappa }00 \\ { - \kappa }{\gamma +\epsilon +\lambda +\nu }0 \\ 0{ - \lambda }{\gamma +\xi +\sigma } \end{array}} \right)\)$$P{W^{ - 1}}=\left( {\begin{array}{*{20}{c}} 0&{\frac{{V\phi }}{{R+S+V}}+\frac{{S\omega }}{{R+S+V}}}&0 \\ 0&0&0 \\ 0&0&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{1}{{\gamma +\kappa }}}&0&0 \\ {\frac{\kappa }{{\left( {\gamma +\kappa } \right)\left( {\gamma +\epsilon +\lambda +\nu } \right)}}}&{\frac{1}{{\gamma +\epsilon +\lambda +\nu }}}&0 \\ {\frac{{\kappa \lambda }}{{\left( {\gamma +\kappa } \right)\left( {\gamma +\epsilon +\lambda +\nu } \right)\left( {\gamma +\xi +\sigma } \right)}}}&{\frac{{\gamma \lambda +\kappa \lambda }}{{\left( {\gamma +\kappa } \right)\left( {\gamma +\epsilon +\lambda +\nu } \right)\left( {\gamma +\xi +\sigma } \right)}}}&{\frac{1}{{\gamma +\xi +\sigma }}} \end{array}} \right)$$(7)The dominant eigenvalue of the \({R_0}\) is known as reproduction number and is give as follow:$${R_0}=\frac{{\gamma \kappa \left( {\beta \phi +\left( {\alpha +\gamma +\tau } \right)\omega } \right)}}{{\left( {\gamma +\kappa } \right)\left( {\gamma +\epsilon +\lambda +\nu } \right)\left( {\alpha \gamma +\left( {\beta +\gamma } \right)\left( {\gamma +\tau } \right)} \right)}}$$(8)Stability analysisTheorem 2The measles free equilibrium points of the model (1) is locally asymptotically stable if \({R_0}b_{3}^{2}+b_{1}^{2}{b_4}\). Thus, DFE of the system (1) is LAS.Estimation of the parametersParameter estimation is a critical component of epidemiological modelling, directly affecting the accuracy of model outcomes and the capacity of the model to represent disease transmission dynamics. Well-estimated parameters enable the model to reflect observed epidemiological trends and enhance predictive performance. In this section, we estimate the proposed model (1), the parameter values by fitting it to reported measles infection data from China from 2005 to 201751. The estimation is performed using the nonlinear least squares method, which determines parameter values by minimizing the difference between the model-predicted infected and observed cases. In particular, the contact rate parameter \(\omega\) was estimated through the nonlinear least squares fitting procedure by calibrating the infected component of the proposed model to the reported measles cases in China from 2005 to 2017. The value of \(\omega\) was obtained by minimizing the residual error between the model-predicted infected cases and the observed epidemiological data. The detail of the parameters is given in Table 1 and Fig. 2.The parameter estimation was performed using the nonlinear least squares method, which determines the optimal value of \(\omega\) by minimizing the sum of squared differences between the model-predicted infected cases and the observed epidemiological data. The objective function minimized during the optimization process is formally defined as:\(\hbox{min} \sum\limits_{{i=1}}^{n} {{{\left( {I\left( {{t_i}} \right) - \hat {I}\left( {{t_i}} \right)} \right)}^2}}\)where \(I\left( {{t_i}} \right)\) represents the model-predicted number of infected individuals at time \({t_i}\)​ for a given value of \(\omega\), and \(\hat {I}\left( {{t_i}} \right)\) denotes the corresponding reported measles cases from the observed data.Fig. 2Full size imageShows the comparison of real measles cases and the model (1) prediction.Numerical simulationWe present the numerical simulation of the measles model (1) by constructing an ANN-based computational framework to approximate the model (1) state variable \(S\left( t \right),\,V\left( t \right),L\left( t \right),I\left( t \right),Q\left( t \right)\) and \(R\left( t \right)\) over time. The model is described by the system of coupled nonlinear ordinary differential equations to evaluate the dynamics of measles in the population. The neural network is employed with the ODEs whose detail is given Algorithm. The neural network is trained to satisfy the governing equation together with the prescribed initial conditions and parameter values to give more accurate approximations. The parameter values, which are used in the simulation, are mentioned in Table 1.AlgorithmFull size imageANN-based Numerical Scheme for the deterministic measles model.The influence of the contact rate \(\omega\) on the transmission of measles is clearly demonstrated in Fig. 3. As shown in figure, as the contact rate values increase, disease transmission accelerates and depletes the susceptible population more quickly while the infected population remains higher and converges to large endemic equilibrium, demonstrating the