Modelling the impact of awareness campaigns on HPV transmission dynamics: a neural network-enhanced mathematical model

Abstract

HPV is the most common sexually transmitted infection; most sexually active individuals are exposed to HPV at some point in their lives, and the majority of these infections resolve naturally through the immune system. However, persistent infection with high-risk HPV types is the leading cause of cervical cancer worldwide, underscoring the importance of vaccination and early detection in reducing disease burden. In this study, the effect of an awareness campaign on the dynamics of human papillomavirus infection has been investigated through a mathematical model using neural networks. The basic characteristic, boundedness, the endemic and disease-free equilibrium point, and the reproduction number of the model have been investigated. Using the reproduction number, the stability of the model is also investigated. Moreover, a sensitivity analysis is performed to identify the most sensitive parameter affecting the spread and control of HPV. The deterministic numerical solution of the HPV transmission model was first obtained using the ode45 solver based on the Runge–Kutta method; the resulting solution trajectories were then used as training data for a feedforward neural network, which serves as a surrogate approximation model to validate and reproduce the system dynamics. The numerical simulation examines mean square error, validation performance, gradient, and mu across varying levels of vaccination coverage, public awareness, and cervical infection rates. Numerical simulation suggests that vaccination and awareness about the risk of human papillomavirus infection play a crucial role in controlling the spread of the virus in the population.

Introduction

Human Papillomavirus (HPV) is of particular concern at this time owing to its status as a vector of concern with regard to public health studies conducted around the world. HPV is a common form of DNA virus that may infect human skin and mucosal cells. There are several subtypes of this virus, and some of them are linked to serious health problems, such as cervical cancer¹. Patients infected with HPV may initially exhibit no symptoms². However, some people get cervical precancerous lesions and invasive cervical cancer because the virus stays within their cells all the time, interrupts intracellular gene regulation, and causes cervical epithelial cells to convert malignantly³. Cervical cancer is mainly promoted by persistent infection caused by HPV, since HPV is generally eliminated by the immune system⁴. It is possible, however, that infections can persist for years, or even a lifetime, increasing cervical cancer risks⁵. In the United States, approximately 43,000 new HPV-associated cancers are diagnosed annually, with cervical cancer being the most common HPV-attributable cancer among women, accounting for approximately 11,500 new cases each year⁶. Cervical cancer is a frequent HPV-induced cancer in females. HPV preventive vaccinations, including bivalent, quadrivalent, and nonavalent, have revolutionized cervical cancer prevention by giving strong immune protection to HPV-exposed people.

To protect against human papillomavirus (HPV) infection and the cellular changes it may cause, the vaccination stimulates the immune system to make antibodies that are unique to each strain of HPV⁷. After the bivalent and quadrivalent vaccinations prevent HPV 6, 11, 16, 18, the nine-valent vaccine prevents HPV types 31, 33, 45, 52, and 58. This avoids over 90% of cervical cancers and more HPV-related malignancies⁸. It is strongly advised that HPV vaccination be given before engaging in sexual activity since it offers immunological protection prior to HPV exposure⁹. In order to prevent HPV infection and related diseases, the WHO and other national health authorities particularly suggest HPV vaccination. They also believe that it is crucial to safeguard person health and advance public health.

Compartmental models in epidemiology utilize ordinary and fractional differential equations to analyse the flow of people among compartments based on infection status^10–18. A compartmental model using vaccine class or state variables can quantify the unique impact of immunization on lowering the vulnerable population fraction and delaying disease spread. The compartmental model may also effectively forecast disease transmission, dissemination, and peak times, enabling prospective evaluation of the epidemic’s spatiotemporal dynamics. The model simulates the effects of isolation, vaccination, and masking on disease transmission to assess the efficacy and cost-effectiveness of control and prevention measures, allowing decision-makers to choose the most effective and cost-effective tools¹⁹. Alaje and Olayiwola²⁰ developed a fractional-order diffusive epidemic model incorporating vaccination and Caputo derivatives to examine the spatiotemporal spread of COVID-19, demonstrating that higher vaccine distribution rates and increased vaccine uptake among susceptible individuals significantly reduce disease transmission and accelerate progress toward epidemic elimination.

