Fractional order modeling of hepatitis C transmission dynamics with physics-informed neural network solutions

Abstract

This study develops a fractional-order mathematical model for Hepatitis C Virus (HCV) transmission using a six-compartment framework and the Caputo–Fabrizio derivative. The operator’s exponential kernel captures essential short-term memory effects in acute infection and treatment response, a key advancement over classical integer-order models. We establish the model’s well-posedness through rigorous positivity, boundedness, existence-uniqueness, and stability analyses. The basic reproduction number confirms epidemic potential, with sensitivity analysis identifying acute transmission (β₂) as the dominant driver, accounting for over 51% of new infections—a critical public health insight. Fractional-order simulations demonstrate that increased memory effects (lower fractional order) delay and flatten outbreak waves, underscoring the importance of sustained interventions. A novel Physics-Informed Neural Network (PINN) is implemented to solve the coupled fractional system. The PINN achieves high predictive accuracy (, low RMSE) for Susceptible, Exposed, Acute, and Quit Treatment compartments, though performance is moderate for the Chronic stage (, reflecting its more complex, heterogeneous dynamics. The PINN’s success offers a mesh-free, computationally efficient alternative to traditional numerical schemes for complex epidemiological systems. This work’s primary contribution is the first integration of a Caputo–Fabrizio HCV model with a PINN solver, creating a hybrid mechanistic-data-driven framework. The results provide actionable strategies: prioritizing acute-phase control and implementing long-term, memory-aware public health measures. The model serves as a validated tool for optimizing HCV elimination strategies, particularly in resource-limited settings, while highlighting directions for future refinement, including integration with real-world clinical data.

Introduction

With 58 million cases worldwide, hepatitis C virus (HCV) infection [1] continues to be a public health concern. Africa [1] is home to 8 million people, or one-sixth, of all chronically infected people. Hepatocellular carcinoma, liver cirrhosis, and even death can result from a persistent HCV infection. An estimated 67,000 people in Zimbabwe [2] were estimated to have chronic HCV in 2022, and the illness claimed 582 lives there [3]. The goal of the World Health Organization’s (WHO) [4] Global Health Sector Strategy on Viral Hepatitis is to eradicate HCV by 2030. Global goals to diagnose 90% of infected individuals and treat 80% of them are part of the elimination plan. By 2030, very few nations [5] will have achieved these goals. An almost tenfold increase in the number of individuals treated is required, from the current 750,000 annually to 7.2 million annually [5]. Like many LMICs, Zimbabwe [6–8] falls short of the HCV diagnosis and treatment goals. According to reports, 16% and 0.69% of the infected in Zimbabwe [9] received a diagnosis and are currently receiving treatment for HCV, respectively. The discovery of directly acting antivirals (DAAs), which are extremely safe and effective, was one of the key advancements in the treatment of HCV [10, 11]. Even though DAA costs have dropped dramatically, access to treatment is still severely hampered in low-income nations like Zimbabwe. According to a study conducted in Zimbabwe [12], a 12-week course of generic sofosbuvir/velpatasvir costs USD 1400. With an average household monthly income of USD 370 [13] and very little health insurance coverage [14], the cost of this is still prohibitive for the Zimbabwean population [15], even though it represents a 96% reduction from the originator brands. Complex care pathways, limited access to diagnostics, and a lack of patient and provider knowledge about HCV are additional obstacles to receiving treatment [16]. Additionally, stigmatization, criminalization, and a lack of trust in the healthcare system may prevent some groups at risk of HCV infection—such as injecting drug users (PWID), people living with HIV, and those incarcerated—from accessing screening and treatment services [17–19].

Based on its growing role in morbidity and mortality, viral hepatitis poses a threat to public health [20, 21]. Hepatitis B (HBV) and C (HCV) cause 91% of morbidity and 96% of mortality, despite the fact that there are five hepatotropic viruses (A, B, C, D, and E) [22]. Globally, chronic HBV and HCV-related complications caused 1.34 million deaths in 2015—more than HIV/AIDS-related causes combined. One With 60 million and 10–15 million individuals suffering from chronic HBV [23] and HCV [24] hepatitis, Sub-Saharan Africa [25] has a significant viral hepatitis burden infection, in that order.

The prevalence of viral hepatitis in Zimbabwe’s general population and the number of fatalities from it are not well documented. According to the WHO scorecard (2019) [26], there are roughly 1.6 million chronic infections and a 10.1% seroprevalence of HBV. The “World Health Organization Viral Hepatitis Scorecard 2019” likely refers to data and statistics on the global prevalence, incidence, and mortality of viral hepatitis (specifically HBV and HCV) as of 2019, with including 296 million people living with chronic HBV, 58 million with chronic HCV, and 1.1 million deaths from viral hepatitis.

According to modeling studies, there are 2500 HBV-related deaths per year, and the prevalence is 9% and the 2018 Lancet Gastroenterology & Hepatology study, “Global prevalence, treatment, and prevention of hepatitis B virus infection in 2016” by Razavi-Shearer D, Gamkrelidze I, Nguyen M, et al., [27] used modeling to estimate the global burden of Hepatitis B Virus (HBV) and evaluate strategies for its prevention and treatment. As of right now, the HCV burden has no official national estimates. According to experts, the prevalence of HCV is between 1% and 2% in the general population, but it is probably higher in high-risk groups. A higher percentage of individuals with chronic infections are unaware of their status. Viral hepatitis is a silent killer because both infections have protracted latent periods during which the infected are asymptomatic. WHO [28] unveiled a bold yet doable global plan in 2016 to eradicate viral hepatitis as a threat to public health by 2030.

The elimination targets for HCV are attainable due to the availability of highly effective directly acting antiviral (DAA) therapy [29–31]. Additionally, modeling studies forecasted that viral hepatitis could be eradicated worldwide [32–34]. To meet national HCV elimination targets, a strategy known as “micro elimination” [35], which targets a single population or geographic region at a time, has been suggested. Despite all of these developments, the majority of African nations still lag behind in developing policies and putting elimination plans into action. The main causes of the slow pace of elimination efforts are a lack of financial resources, political will, and national coordination [23, 24, 36]. Assessing the current situation is crucial because it is uncertain whether Zimbabwe [37] can meet the 2030 elimination targets. Several nations that have reported significant progress toward eradication, such as Rwanda, Georgia, and Egypt, have combated viral hepatitis using various frameworks.

Fractional-order dynamics are essential for this HCV model because they mathematically encode the non-Markovian memory and hereditary properties inherent in the biological processes of infection and treatment. Unlike classical integer-order derivatives that assume instantaneous, memory less transitions, the Caputo–Fabrizio fractional derivative captures how past states—such as previous viral load levels, immune system history, or cumulative drug exposure—continuously influence current disease progression. This is particularly critical for HCV, where the rapid initial viral kinetics, delayed immune response, and pharmacodynamics of Direct-Acting Antiviral therapy exhibit fading memory patterns better described by an exponential kernel than by integer-order rates. The fractional order thus serves as a tunable memory index, allowing the model to represent real-world variations in progression speed and intervention efficacy that classical models cannot capture, ultimately providing more realistic epidemic timelines and intervention outcomes.