sustained impact of frequent contacts on infection persistence. The error distribution in Fig. 4(a) is sharply concentrated around zero, confirming high numerical accuracy. Figure 4(b) shows excellent convergence behaviour with low training and testing errors. The regression plots in Fig. 4(c & d) yield a perfect correlation (R = 1), validating the robustness and generalisation capability of the neural network model. Figure 4 shows the impact of vaccination rate \(\beta\) on the dynamics of measles disease in the population. From Fig. 5 it is observed that as the value of the \(\beta\) increases the susceptible population declines more rapidly indicating that higher vaccination rate effectively decreases the number of individuals susceptible to infection while the number of vaccinated individuals increases. Figure 6 illustrate the performance of the neural network mode. The error histogram in Fig. 6(a) reveals that prediction error are tightly clustered around zero indicating precise learning of the system dynamics. Figure 6(b) display stable convergence of the training and testing error with low mean squared error achieved within a limited number of epochs. The regression plots in 6(c) and 6(d) show an excellent match between predicted and target value with R = 1 confirming the accuracy and reliability of the model. Figure 7 illustrate the effect of progression rate \(\kappa\) on the dynamics of the infected and exposed population. As shown in the Fig. 7, an increase in the \(\kappa\) value results in a faster rise and higher steady state level of the infected population indicating an accelerated progression from exposure to active infection. Conversely, lower values of \(\kappa\) delay the growth of the infected class. The predictive performance of the neural network model for \(\kappa\) is assessed in the Fig. 7. The error histogram in Fig. 8(a) shows that the error are symmetrically distributed around zero indication the stable learning behaviour. Figure 8(b) indicates stable convergence of training and testing errors with low mean squared error achived after 171 epochs. The regression plots in Fig. 8(c) and 8(d) confirm a perfect linear correlation between predicted and target values (R = 1). These results confirm the reliability and robustness of the neural network approach in capturing the effect of the exposed-to infected transition rate. Figure 9 shows the impact of \(\lambda\) on the dynamics of measles in the population. In the Fig. 9 it is observed that when the values of \(\lambda\) increases the number of infected individual decreases while the quarantine individuals increases. Figure 10 demonstrates the reliability and accuracy of the neural network in capturing the nonlinear dynamics of the model.Fig. 3Full size imageShow the influence of contact rate \(\omega\).Fig. 4Full size imageShows the Error analysis and convergence of the neural network for \(\omega\).Fig. 5Full size imageShow the influence of vaccination rate \(\beta\).Fig. 6Full size imageShows the Error analysis and convergence of the neural network for \(\beta\).Fig. 7Full size imageShow the influence of \(\kappa\) rate.Fig. 8Full size imageShows the error analysis and convergence of the neural network for \(\kappa\).Fig. 9Full size imageShow the influence of \(\lambda\) rate.Fig. 10Full size imageShows the Error analysis and convergence of the neural network for\(\lambda\).ConclusionIn this study, a deterministic SVLIQR measles transmission model was developed to investigate the effects of vaccination, latency, quarantine, and treatment on disease dynamics. The mathematical analysis showed that the solutions of the model were positive and bounded and the disease free and endemic level were deduced. The reproduction number was derived as a basic number to describe the threshold behaviour in the new measles transmission, and that the stability was discussed at the point where the reproduction number is below one, the disease-free state was found to be stable. On top of that, estimation of parameters through measles data that was reported in China showed that the model can capture the observed epidemiological trend and there is good correspondence between the model and trend. One of the significant contributions made to this work is the application of an artificial neural network (ANN) to compute the approximation of the solution of the nonlinear measles model as a calculation tool. The impact of applying ANN is that, it enhances numerical approximation of the model trajectories, in addition to effectively approximating the nonlinear relationship