Many mathematical studies have examined HPV and cervical cancer transmission processes in recent years, emphasizing the need for prevention, vaccination, and therapy. Okware et al.²¹ used the SVICR model to study HPV dynamics under vaccination. They included those who obtained permanent recovery via vaccination and the immune system, making them no longer vulnerable. HPV secondary infections are not included in the model. In their qualitative deterministic model, Omame et al.²² included HPV vaccination and therapy for both sexes. The results demonstrate that widespread use of condoms, even in conjunction with moderate or low vaccination rates, may lead to the extinction of HPV from the community. In order to reduce the incidence of cervical cancer in India, Rajan et al.²³ developed an SICR model to describe how HPV causes cervical cancer. They used actual data of HPV cases in India and used least squares for parameter fitting. Several mathematical modelling studies have investigated the role of vaccination, treatment, and awareness-based interventions in controlling HPV and related diseases. Elbasha and Dasbach²⁴ developed a multi-type HPV model incorporating vaccination and demonstrated that even partial vaccine coverage significantly reduces infection prevalence at the population level. Sharomi and Malik²⁵ extended compartmental HPV models by incorporating treatment compartments for both sexes, showing that combined vaccination and treatment strategies are substantially more effective than either intervention alone. Liu et al.²⁶ analysed HPV transmission dynamics under vaccine intervention using real cervical cancer data from Hungary, demonstrating that dynamic modelling with empirical calibration can reliably guide public health policy. Beyond vaccination and treatment, awareness-driven behavioural changes have increasingly been recognized as an important mechanism in epidemic control. Misra et al.²⁷ showed in an infectious disease context that awareness campaigns can meaningfully reduce the susceptible population by prompting individuals to adopt preventive behaviours, effectively acting as a non-pharmaceutical intervention. while Agaba et al.²⁸ demonstrated that increased public awareness can significantly lower peak infection levels in epidemic models. These studies highlight the important roles of vaccination, treatment, and behavioural awareness in controlling HPV transmission; however, most existing models examine these mechanisms separately and rarely combine them with modern computational approaches such as artificial neural networks, which motivates the development of the present HPV transmission model integrating vaccination, treatment, awareness, and ANN-based numerical analysis within a unified framework.

Artificial neural networks (ANNs) enable a wide range of applications, including pattern recognition, natural language processing, and image recognition. They can be trained using different learning strategies, including supervised learning, unsupervised learning, and reinforcement learning. However, ANNs have significant weaknesses. These models always need an adequate amount of training data to be effective. Moreover, ANNs need as much as required in terms of computation. For instance, ANNs tend to have a high possibility of overfitting. Overfitting in ANNs means the network learned the data perfectly but failed to look at new unseen data. It is notable that ANNs have significant versatility to tackle various computing problems^29–41.

As motivated by the above literature, this study investigates the impact of awareness campaigns on the transmission dynamics of human papillomavirus (HPV) using a mathematical modelling framework enhanced with neural network techniques. The proposed model extends existing HPV compartmental formulations by incorporating a public awareness parameter that reduces the effective transmission rate, thereby capturing the behavioral impact of health campaigns within a unified VSICR structure. The fundamental qualitative properties of the model are analysed, including boundedness of solutions, derivation of the basic reproduction number, and rigorous local and global stability analysis of the disease-free equilibrium, together with a normalized sensitivity analysis to identify the most influential parameters governing HPV spread. In addition, a feedforward neural network trained using the Levenberg Marquardt algorithm is integrated with the classical ode45 solver to approximate and validate the system dynamics. The close agreement between the deterministic solutions and neural network predictions demonstrates the reliability of this hybrid deterministic–neural framework for studying HPV transmission dynamics.

Mathematical formulation of HPV model

N(t) represents the total population. It is further divided into five classes such as vaccinated V(t), susceptible , infected , cervical patientC(t), and recovered R(t). The variables and parameters of the system are all non-negative.

Modelling the awareness campaign

In this model, the awareness campaign is represented by the parameter σ (0 ≤ σ ≤ 1), which captures the proportion of the susceptible population that modifies their behavior in response to public health messaging. Specifically, the campaign affects the S to I transmission pathway: aware individuals reduce high-risk sexual contact and are more likely to seek vaccination, thereby lowering their effective probability of acquiring HPV. This is incorporated into the model through the modified transmission term . When σ =0, no awareness exists and transmission proceeds at the baseline rate . As , the campaign approaches maximum effectiveness and transmission is suppressed. This formulation follows established approaches in behavioral epidemiology, where awareness or education interventions are modeled as a multiplicative reduction in the contact/transmission rate.

Based on the description above, an HPV disease model can be constructed as follows:

1

Initial conditions for model (1):

2

Positive invariant set

The solution of the HPV model is uniformly bounded in a proper subset

At any given time t total population is given as follow:

3

4

5

The feasible solution set in the model (1) will be:

6

Disease-free equilibrium points

Disease-free point of the model (1), when the HPV infection does not exist in the population, is determined by putting I=C=0^42,43.