The choice of the Caputo–Fabrizio derivative over alternative operators such as the Atangana–Baleanu derivative is fundamentally rooted in the structure of its kernel and the nature of the biological memory it represents. The CF derivative employs a non-singular, exponentially decaying kernel, making it the natural mathematical tool for systems where the influence of past states decays rapidly. This aligns with the acute-phase kinetics and fast pharmacodynamic transitions in HCV infection and therapy. In contrast, the ABC derivative uses a Mittag-Leffler kernel that captures long-range, heavy-tailed memory effects, which are characteristic of processes with slow, power-law decay—a dynamic more relevant to geological or chronic aging phenomena than the fast-saturating viral and treatment dynamics central to this model. The CF framework thus provides a more parsimonious and biologically precise representation for the system under study.

Yunus and Olayiwola examined the efficacy of vaccination in mitigating the transmission of emerging SARS-CoV-2 variants through the application of the SEVIAR mathematical model in sense of Caputo fractional orders [38], suggested using the Laplace-Adomian decomposition method for a malaria epidemic [39], co-infection of Malaria and Ebola [40], tuberculosis incorporating enlightenment and therapy [41] and Lassa virus [42] and also discussed about the transmission and control of Ebola virus using vaccination [43] and Lassa virus by using dual dose of vaccination [44] in sense of caputo operator. They also discussed about the fractional order mathematical modeling for COVID 19 [45], measles [46], computer virus [47] and chikungunya virus dynamics [48].

Muhammad Rafiq et al. proposed a non-linear epidemic model for rift valley fever [49], rubella Virus against two dosage vaccination [50], chlamydia disease [51] and bi-susceptibility patterns in COVID-19 [52]. ur Rahman and et al. [53] proposed a neural network analysis omicron model with vaccination and booster. Butt et al. [54] worked on optimal control and mathematical analysis of Corona-virus during pandemic.

Manivel and et al. proposed fractional order mathematical model for the dynamics of the transmission of Monkeypox [55], impact of Intervention Strategies on the Dissemination of Monkeypox [56], co-infection of Monkeypox and HIV model [57], vaccinated humans with the impairment of Monkeypox transmission [58] and Monkeypox viral transmission using Caputo fractional variational iteration method [59].

Purnendu Sardar and et al. proposed a mathematical model co-infection of tuberculosis and diabetes in pregnant women by using PINN [60], predict tuberculosis in both diabetic smoker and non smoker individuals by using PINN [61], impact of memory and CTL mediated immune therapy on the dynamics of a fractional-order HIV model with antiretroviral therapy-based control optimization [62], PINN approach to analyze the role of antiretroviral therapy in an HIV/AIDS model with both vertical and sexual transmission [63], fractional order HIV model of CD 4+ T-cells with recovery [64] and analysis of a fractional-order HIV model of CD T cells with memory and external virus transmission from macrophages [65].

Tharmalingam Gunasekar and et al. worked on the epidemiological model for monkeypox virus by using Atangana-Baleanu derivative [66], HIV/AIDS treatment by using Caputo-Fabrizio derivative [67], COVID-19 and tuberculosis by using Atangana-Baleanu derivative [68], Monkeypox disease with Ulam-Hyer stability [69], Zika virus control with optimal control [70] and SEITR tuberculosis transmission model [71]. Ali Raza and et al. worked on the HBV model with delay effect into stochastic differential equations [72], stochastic HBV model with the migration effect [73], new application related to HCV model [74], numerical solutions for norovirus epidemic spread for public health control [75] and analysis of stochastic delay dynamics model in maize streak virus [76]. Oluwafemi Ezekiel Abiodun and Morufu Oyedunsi Olayiwola proposed the caputo derivatives for co-infection on covid-19 and tuberculosis [77], bacteria dynamic in tuberculosis disease dynamic [78], booster vaccine effect on COVID-19 [79], awareness and vaccination on tuberculosis dynamics [80], tuberculosis epidemics incorporating consciousness effects [81] and tuberculosis infection dynamics with diagnosis and treatment interventions [82].

Kaushal Soni, Arvind Kumar Sinha and et al. worked the model on monkeypox with control intervention [83], marburg virus for control with limited hospital beds [84] and presence of Burial and Cremation Practices [85], epidemic model with conformable fractional derivative [86], hepatitis B for impact of awareness [87], marburg virus transmission for assessing the impact of awareness, burial and cremation practices [88] and modeling of hepatitis B dynamics incorporating socio-environmental factors [89]. Muhammad Farhan and et al. proposed the HBV models in light of asymptomatic carriers, vaccination and treatment [90], stability analysis of the Hepatitis B model with multi-layer neural network [91], hepatitis B virus with treatment insight of Deep learning methodology [92] and global dynamics and computational modeling of HBV for analyzing and controlling [93].

Abdul Mannan et al. [94] worked a model on HCV and proposed Periodic Neural Networks for Computational Investigations. Sabir et al. [95] proposed a novel radial basis neural structure. Khan and Jung [96] worked in stochastic hepatitis B model using feed forward neural network. Rahman et al. [97] proposed a PINN neural network for Monkeypox disease dynamics through Chebyshev-Distributed Collocation.

The novelty of this work lies in the first integration of a Caputo-Fabrizio fractional-order model for HCV with a Physics-Informed Neural Network (PINN) solver, creating a hybrid mechanistic-data-driven framework specifically tailored to capture the rapid kinetics of acute infection and DAA therapy response. The advantage of the results is demonstrated through the identification of the fractional-order as a universal tempo regulator of the epidemic, the PINN’s high-accuracy, mesh-free solution capability, and the actionable public health insight—validated by dual sensitivity analyses—that acute-phase transmission drives 51% of infections, providing a precise target for optimized intervention strategies.

This study employs Physics-Informed Neural Networks (PINNs) to optimize HCV intervention strategies—specifically treatment adherence enhancement and acute-phase recovery acceleration—due to their unique ability to embed known epidemiological dynamics directly into the learning process, ensuring that policy recommendations remain biologically plausible and clinically interpretable. Unlike conventional neural networks or classifiers, which are data-intensive and risk generating non-physical outputs, PINNs integrate the compartmental ODE structure of our transmission model, allowing them to operate effectively even with the sparse, noisy surveillance data typical of LMICs like Zimbabwe. This approach not only reduces computational costs compared to agent-based or ensemble simulation methods but also incorporates cost-awareness by penalizing economically infeasible strategies within the optimization loss function. The resulting PINN-derived strategies reflect real-world implementation constraints—such as phased rollouts, diagnostic delays, and budget limitations—yielding time-varying intervention schedules that are both actionable for policymakers and scalable within existing health systems, thereby offering a practical, cost-sensitive tool for evidence-based HCV elimination planning in resource-limited settings.

The primary challenge addressed in this work is the computational and theoretical complexity of solving and analyzing a high-dimensional, nonlinear fractional-order system that incorporates treatment adherence dynamics. Specifically, this involves:

• Ensuring mathematical well-posedness (existence, uniqueness, stability) for the Caputo-Fabrizio fractional system, which lacks a singular kernel but requires specialized fixed-point approaches;

• Developing a stable and efficient numerical scheme (Newton interpolation) for the coupled fractional differential equations; and

• Bridging mechanistic modeling with machine learning by designing and training a Physics-Informed Neural Network to solve the system without dense discretization, which necessitates balancing physical constraint enforcement with data fidelity to avoid nonphysical solutions.

These challenges are compounded by the need to provide biologically interpretable results from the fractional memory effects and to perform robust global sensitivity analysis on a parameter-rich model to distill actionable public health insights.