between time and epidemic compartments. Under varying parameters, the ANN outcomes were just a bit shy of the deterministic numerical results, whereas the error histograms, regression plot, and the mean squared error values showed high accuracy, the strong convergence, and the high predictive power. Specifically, ANN framework demonstrated valid approximations in cases where contact rate, vaccination rate, progression rate and quarantine rate were varied, and thus it is potential in carrying out sensitivity analysis, and in investigating the effects of control measures on measles transmission. All in all, the synthesis of deterministic modelling, actual epidemiological data, and ANN-based computation offers a more robust framework of comprehending and anticipating the dynamics of measles. All the results suggest that the best approaches to the minimization of transmission are vaccination and quarantine, and the ANN improves the quality, accuracy, and reliability of the numerical analysis. Thus, ANN is as much a supporting computational method, as it can be a significant and powerful component, which adds to the strength of the model in analysing the complex epidemic behaviour. Further investigations in the field can expand on this framework by introducing delay effects or using more sophisticated deep learning models like PINNs or LSTM models.Data availabilityThe datasets used to analyzed during the current study are available from the corresponding author on reasonable request.ReferencesMoltz, H. Fever: causes and consequences. Neurosci. Biobehavioral Reviews. 17 (3), 237–269 (1993).Article  Google Scholar Bester, J. C. Measles and measles vaccination: a review. JAMA Pediatr. 170 (12), 1209–1215 (2016).Article  PubMed  Google Scholar UNICEF, W. 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Available from: https://www.who.int/data/gho/data/indicators/indicator-details/GHO/measles---number-of-reported-casesDownload referencesAcknowledgmentsThe authors extend their gratitude to the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University, Kingdom of Saudi Arabia 1447/R/2026.Author informationAuthors and AffiliationsSchool of Information Engineering, Qujing Normal University, Qujing, 655001, ChinaKamil Shah & Changqing DuDepartment of Electronics and Communication Engineering, Saveetha School of Engineering, SIMATS, Chennai, Tamilnadu, IndiaAli AkgülDepartment of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam bin Abdulaiz University, Al-Kharj, 11942, Saudi ArabiaFarad Sameer AlshammariDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh, 11671, Saudi ArabiaSanaa Ahmed BajriDepartment of Mathematics, College of Science, Qassim University, Buraydah, 51452, Saudi ArabiaHamiden Abd El-Wahed KhalifaFaculty of Science, Department of Computer Sciences, Karadeniz Technical University, Trabzon, TürkiyeAli AkgülArt and Science Faculty, Department of Mathematics, Siirt University, Siirt, 56100, TürkiyeAli AkgülDepartment of Computer Engineering, Biruni University, Topkapı, Istanbul, 34010, TürkiyeAli AkgülMathematics Research Center, Department of Mathematics, Near East University, Near East Boulevard, 99138, Nicosia /Mersin 10, TürkiyeAli AkgülApplied Science Research Center, Applied Science Private University, Amman , JordanAli AkgülAuthorsKamil ShahView author publicationsSearch author on:PubMed Google ScholarChangqing DuView author publicationsSearch author on:PubMed Google ScholarAli AkgülView author publicationsSearch author on:PubMed Google ScholarFarad Sameer AlshammariView author publicationsSearch author on:PubMed Google ScholarSanaa Ahmed BajriView author publicationsSearch author on:PubMed Google ScholarHamiden Abd El-Wahed KhalifaView author publicationsSearch author on:PubMed Google ScholarContributionsK.S. developed the methodology, performed simulations, analyzed results, and wrote the manuscript. C.D. supervised the study and contributed to validation and manuscript revision. A.A., F.S.A., S.A.B., and H.A.E.-W.K. contributed to model development, data handling, interpretation of results, and critically reviewed and approved the final manuscript.Corresponding authorsCorrespondence to Kamil Shah or Farad Sameer Alshammari.Ethics declarationsCompeting interestsThe authors declare no competing interests.Publication consentThis manuscript has been accepted for publication by all authors.Additional informationPublisher’s noteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Rights and permissionsOpen Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. 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