7

Endemic equilibrium points

To determine the endemic equilibrium points of model (1), we set the right-hand side of the system in (1) equal to zero. After simplifying the resulting algebraic equations, we obtain:

8

Where

Reproduction number

The basic reproduction number is a crucial quantity in epidemic modelling that determines whether a disease will spread or eliminated from the population. If the disease-free equilibrium is stable and the disease will be eliminated from the population and then disease become persist in the population. The next-generation matrix method is employed to derive the reproduction number for the HPV model (1)^44,45. Consider the infected compartments given below to calculate the reproduction number.

9

Let

The Jacobian matrices of F and V by at then:

10

11

Where

12

Then

13

14

Therefore, the basic reproduction number of the HPV model (1) is determined as the spectral radius of the next-generation matrix given as follows:

15

Local stability analysis of disease-free equilibria

Theorem

When, disease-free equilibrium point is LAS, otherwise, it is unstable.

Proof

In order to calculate the local stability of HPV equilibrium points, the Jacobian matrix of the HPV model (1) is as follows:

16

17

18

19

20

21

Which implies that or . Since the parameter’s values are assumed positive, then the first root of the characteristic Eq. (21), denoted as .

So

22

Thus for all the eigenvalues of the system at the DFE point have real parts. As a result, HPV equilibrium point is locally asymptotically stable.

Global stability of HPV-free equilibria

Theorem

If then the DFE in the interior of is GAS.

Proof

Considered Lyapunov function as follows:

23

24

25

26

After simplification, the above equation can be written as follows:

27

28

Where, p₁ = –1, based on the solution of the differential equation system in (1) , if R₀< 1 and

29

Therefore, the highest compact invariant set in I is the singleton . According to the invariance principle is GAS in .

Sensitivity analysis

Through sensitivity analysis of the model parameters, we identified those with the greatest influence on disease transmission. To identify the most sensitive parameter, we applied the following relation:

30

where is parameters and is the reproductive number.

For q = τ

For q = σ

For q = β

For

For

For q=τ

For

For q = τ

Table 2.

Sensitivity indices of R₀.

No.	Parameters	Analytical sensitivity analysis	Numerical sensitivity
1.	π	1	\:1
2.	σ
3.		1	1
4.
5.			\:0.819000819000819
6.	τ
7.
8.	ϕ

A strong sensitivity dependence exists among certain parameters and the reproduction number, as illustrated in Fig. 1. The parameters ϕ,,τ, and σ contribute to reducing the spread of infection. In contrast, increasing and ​ leads to a higher transmission level and therefore increases the spread of the infection. The transmission parameter ​ exhibits a sensitivity index of + 1, indicating that any increase in ​ directly raises the basic reproduction number ​. This highlights the importance of minimizing high-risk contact behaviors and strengthening public health interventions to control disease spread. The vaccination rate ϕ shows a sensitivity index of − 0.90, demonstrating that increased vaccination coverage significantly reduces . Consequently, large-scale HPV vaccination initiatives should be strongly encouraged by public health authorities. Similarly, the progression rate ​ has a sensitivity index of − 0.99, suggesting that prompt medical intervention and early treatment of HPV-infected individuals can substantially reduce the likelihood of progression to cervical disease. In addition, the awareness parameter σ has a sensitivity index of − 0.11, indicating that public awareness programs, although moderate in impact, contribute to lowering transmission by encouraging preventive measures such as vaccination and early screening. Overall, these results indicate that an integrated strategy combining widespread vaccination, awareness campaigns, early detection, and reduced transmission risk is essential for effectively controlling HPV transmission within the population.

Fig. 1.

Illustrates the sensitivity analysis of the .

Numerical simulation and discussion

A numerical simulation of the HPV dynamics of model (1) in populations is presented in this section. In order to solve the state equations over time. Table 1 shows the parameter values of the HPV models that are used for numerical simulation and initial conditions are over the simulation period t ∈ [0,40]. Ordinary differential equations are solved using ode45. The neural network component is employed in this study as a surrogate computational framework to approximate the numerical solution of the proposed HPV transmission model. The training dataset was generated using the deterministic numerical solution of the system of differential equations obtained via the Runge–Kutta method over the simulation time horizon. In this framework, the time variable t is used as the input to the neural network, while the outputs correspond to the five state variables of the model as defined in "Mathematical formulation of HPV model", namely the vaccinated, susceptible, infected, cervical patient, and recovered compartments. The generated dataset was divided into three subsets consisting of 70% for training, 15% for validation, and 15% for testing in order to ensure reliable model performance and to prevent overfitting. A feed-forward neural network architecture with a single hidden layer was adopted, and the network parameters were optimized using the Levenberg–Marquardt backpropagation algorithm. The training process was terminated when the validation performance failed to improve for a predefined number of consecutive iterations or when the mean squared error reached a sufficiently small value. To further evaluate the predictive capability of the neural network, an out-of-sample validation experiment was conducted using parameter values. The neural network predictions were then compared with the numerical solution of the model, demonstrating strong agreement and confirming the robustness and generalization ability of the proposed computational approach.