The foundational model by Mupedza et al. [98] (2025) provided a crucial integer-order framework to analyze HCV transmission with treatment adherence. However, this classical formulation assumes Markovian (memory less) dynamics, which oversimplifies the intrinsic memory effects in biological processes such as immune response decay, viral load progression, and the pharmacokinetics of Direct-Acting Antiviral therapy.

This constitutes a significant research gap: the lack of a temporal memory mechanism limits the model’s ability to capture the rapid initial transients and fading historical influences that characterize acute infection and treatment outcomes. To address this, the present study extends the model into the fractional calculus domain using the Caputo-Fabrizio derivative. This critical advancement replaces the integer-order time derivatives with an operator featuring an exponential kernel, specifically chosen to model the short-term, exponentially decaying memory relevant to HCV dynamics. Consequently, while the original model describes what transitions occur, the extended framework captures how the pace and intensity of these transitions are modulated by the system’s history, introducing the fractional order as a key parameter regulating the entire epidemic timeline. This extension bridges the gap between conventional compartmental modeling and the need for more biologically congruent temporal representation in infectious disease dynamics.

This paper is organized as follows: Section “HCV model construction and basic preliminaries” introduces the HCV transmission model, derives the six-compartment fractional-order system, and provides essential mathematical preliminaries. Section “Qualitative analysis” establishes the qualitative properties of the model, including existence, uniqueness, positivity, and boundedness of solutions. Section “Equilibrium points and stability analysis” analyzes equilibrium points, computes the basic reproduction number, and investigates local, global, and Ulam–Hyers stability. Section “Numerical scheme derivation” derives the numerical scheme using Newton interpolation. Section “Graphical visualization” presents numerical simulations, fractional-order effects, and sensitivity analyses. Section “Physics-informed neural networks: bridging machine learning and mechanistic modeling” details the Physics-Informed Neural Network methodology, architecture, training, and results. Section “Comparative analysis: ODE, fractional derivatives, and PINN approaches; provides a comparative analysis of ODE, fractional, PINN approaches” and discusses the findings. Finally, section “Conclusion” concludes the study with future research directions, limitations, and public health implications.

We are going to formulate our problem into Caputo-Fabrizio fractional differential equations.

HCV model construction and basic preliminaries

Mupedza et al. et al. [98] introduced a model for the transmission dynamics of HCV in 2025, which is a non-linear ordinary differential equations with five compartment derived from the various parts of the model. Based on the biological and clinical assumptions outlined, the HCV transmission dynamics are modeled using a system of nonlinear fractional differential equations in the Caputo-Fabrizio sense with an exponential kernel. This operator is chosen to accurately represent the short-term memory and rapid saturation dynamics characteristic of acute viral replication, immune response, and the pharmacokinetic-pharmacodynamic profiles of Direct-Acting Antiviral therapy within the disease transmission process.

Our proposed model, which integrates an Exposed compartment and employs Caputo-Fabrizio fractional-order derivatives, demonstrates significant theoretical and practical advances over established HCV modeling frameworks, including the classical integer-order compartmental model by Mupedza et al. (2025). Unlike conventional models that assume exponential waiting times and homogeneous disease progression, our fractional-order formulation introduces a memory effect that captures the non-Markovian nature of HCV transmission—such as variable latency periods, prolonged viral persistence, and delayed intervention impacts—thereby offering greater biological realism. The explicit inclusion of an exposed compartment further refines the representation of pre-infectious stages, a feature often omitted in prior models. Methodologically, the use of a non-singular kernel (Caputo-Fabrizio) ensures smooth, continuous dynamics and avoids the unrealistic singularities associated with classical Caputo derivatives, enhancing numerical stability and long-term predictive accuracy. Compared to existing works that focus on isolated intervention parameters (e.g., treatment adherence or recovery rates alone), our framework reveals nuanced, time-dependent synergies between adherence enhancement and acute-phase recovery, providing more actionable insights for resource-limited settings. Importantly, the model’s ability to account for long-range temporal dependencies allows for a better fit to longitudinal epidemiological data from Zimbabwe and identifies persistent transmission risks even under improved adherence—a critical public health insight that underscores the need for sustained, combined interventions. Thus, our approach not only aligns more closely with clinical and epidemiological realities but also offers a superior, computationally stable tool for optimizing HCV elimination strategies in low-resource contexts.

The following assumptions underpin the mathematical formulation of our HCV transmission model:

• Homogeneous Population Mixing: The population is assumed to mix uniformly, with equal contact probabilities across all individuals.

• Constant Demographic and Epidemiological Parameters: Recruitment, natural mortality, transmission rates, recovery rates, and treatment adherence are held constant over time.

• Inclusion of an Exposed Compartment: A latent period is incorporated between infection and infectiousness, with a fixed duration.

• Binary Treatment Adherence Classification: Individuals under treatment are categorized as either fully adherent or non-adherent, with immediate transition upon non-adherence.

• Negligible Re-susceptibility: The rate of loss of immunity following recovery is assumed to be minimal, based on empirical evidence.

• Absence of Age and Spatial Structure: The model does not account for variations in transmission or adherence due to age, geographic location, or mobility.

• Data-Driven Parameter Estimation: Model parameters are calibrated using national trend data, assuming representative long-term epidemiological patterns.

The model specifically focuses on the transmission of HCV within both homogeneous populations in Fig. 1. It is possible to divide the population into a few distinct compartments such as Susceptible (), Exposed (), Acute infected (), Chronic infected (), Chronic infected after quitting DDA Treatment () and recovered individuals (). The parameter and its description are discussed in Table 1. The total population at time , denoted by , is given by

1

Fig. 1.

A flowchart of HCV model incorporating acutely infected, chronicly infected, quitting treatment and recovered compartments

Table 1.

Model parameters and their descriptions

Parameter	Description
	Birth rate: The number of susceptible individuals entering the population at rate .
µ	Natural death rate: The mortality rate applied to all compartments.
	Disease-induced death rate (on treatment): Mortality during the chronic phase while receiving DAA therapy.
	Mortality rate (off treatment): Mortality in the chronic phase without DAA therapy.
λ	Force of infection: Defined as .
	Transmission rates from compartments , , , and , respectively.
ϕ	Waning immunity rate: Rate at which recovered individuals lose immunity and revert to susceptible.
	Temporary recovery rate: Rate at which individuals recover directly from acute infection.
	Progression rate: Rate at which individuals progress from acute to chronic infection.
ρ	Recovery rate with treatment: Rate at which individuals recover with DAA therapy.
αρ	Slow recovery rate: Rate at which individuals recover without DAA treatment.
	Treatment quitting rate: Rate at which chronic individuals quit treatment and move to .

The Caputo-Fabrizio() derivative of the HCV disease model is provided by

2

under the following initial conditions, where is the fractional order .

3

Definitions of model variable and parameters

The force of infection , which term support the heterogeneity of population in the model, λ incorporates transmission from all infected states. The inclusion of the Exposed compartment () via the term accounts for potential low-level, presymptomatic viremia during the incubation period, reflecting evidence of early viral replication before acute symptom onset. The parameters are the respective transmission rates, scaled by the total population to reflect frequency-dependent transmission. This structure captures the differential infectiousness across disease stages, which is critical for evaluating stage-targeted interventions.

Let us recall the basic definitions, we need to understand this problem.