Table 1.

Parameters Description and its values of the HPV model.

Parameters	Description	Units	Values	Source
ϕ	Vaccination rate for susceptible individuals	year− 1	0.6	26
	Vaccination rate for HPV recovered individuals	year− 1	0.02	Assumed
	Mortality Rate	year− 1	0.006	Assumed
	Rate at which vaccinated individuals join susceptible individuals	year− 1	0.01	26
π	Recruitment rate	Person.year− 1	1000	26
	Partial immunity rate	year− 1	0.0001	26
	Transmission rate	Person.year− 1	0.0002	26
σ	Awareness rate	Dimensionless	0.1	Assumed
τ	Rate of treatment	year− 1	0.001	26
	HPV infected become cervical infected	year− 1	0.7	Assumed
ξ	Death rate of cervical patients	year− 1	0.003	Assumed

In MATLAB, a feedforward neural network (FFNN) is constructed using the train function, which employs the Levenberg-Marquardt algorithm as the default optimization method instead of stochastic gradient descent. The network consists of a single hidden layer with 25 neurons, and it is trained for 100 epochs. Figure 2 describes architecture; training progress (loss decay). As the model learns and advances, the loss lowers dramatically early in the training process. As epochs increase, the rate of progress slows, and loss curve begins to level out, indicating that the model is nearing full optimisation. During the training phase, optimization of the model’s loss parameters is performed. Effective training enhances the model’s predictive accuracy through continuous reduction of the loss function. Figures 3, 4, 5, 6 and 7 collectively present the numerical results of the proposed HPV transmission model and demonstrate the effectiveness of the neural network approximation in capturing the system dynamics under different epidemiological scenarios. Figure 3 illustrates the temporal evolution of the model compartments obtained from both the deterministic numerical solution and the neural network predictions. The results show a strong agreement between the two approaches, indicating that the neural network successfully learns and reproduces the nonlinear dynamics of the HPV transmission model. This close correspondence confirms the reliability of the neural network framework as a surrogate modelling technique for solving complex epidemiological systems. Figure 4 provides further validation of the neural network model through regression analysis, training performance, and error evaluation. The regression results demonstrate a strong correlation between the predicted outputs and the target values obtained from the deterministic solver, while the small mean squared error values indicate accurate model training and convergence. Figures 5, 6 and 7 investigate the impact of key epidemiological parameters on the transmission dynamics of HPV in the community. Specifically, Fig. 5 demonstrates the influence of the vaccination rate on the dynamics of HPV transmission in the population. As the vaccination parameter increases, a significant decline in the number of infected individuals is observed. This result indicates that vaccination plays a critical role in reducing the transmission of HPV and protecting susceptible individuals from infection. At higher vaccination levels, the vaccinated population increases steadily, while the infected population decreases correspondingly. This behavior highlights the effectiveness of immunization programs in controlling HPV spread. The results therefore support the implementation of large-scale vaccination campaigns as a key public health strategy for reducing HPV prevalence and preventing associated complications such as cervical cancer. Figure 6 examines the influence of public awareness on disease dynamics and shows that increased awareness leads to greater adoption of preventive measures such as vaccination and early medical consultation. Consequently, the infected population declines while the protected population increases. Finally, Fig. 7 examines the effect of the progression rate from HPV infection to cervical disease. The simulation results indicate that when the progression rate increases, the number of cervical disease cases rises correspondingly. This outcome reflects the potential consequences of delayed diagnosis or lack of treatment among HPV-infected individuals. The results emphasize the importance of early detection, screening programs, and timely medical intervention. By identifying HPV infections at an early stage and providing appropriate treatment, the transition to severe cervical conditions can be significantly reduced. Therefore, strengthening screening strategies and improving access to healthcare services are essential components in minimizing HPV-related health complications. Overall, the numerical simulations highlight the significant roles of vaccination strategies, awareness campaigns, and medical interventions in mitigating HPV transmission and reducing the burden of cervical disease in the population (Tables 2, 3, 4, 5).

Fig. 2.

Depicts a neural network architecture with one hidden layer containing 25 nodes and an output layer consisting of five nodes.

Fig. 3.

The dynamics of the HPV model in the population with deterministic and neural networks.

Fig. 4.

The regression, training, instance, and performance for the HPV model.

Fig. 5.

The dynamics of the deterministic HPV model with NNs for different values of ϕ.

Fig. 6.

The dynamics of the deterministic HPV model with NNs for different values of σ.

Fig. 7.

The dynamics of the deterministic HPV model with NNs for different values of .

Table 3.

Neural network performance metrics under varying vaccination rate ϕ.

Parameter ϕ	Mean square error			Performance	Mu	Gradient	Epoch
	Training	Validation	Testing
0.6	7.9e-08	6.4e-08	5.9e-08	5.34e-11	1e-11	3.92e-07	100
0.7	8.1e-08	9.4e-08	7.2e-08	5.36e-11	1e-11	4.24e-07	100
0.8	7.7e-08	8.2e-08	6.6e-08	5e-11	1e-11	3.68e-07	100
0.9	6.7e-08	7.1e-08	5.5e-08	5.09e-11	1e-11	3.29e-07	100

Table 4.

Neural network performance metrics under varying awareness rate σ.

Parameter σ	Mean square error			Performance	Mu	Gradient	Epoch
	Training	Validation	Testing
0.1	7.9e-08	6.4e-08	5.9e-08	5.34e-11	1e-11	3.92e-07	100
0.3	7.1e-08	8e-08	5.6e-08	4.89e-11	1e-11	3.75e-07	100
0.5	9.3e-08	5.9e-08	1.1e-07	5.36e-11	1e-11	3.85e-07	100
0.6	2.5e-07	2e-07	2.5e-07	1.02e-10	1e-11	7.4e-07	100

Table 5.

Neural network performance metrics under varying HPV to Cervical progression rate .

Parameter	Mean square error			Performance	Mu	Gradient	Epoch
	Training	Validation	Testing
0.001	7.9e-08	6.4e-08	5.9e-08	5.34e-11	1e-11	3.92e-07	100
0.003	7.6e-08	7.1e-08	5.6e-08	5.21e-11	1e-11	3.78e-07	100
0.005	7.5e-08	5.9e-08	5.6e-08	5.21e-11	1e-11	3.79e-07	100
0.007	7.5e-08	5.8e-08	5.6e-08	5.21e-11	1e-11	3.8e-07	100

Conclusion

The dynamics of the HPV model with the vaccination effect through Neural networks is investigated. The model’s boundedness, disease-free and endemic equilibria and the reproduction number is investigated. The disease-free equilibrium point stability is evaluated using . As , the DFE demonstrates stability on both a global and local scale. Sensitivity analysis for the parameters which are involved in the reproduction is also performed. This study employs a single-hidden-layer feedforward neural network, trained using the Levenberg–Marquardt algorithm on solution trajectories generated by the ode45 solver, as a surrogate model to approximate and validate the deterministic solution of the HPV transmission system. This unique methodological synthesis combines modern computer methods with epidemiological modelling methodologies to show that neural networks can accurately analyse virus trends. The integration of neural networks in our study represents a significant methodological advance, offering a fresh perspective on HPV transmission patterns. Using these new computational methods, deterministic models may be assessed with great precision and flexibility. Combining neural networks with traditional numerical techniques for solving ordinary differential equations provides multiple major advantages. A numerical simulation is also performed to determine how each parameter affects HPV dynamics. According to numerical simulation, HPV infected individuals who refuse treatment and develop cervical cancer contribute to the infection’s spread across the community. Additionally, the dynamics of HPV in the population decrease as the values of the vaccine and awareness parameters increase. Policymakers may utilize the model’s output to reduce the prevalence of illness in the general population. This work has the potential for future expansion into fractional derivatives.

Author contributions

K.S: and A.A.: Conceptualization, Methodology, Software, Writing- Original draft preparation, Review & editing, Numerical computations, results and discussion; C.D & A.K: Methodology, Software; F.S.A: Writing- Original draft preparation. S.A.B & H.W.K: Simulations, Visualization, Investigation. All the authors reviewed the manuscript.

Funding

Acknowledgment: The authors extend their gratitude to the Deanship of Scientific Research at Prince Sattam Bin Abdulaziz University, Kingdom of Saudi Arabia 1447/R/2026.

Data availability

The corresponding author will provide the study data upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.