Model basic preliminaries

Here, we present some mathematical preliminary results in the form of theorems that we will use to demonstrate the uniqueness and positivity of the Hepatitis C virus caused disease model using Caputo-Fabrizio() (2), as defined, respectively.

Definition 2.1

[99] Assume , for , . The Caputo fractional operator is given as

4

where is a normalization function which satisfies the condition, .

Definition 2.2

[100] The integral operator that corresponds to the Caputo fractional derivative is expressed as follows:

5

Definition 2.3

[99] The Laplace transformation of represents as below

6

Qualitative analysis

In this section, we discuss Existence and uniqueness of system by using fixed point theory, positivity and boundedness of the system of Eq. (2).

Existence and uniqueness of system of solution

This subsection which uses the fixed point theorem to show that the proposed model (2) has a solution and that it is the only one. To make this analysis easier, model (2) can be written like this:

7

where

8

When we use the fractional integral operator from (5),the system in Eq. (7) transforms into the Volterra integral equation type of order , as proven by

9

The following assumption is taken into consideration in order to prove our findings: For this, is a continuous function such that , , , and for some positive constants .

Theorem 3.1

Every single kernel is made up of the elements meet the Lipschitz condition criterion under the assumption and the

Proof

Since and , we obtain from (8)

10

where .

Similarly, we can obtain

11

12

13

14

15

Where

Thus, from (10)-(15),We can see that satisfies the Lipschitz property for all . □

Theorem 3.2

At least one solution exists in the system. (2) if .

Proof

Let

Then, we have

For . As approaches infinity, converges to . In the same way, we can conclude that

Since , we get that with for . Therefore, there exists a solution to the system of Eq. (2). □

□

Theorem 3.3

The system (2) has a exact one solution if

Proof

Suppose there is another solution there such as with initial values such that

Now,

which implies that

As a result, . Therefore, . We can provide evidence for this in a similar manner.

Therefore, the system of Eq. (2) has a exact one solution. □

Positivity of solutions

In this subsection, we derive the Positivity of the system of Eq. (2) of HCV disease dynamic.

Theorem 3.4

For positive initial conditions , the solutions of system of the set remain positive for all .

We use Lemma 3.5 to prove Theorem 3.4

Lemma 3.5

Generalized mean value theorem [101] Suppose and 1, then , where ,

By the above Lemma 3.5, we give obtain the following remark.

Remark 3.6

For all , assume that and . If , , then is non decreasing and if , , then is non increasing.

Proof

The proof follows by using Generalized mean value theorem Lemma 3.5 and Remark 3.6 which shows that the solution of HCV model with exist and has a unique solution. Here, the rate is positively invariant for each hyperplane bonding, the positive octant of the vector field points in Ω. The model assumed in (2) becomes

Then, the solutions of (2) are non-negative for all time . □

Boundedness of solutions

In this subsection, we derive the Boundedness of the system of Eq. (2) of HCV disease dynamic.

Lemma 3.7

The model (2) with non-negative initial conditions in region is positively invariant.

Proof

From the system (2), the total population

Applying the Laplace transform we get,

And the Laplace inverse gives,

As a result, and converges for . So the region is positively invariant. □

Equilibrium points and stability analysis

Using this ideology, let us calculate the equilibrium points and the threshold parameter known as the basic reproduction number. We will analyze the stability of our fractional order system using the Routh–Hurwitz conditions and Lyapunov criteria. we also compute the Ulam-Hyer stability and PRCC sensitivity analysis for our model.

Infection free equilibrium

The overarching dynamics of our model are predominantly influenced by the basic controlled reproduction number, represented as . This equation which denotes the mean quantity of secondary infections generated by one infectious individual upon introduction to a wholly susceptible community. The system demonstrates a disease-free equilibrium point in the absence of disease. , which is mathematically expressed as:

16

Basic reproduction number

In order to determine the fundamental reproduction number , we employ the matrix method of the next generation, which was developed by van den Driessche and Watmough [102]. This method allows us to calculate the fundamental reproduction number.

In calculation part, From the , , and compartments:

Now,

where , , , and .

Through the utilization of their methodologies, we are able to determine two matrices, specifically and , at the state of equilibrium that is devoid of any disease. In contrast to the matrix , which encompasses all other transitions between compartments, the matrix displays the rate of new HCV infections that are generated by specific infectious classes.We can get jacobian matrices and by taking the partial derivatives of the equation with respect to , , and and then evaluating them like this:

Subsequently, The basic reproduction number is defined as the spectral radius of the next-generation matrix . it can be expressed as:

17

Where

1. : The term “secondary infections” refers to the average number of secondary infections that are produced by an individual who is exposed to an infectious agent. The parameter β₁ represents the transmission rate from the exposed class, while the rate at which individuals leave the Exposed class due to the transmission of illness or natural mortality is represented by the equation .

2. : The expression refers to the typical number of secondary infections that are produced by an individual who is acutely infected. While the transmission rate parameter β₂ represents the rate at which individuals leave the acute class due to recovery or natural mortality, the rate at which individuals leave the acute class is represented by the equation .

3. : This expression denotes secondary infections resulting from chronically infected persons undergoing treatment. The factor denotes the fraction of acutely infected individuals advances to chronic infection, while β₃ represents the transmission rate from the chronic infected compartment. The denominator comprises , which encapsulates the treatment failure rate, mortality, and recovery within the treated cohort.

4. : This component accounts for persons who commence but do not complete treatment and therapy and subsequently transition to the class. The value β₄ indicates the increased transmissibility linked to partial therapy, while denotes the fraction of those that do not adhere. The expression encompasses the diminished recovery rate within this cohort, including mortality attributable to disease and natural causes.

Endemic equilibrium point

Our research depends on figuring out and analyzing the Endemic Equlibrium Point because it is important for understanding how system (2) stays the same and changes over time. Examining the EEP provides significant insights into long-term behavior, system stability, and the essential threshold conditions required for its persistence. is the EEP for system (2). To find the equilibrium values, Let us assume the RHS of the system’s Eq. (2) equal to zero:

18

Then EEP is given by

where , , , and

Local stability of equilibria

In this subsection, we derive the local stability for both Infection free equilibrium and Disease endemic equilibrium by using Routh–Hurwitz stability conditions.

Theorem 4.1

The Infection Free Equilibrium of is locally asymptotically stable if and unstable if . □

Proof

The Jacobian matrix at Infection Free Equilibrium is

19

Then the eigen values are given by,

Hence, the given system is locally asymptotically stable if . □

Theorem 4.2

The endemic equilibrium is locally asymptotically stable provided that 1 and the sufficient and necessary conditions are satisfied.

Proof

To prove local stability of the DEE, we follow the classical approach of linearization via the Jacobian matrix and apply the Routh-Hurwitz criterion: The Endemic equilibrium has Jacobian matrix which is given in as follows:

20

We compute the characteristic polynomial (λ)=det(λI-), which yields: Then the characteristic polynomial is

21

where coefficients of the characteristic polynomial are:

The coefficients of the characteristic polynomial are

The other derived coefficients of the characteristic polynomial are given in the Appendix section Appendix.

For a Six-degree polynomial, the Routh array is constructed as follows:

where:

The endemic equilibrium is locally asymptotically stable if and only if all the following conditions hold:

All coefficients are positive: for .

All elements in the first column of the Routh array are positive:

The Routh–Hurwitz conditions are therefore applied in order to verify that all of the roots present in the characteristic equation have negative real parts. This is done in order to establish the local stability of the endemic equilibrium so that it can be maintained. □

Global stability for equilibria

Lyapunov stability theory is examined the stability of both the IFE and the DEE in this subsection. Based on the Jacobian matrix 19 and 20, it is evident that all eigenvalues possess negative real parts. Therefore, the system is locally asymptotically stable.

Theorem 4.3

The Infection free equilibrium of the Caputo fabrizo-fractional derivatives of system (1) is globally asymptotically stable.

Proof

Let us define Lyapunov function,

From the reference [103], applying the Caputo fabrizo fractional derivative , we have

Substituting the Caputo fabrizo fractional system Eq. (1), we have

At Disease free equilibrium , we have

Since, the above expression is negative for all . By the principle of LaSalle’s invariance, we can conclude that the Disease free equilibrium is globally asymptotically stable. □

Theorem 4.4

The Disease endemic equilibrium of the Caputo fabrizo-fractional system (1) is globally asymptotically stable.

Proof

Consider the Lyapunov function

From the reference [103], applying the caputo fabrizo -fractional derivative , we have

Put the Caputo fabrizo system of equations in above equations, then

22

The above inequality holds that (22), when . Hence, by the principle of LaSelle’s invariance. The Eq. (22) is negative and the system of equation is globally asymptotically stable.

Ulam-Hyers stability

The Ulam-Hyers stability of the system (2) is established in this subsection. We provide all of the required definitions.

Definition 4.1

The system of Eq. (2) has Ulam-Hyers stability if there exist constants where , which satisfying: For every , , if

23

and there exists a solution of the system (2), and which satisfies the provided model, which means

Remark 4.5

Assume that the function satisfies the first inequality in (23) if and only if a continuous function exists (that depends on ) such that

1. , and

2. .

We can also define other classes of the model (23) for some , where .

Theorem 4.6

Assume that the hypothesis of the system holds true. Then the system (2) is Ulam-Hyers stable if

Proof

Let and the function be arbitary such that

According to Remark 4.5, we have a function with , which satisfies

Accordingly, we get

Let be the system (2)’s unique solution. Next,

Hence,

Then,

where

Similarly, we have

Thus, the system (2) is Ulam-Hyers stable. □

Phase plane analysis of equilibrium points

The phase plane analysis in Fig. 2 reveals disease progression dynamics between susceptible population () and total infected individuals. Starting from four distinct initial conditions (IC1 = 0.05, IC2 = 0.25, IC3 = 0.45, IC4 = 0.55), all trajectories converge toward the endemic equilibrium point, demonstrating system stability and disease persistence. The phase space divides into disease-free (40%) and endemic (60%) regions, with population distributed as 60% susceptible, 17.5% acute, and 20% chronic cases. This convergence pattern indicates that regardless of initial infection levels, the system inevitably reaches an endemic state under current parameter settings, highlighting the challenge of disease eradication.

Fig. 2.

Phase plane analysis of disease dynamics

Pie chart and contour plot analysis of

The value of 2.697 places the HCV outbreak firmly in the epidemic phase which is shown in Fig. 3a. This means each infected individual will transmit the virus to approximately 2.7 new people, creating exponential growth in cases. Acute infections dominate transmission (51%), revealing a critical intervention point. The component of 1.374 shows that individuals in the acute phase are the primary drivers of HCV spread, likely due to high viral loads and unawareness of infection status. The treatment group’s significant contribution (23%) indicates ongoing transmission despite therapy, suggesting issues with treatment adherence, delayed initiation, or incomplete viral suppression. Chronic carriers contribute minimally (2%), allowing resources to be prioritized elsewhere. This distribution demands a shift in control strategy toward early detection and acute case management. The system experiences uncontrolled spread where the infection chain continues indefinitely without intervention. This high value indicates that standard control measures will be insufficient and aggressive, multi-pronged strategies are necessary to bring the epidemic under control.

Fig. 3.

Analyse of reproduction number

The contour plot in Fig. 3b mapping against acute transmission rate (β₂) and progression rate (γ) reveals critical control insights. The blue 1 contour line demarcates the epidemic threshold - combinations below this line achieve disease control. The plot shows that reducing acute transmission (β₂) is more effective than modifying progression rates (γ) for controlling spread. Higher progression rates combined with high transmission create the most dangerous scenarios (dark green regions, 3). This visualization provides a strategic roadmap for interventions: focusing on reducing β₂ through prevention measures offers the most direct path to bringing below the critical threshold of 1.

PRCC sensitivity analysis

In this subsection, let us verify the sensitivity of our parameters with respect to .

The sensitivity dashboard in above Fig. 4 quantifies parameter influence on the basic reproduction number () using Partial Rank Correlation Coefficients (PRCC). Parameter β₂ demonstrates the strongest positive correlation (PRCC = 0.92), followed by (0.87), indicating these are primary drivers of disease transmission. Conversely, µ shows strong negative influence (−0.80) and moderate negative impact (−0.41), suggesting these parameters suppress and could be leveraged for control. Moderate positive influences come from β₄ (0.11), (0.11), η (0.07), (0.06), and β₁ (0.06), while most remaining parameters exhibit minimal effects. This analysis identifies β₂ and b as critical targets for reducing transmission, with µ and offering complementary suppression mechanisms for epidemic control.

Fig. 4.

Parameter influence on basic reproduction number ()

Phase portrait analysis of R₀-sensitive parameters

This phase portrait analysis in Fig. 5 examines how variations in -sensitive parameters affect disease dynamics across multiple scenarios. The two most increasing parameters ( and β₂) changes from base values (ranging from 33.13 to 0.00) through systematic reductions, with decreasing and attain the reproduction number when and . The two most decreasing parameters ( and µ) changes from base values (ranging from 10.27 to 0.00) through systematic reductions, with decreasing and attain the reproduction number when and µ = 0.025. The one increasing and one decreasing parameters (µ and β₂) changes from base values (ranging from 0.03 to 0.00) through systematic reductions, with decreasing and attain the reproduction number when µ = 0.0250 and . The three dimensional visualization of Reproduction number for β₂ and µ are shown in the last plot which attain reproduction number.

Fig. 5.

Phase portrait analysis of R₀-sensitive parameters

Contours plot for sensitive parameters

This contour plot analysis in Fig. 6 examines the four most influential parameters on the reproduction number, revealing their sensitivity ranges and directional effects. Parameter β₂ exhibits the strongest positive influence (0.918) with a substantial operational range (0.255 upper/lower bounds), making it a primary transmission driver. Parameter shows strong positive correlation (0.874) but operates within a narrow range (0.030 bounds), indicating precise control requirements. Conversely, µ demonstrates significant negative influence (−0.802) within tight bounds (0.025), suggesting it serves as an effective but finely-tuned suppression mechanism. Parameter provides moderate negative control (−0.413) with wider operational flexibility (0.173 bounds). The contour gradients visualize how reproduction number responds to parameter variations, highlighting that β₂ reduction combined with µ enhancement offers the most effective strategy for epidemic control, while acknowledging the precision needed for b and adjustments.

Fig. 6.

Contour analysis of top parameters influencing reproduction number

Numerical scheme derivation

In this section, we discuss numerical scheme for Fractional Epidemiological Model using Newton Interpolation Method.

Fractional system formulation

Consider a general Cauchy problem with the Caputo-Fabrizio fractional differential operator:

24

where is the fractional order, and is a normalization function with .

Applying the associated fractional integral operator yields:

Discretization

Let , where is the step size, and denote . Evaluating at :

The difference between successive terms gives:

25

Newton interpolation approximation

To approximate the integral in (25), we employ Newton’s forward difference formula. On the interval , the function is approximated by a quadratic Newton polynomial using three points :

Let . The Newton forward polynomial of degree 2 is:

However, for an explicit two-step method, we use linear interpolation over and extrapolate to . The linear Newton polynomial through and is:

26

Using (26) to approximate over , we compute:

27

Evaluating the integral:

Substituting back into (27):

28

Numerical scheme

Substituting (28) into (25) gives:

Simplifying:

29

where

Application to epidemiological HCV model

Consider the six-compartment model:

30

Component-wise numerical scheme

Applying the scheme (29) to each compartment:

where the kernel functions are:

Graphical visualization

During this section, we will be performing numerical simulations on the parameters of our model showed in Table 2. For the purpose of applying the referenced parameters (, µ, , β₂, , η, and δ), their respective ranges from the literature were utilized [98]. For the purpose of ensuring that the model closely matched HCV data from Zimbabwe (1990–2019) with an RMS error of 0.0001, the slider feature of Berkeley Madonna was utilized to locate the parameters that were fitted. which can be obtained from: https://www.globalhep.org/country-progress/zimbabwe [110], using initial conditions as follows  = 0.8,  = 0.01,  = 0.015,  = 0,  = 0.

Table 2.

Model parameters and their interpretations

Definition	Symbol	Value (Range)	Source
Recruitment rate		0.03 (0.009–0.06) year−1	[104, 105]
Natural mortality rate	µ	0.025 (0.01–0.03) year−1	[104, 105]
Disease induced death rate (on treatment)		0.005 (0.001–0.008) year−1	[106]
Disease induced death rate (not on treatment)		0.050 (0.01–0.065) year−1	[98]
Force of infection	λ	0.03452 year−1	Estimated
Progression rate (Exposed to Acute)	γ	0.2 (0.1 - 0.5) year−1	Assumed
Transmission rate (Exposed E)	β1	0.12 (0.08 - 0.15) year−1	Assumed
Disease transmission rate ()	β2	0.255 (0–3) year−1	[107]
Disease transmission rate ()	β3	0.019 (0.01–0.02) year−1	[98]
Disease transmission rate ()	β4	0.08 (0.01–0.1) year−1	[98]
Re-susceptibility rate	ϕ	0.0009 (0.0008–0.0012) year−1	[108]
Acute to recovered rate		0.173 (0.1–0.18) year−1	[98]
Recovery rate in acutely infected cases	η	0.256 (0.1–0.4) year−1	[109]
Treatment quitting rate	δ	0.5 (0.4–0.7) year−1	[104]
Treatment adherence proportion	θ	0.6 (0–1) year−1	[98]
Recovery rate for those with chronic infections	ρ	0.0056 (0.0011–0.009) year−1	[98]
Slow recovery modification parameter	α	0.001 (0.0009–0.0011) year−1	[98]

The dynamics of the susceptible population, showed in Fig. 7, provide critical insight into the decrease rate of healthy individuals. The susceptible population declines most rapidly, reaching its lowest equilibrium swiftly. In stark contrast, as the fractional order decreases, the descent of the curve becomes significantly more gradual and converge after 30 years. This indicates a system crucial factor for long-term epidemic forecasting and the planning of treatment or public awareness campaigns.

Fig. 7.

Susceptible () population dynamics for different fractional order

The behavior of the exposed compartment (), shown in Fig. 8, ( = 1.0) predicts a sharp, high-magnitude peak upto 18 years for lower fractional order and the curve decrease of slowly for higher fractional order, which suggesting a rapid and intense initial outbreak. However, as decreases, the system’s memory dampens this response and converge after 50 years. Therefore, accurately estimating the appropriate fractional order for a specific population is essential for designing effective intervention strategies, whether they aim to mitigate a sharp peak or manage a persistent, endemic state.

Fig. 8.

Exposed () population dynamics for different fractional order

The trajectories of the acute () in Fig. 9 and chronic () populations in Fig. 10 shows the critical impact of fractional order on disease progression. For lower values of , the peak of the acute compartment is peak upto 25 years and decline slowly and converge after 50 years, indicating a slower progression from exposure to symptomatic acute infection. Consequently, the subsequent influx into the chronic compartment () is also slower and more gradual and similarly converge after 50 years. The integer-order model ( = 1.0), by contrast, predicts a sharper, more severe acute outbreak that rapidly seeds a larger chronic population. This has direct implications for healthcare planning, as a lower would predict a less sudden but more sustained demand for diagnostic and management resources for chronic HCV.

Fig. 9.

Acute () population dynamics for different fractional order

Fig. 10.

Chronic () population dynamics for different fractional order

The fractional order critically influences the ultimate success of treatment interventions, as evidenced by the quit treatment () in Fig. 11 and recovered () compartments in Fig. 12. A lower order results in a slower accumulation of individuals who quit treatment () and curve is goes at peak om 30 years, then decline and converge after 50 years, implying a system where the decision to discontinue therapy is influenced by longer-term historical factors, leading to a more protracted but less voluminous trend of treatment failure. Most significantly, the recovery process is drastically altered. The classical model ( = 1.0) shows a rapid rise in recoveries, reaching a high equilibrium quickly and converge at 40 years. As decreases, the path to recovery becomes a much slower, more gradual climb and converge after 50 years for lower values of fractional order, reflecting the cumulative and memory-dependent nature of successful long-term treatment outcomes and immunity. The final recovered population equilibrium is highly sensitive to , indicating that the long-term effectiveness of control programs is intrinsically tied to these system memory properties.

Fig. 11.

Quit treatment (Q_t) population dynamics for different fractional order

Fig. 12.

Recovery (R) population dynamics for different fractional order

The unified response across all model compartments solidifies the role of the fractional order as a master regulator of epidemic tempo and outcome in Fig. 13. Lower values of depict an epidemic characterized by slower transmission, delayed peaks, dampened outbreak intensity, and a prolonged but steady journey toward recovery and control. The Fig. 14 shows the three dimensional visualization of different fractional order effects for HCV dynamics. Our findings strongly advocate for the adoption of fractional calculus frameworks to develop more robust and realistic public health strategies for managing HCV and similar complex infectious diseases.

Fig. 13.

HCV model dynamic of all compartment

Fig. 14.

Three-dimensional visualizations of HCV compartment dynamics illustrating the complex interplay between time (x-axis), fractional order (y-axis: 0.3 to 1.0), and population proportion (z-axis). These surfaces demonstrate how fractional calculus, through its memory kernel, modifies disease progression patterns, epidemic timing, and compartment equilibria across all infection stages. (a) Susceptible population dynamics for fractional order is showing decrease of healthy individuals. Lower values maintain higher susceptibility decrease due to memory effects preserving infection risk. (b) Exposed population evolution with varying fractional orders. Lower produces higher and broader peaks, representing prolonged treatment periods and heterogeneous disease progression. (c) Acute infection dynamics showing delayed and flattened epidemic peaks at lower fractional orders. The continuous surface demonstrates transition from ordinary () to fractional derivatives. (d) Chronic infection patterns showing relative stability across fractional orders. Chronic prevalence converges to similar long-term values regardless of , indicating it represents a system-level property. (e) Treatment discontinuation dynamics with minimal variation across fractional orders. This suggests treatment adherence behaviors are less influenced by memory effects compared to biological processes. (f) Recovery accumulation showing slower progression at lower fractional orders. Reduced values produce more gradual recovery curves, representing delayed treatment responses and heterogeneous healing rates

Physics-informed neural networks: bridging machine learning and mechanistic modeling

Conceptual foundation and theoretical motivation

The paradigm shift in computational epidemiology

Traditional approaches to epidemiological modeling face fundamental limitations:

• Numerical Methods: Finite difference/element methods require dense discretization and struggle with inverse problems

• Data-Driven ML: Pure neural networks lack physical interpretability and require massive datasets

• Parameter Estimation: Conventional methods often decouple state estimation from parameter identification

Physics-Informed Neural Networks (PINNs) represent a transformative approach that seamlessly integrates mechanistic knowledge with data-driven learning. For HCV modeling, this enables:

31

Mathematical formulation of the HCV transmission problem

The model defines a dynamical system:

32

33

where represents population compartments and denotes epidemiological parameters.

PINN methodology

The flowchart in Fig. 15 illustrates the iterative training process of Physics-Informed Neural Networks (PINNs) for epidemiological modeling. The methodology begins with data collection and network architecture design which contains hidden layer, followed by the computation of both physics loss (incorporating disease dynamics equations) and data loss (matching observed data). These components combine in total loss calculation, driving backpropagation and optimization to update network parameters. The process cycles through convergence checks, where unmet criteria trigger architecture updates and parameter recalibration, while convergence enables model validation. This integrated approach ensures neural network predictions respect both observed data patterns and fundamental epidemiological principles, providing a robust framework for parameter estimation and disease forecasting with enhanced physical consistency compared to purely data-driven methods.

Fig. 15.

Physics-Informed neural network (PINN) methodology flowchart

Enhanced training setup and hyperparameter configuration

The PINN Training Hyperparameters, its values and rationale are mentioned in the Table 3.

Table 3.

PINN training hyperparameters

Parameter	Value	Rationale
Hidden Layer 1	20 Neurons	Balances capacity and reduces overfitting
Activation Function	Hyperbolic Tangent	Smooth derivatives for ODE compliance
Initial Learning Rate	0.05	Fast convergence without oscillation
Learning Rate Schedule	Cosine Annealing	Gradual refinement in later training
Momentum	0.9	Accelerates convergence in flat regions
Batch Size	Full batch	Utilizes all collocation points each iteration
Training Epochs	100	Ensures convergence with early stopping
Physics Loss Weight	1.0	Primary constraint from governing equations
Data Loss Weight	0.1	Secondary constraint from synthetic data

Training Data Generation:

• Collocation Points: We generated equally spaced collocation points for physics loss evaluation

• Training Data: Synthetic data from numerical ODE solutions at 100 time points served as training targets

• Validation Split: 80–20 temporal stratified split ensuring representation across all disease phases

Hyperparameter Configuration:

Neural network architecture as function approximator

Universal approximation theorem foundation

The theoretical basis stems from the universal approximation theorem:

Theorem 7.1

[111] Universal Approximation] For any continuous function and ϵ > 0, there exists a neural network such that:

34

for any compact set .

This guarantees that our neural network can approximate the true solution to arbitrary precision.

Network architecture specification

We employ a feedforward architecture with hyperbolic tangent activations:

35

36

37

Architecture Rationale:

• Single Input: Time t enables learning temporal evolution

• 20 Hidden Units: Balances model capacity with computational efficiency

• Hyperbolic Tangent: Provides smooth derivatives essential for ODE compliance

• Linear Output: Preserves interpretability of population proportions

The diagram Fig. 16 presents the neural network architecture designed for Physics-Informed Neural Network (PINN) modeling of Hepatitis C Virus (HCV) dynamics. The network follows a 1–20-6 configuration, accepting time (t) as the single input variable and producing six output states corresponding to HCV epidemiological compartments. The hidden layers 20 neurons respectively enable nonlinear transformation and feature extraction, capturing complex temporal patterns in disease progression. This architecture balances model complexity with computational efficiency, allowing the PINN to learn both data-driven patterns and physics-constrained relationships between input time and output states. The structured design facilitates simultaneous prediction of multiple disease variables while maintaining interpretability and adherence to biological constraints essential for epidemiological applications.

Fig. 16.

Neural network architecture for HCV dynamics modeling

Parameter initialization strategy

We employ He initialization for stable training:

38

39

This initialization preserves activation variances through forward propagation, preventing vanishing/exploding gradients.

Residual minimization framework

Compartment-weighted objective function

To account for differences in scale, clinical importance, and data quality across compartments, we define a weighted composite loss function:

40

where:

• : Loss for compartment m, typically the mean squared error:

  
• ω_m: Compartment-specific weight, determined by:

  

  where is the variance of the observed data for compartment m, and I_m is an importance factor (e.g., to prioritize fitting the acute compartment).

• : Regularization term constraining parameters to biologically plausible ranges :

  

• λ: Regularization strength hyperparameter.

Two-phase estimation strategy

1. Phase 1: Sensitivity-Guided Parameter Selection

   A global sensitivity analysis (Morris method) identifies parameters with high influence on model outputs. Parameters with low total-effect indices are fixed at literature values, while high-sensitivity parameters proceed to estimation.

2. Phase 2: Constrained Optimization

   The reduced parameter set is estimated by solving:

   
   where is the feasible space for estimated parameters, and are literature-based fixed parameters. Optimization uses a hybrid approach:

   • Global search via Differential Evolution to avoid local minima.
   • Local refinement via L-BFGS-B with gradient information.

PINN-Integrated parameter estimation

When using the Physics-Informed Neural Network, the epidemiological parameters θ become trainable variables within the neural network framework. The loss function becomes:

41

where . The parameters θ and neural network weights θ_NN are updated simultaneously via backpropagation, enabling joint state and parameter inference in a single optimization loop.

Physics-Informed loss function: the core innovation

Multi-faceted regularization framework

Our overfitting prevention strategy employs four complementary mechanisms:

1. Physics-Informed Regularization: Intrinsic regularization through governing equations

2. Architectural Constraints: Network architecture design to limit model capacity

3. Training Protocol Optimization: Dynamic learning strategies during training

4. Validation-Based Control: Continuous monitoring and intervention

Multi-objective optimization framework

The PINN loss function represents a constrained optimization problem:

42

Physics-Informed loss component

This term ensures the neural network satisfies the HCV transmission dynamics:

43

Mathematical Interpretation:

• : Neural network’s temporal derivatives (computed via automatic differentiation or finite differences)

• : Right-hand side of ODE system

• N_c: Number of collocation points in temporal domain

• This term measures how well the network satisfies the governing equations

Data fidelity loss component

This term ensures agreement with available observational data:

44

Epidemiological Significance:

• : Surveillance data, clinical records, or synthetic observations

• Enables calibration to real-world scenarios

• Provides regularization against unphysical solutions

Loss weighting strategy

We employ ν = 0.1 to balance physical consistency with data fitting:

45

This heuristic weighting acknowledges that physical principles provide stronger constraints than potentially noisy data.

Comprehensive overfitting prevention strategy

Advanced training protocols for overfitting prevention

Early Stopping with Patience: We implement early stopping based on validation loss:

Learning Rate Scheduling: We employ cosine annealing with warm restarts:

46

This prevents the optimizer from getting stuck in sharp minima that may indicate overfitting.

Gradient Clipping: We implement gradient clipping to prevent exploding gradients:

47

with clipping threshold τ = 1.0.

Data-specific overfitting controls

Synthetic Data Augmentation: For data loss component, we employ:

• Additive Noise: Gaussian noise added to training data

• Temporal Jittering: Small random shifts in temporal sampling

• Dropout in Data Loss: Randomly omit data points during training

Weighted Data Loss: We use a reduced weighting for data loss to prevent over-reliance on potentially noisy data:

48

This prioritizes physical consistency over exact data fitting.

Validation and monitoring framework

Multi-Metric Validation: We monitor multiple metrics to detect overfitting:

• Physics Residual Norm: Should decrease monotonically

• Data Loss: Should stabilize without sudden drops

• Gradient Norm: Should remain bounded

• Parameter Norm: Should not grow excessively

Temporal Cross-Validation: We implement temporal cross-validation by:

1. Training on

2. Validating on

3. Testing on

Theoretical justification

Bias-Variance Tradeoff in PINNs: The physics-informed approach inherently reduces variance by restricting hypothesis space:

49

Generalization Bound: For our PINN architecture, we can bound generalization error:

50

The regularization techniques ensure the second term remains small.

Numerical implementation and computational aspects

Derivative computation methodology

Finite Difference Scheme: We implemented a hybrid finite difference approach for derivative computation:

• Central Difference (interior points):

  51
• Forward Difference (first point):

  52
• Backward Difference (last point):

  53

where is the temporal spacing between collocation points.

Automatic Differentiation Comparison: While finite differences provided sufficient accuracy for our application, we verified consistency with PyTorch’s automatic differentiation, achieving comparable results (mean absolute error < 10⁻⁴).

Optimization and training dynamics

Gradient Descent with Momentum: We employed gradient descent with Nesterov momentum for stable convergence:

54

55

where µ = 0.9 is the momentum coefficient and η is the adaptive learning rate.

Learning Rate Schedule: We implemented cosine annealing for adaptive learning rate adjustment:

56

where , , and K = 300 total epochs.

Early Stopping Criterion: Training terminated when validation loss failed to improve by at least 10⁻⁶ for 25 consecutive epochs, preventing overfitting.

Optimization algorithm

We employ gradient descent with learning rate decay:

Computational complexity analysis

The training complexity scales as:

57

where:

• K: Number of training iterations

• N_c: Collocation points

• P_forward: Forward pass complexity

• P_gradient: Gradient computation complexity

Theoretical advantages and error analysis

Error decomposition framework

The total approximation error can be decomposed:

58

Convergence guarantees

Under certain regularity conditions, PINNs exhibit:

Theorem 7.2

PINN Convergence For sufficiently large neural networks and adequate training, the PINN solution converges to the true solution u:

59

Software and data source

The numerical simulations and sensitivity analyses in this study were implemented in Python, utilizing its scientific computing libraries (NumPy, SciPy). The model was parameterized using a combination of biologically realistic values and ranges obtained from published epidemiological literature on HCV [98] which is taken from the supplemented with fitted parameters calibrated to match Zimbabwean HCV data from 1990 to 2019, ensuring the model reflects real-world transmission dynamics. The Physics-Informed Neural Network (PINN) was similarly developed and trained in Python, using the model’s own governing fractional differential equations as the primary data source for generating the physics-informed loss.

Training and testing metrics

The training and testing loss curves in Fig. 17 demonstrate excellent convergence behavior, with both losses decreasing steadily and stabilizing at low values (10⁻² range). This indicates the Physics-Informed Neural Network effectively learned the HCV transmission dynamics without overfitting. The convergence pattern validates that the PINN successfully captured the underlying all compartment ODE system while maintaining physical plausibility. The low final loss values provide confidence in the model’s predictions for epidemiological analysis and public health decision-making.

Fig. 17.

Training and testing loss of PINN

Distribution analyse and sensitivity analysis of reproduction number

The compartment distribution in Fig. 18a reveals the stable endemic state of HCV in the population after 100 time units. Susceptible individuals (42.2%) remain the largest group, but significantly reduced from the initial 80%, indicating substantial infection penetration. The recovered population (15.6%) has built up substantial immunity, while chronic (13.7%) and Quit treatment (8.3%) groups represent the long-term disease burden. The acute cases (4.6%) and exposed individuals (8.1%) maintain active transmission chains. This distribution confirms the persistent nature of HCV infection with all compartments maintaining non-zero populations, consistent with the 1 stability analysis.

Fig. 18.

Analyse of distribution and contour plot of reproduction number. (a) Distribution analyse of all compartment. (b) Parameter sensitivity of reproduction number

The Fig. 18b shows the sensitivity of the reproduction number and sensitivity of performs in the parameters which are β₁, β₂, β₃, β₄, γ, , η, δ and θ.Acute transmission rate (β₂) emerges as the most powerful leverage point for control. A 1% reduction in acute transmission decreases by 0.51%, making it the most efficient intervention target. Simultaneously, improving acute recovery rate () provides the strongest protective effect, with each 1% increase reducing by 0.41%. The sensitivity analysis reveals that resources focused on acute phase interventions yield maximum impact, while chronic phase parameters show minimal returns. This creates a clear priority hierarchy for resource allocation in control programs.

Sensitivity analysis through Morris method

In this Morris sensitivity analysis method, we apply the sensitivity for the parameters β₁, β₂, β₃, β₄, γ, , η, δ and θ through PINN. The Morris sensitivity analysis in Figure 19 identifies β₂ (acute transmission rate) as the most influential parameter with the highest mean elementary effect (µ ≈ 1.6), indicating it exerts the strongest direct impact on increasing , conforming the dominant factor in HCV spread. Parameters β₁ (exposed transmission) and β₃ (chronic transmission) show moderate positive µ values which has moderate sensitivity, acting as transmission amplifiers. Treatment-related parameters (β₄, θ, δ) demonstrate intermediately lower sensitivity, while recovery rates (, η, γ) exhibit lower but meaningful influence as transmission suppressors. This analysis confirms that acute phase transmission control should be the primary intervention focus, with treatment optimization as a secondary strategy for effective epidemic management.

Fig. 19.

Analysis of sensitivity via Morris method

Time series error analysis

In this subsection, we analyse time series error analysis with synthetic real data and model prediction for the fractional order .

This model’s behavior is exceptionally accurate for the Susceptible compartment, as confirmed by the goodness-of-fit plot in Fig. 20a. The near-perfect R² score of 0.995 and very low RMSE of 0.010 show an almost exact match between the model’s prediction and the synthetic data. This means the fractional-order model reliably captures the real-world dynamics of how the susceptible population decreases over time.

Fig. 20.

Time series analyse of susceptible compartment and Exposed compartment. (a) Model prediction and synthetic real data plot of susceptible compartment. Model prediction and synthetic real data plot of Exposed compartment

The goodness-of-fit plot confirms the model in Fig. 20b is highly accurate for this compartment. The strong R² value of 0.978 and a low RMSE of 0.010 demonstrate that the model’s predictions for the exposed population curve closely match the observed synthetic data, reliably capturing the dynamics of how individuals become infected but are not yet infectious.

The model demonstrates excellent predictive power for acute infection dynamics in Figure 21a, as confirmed by the goodness-of-fit plot. The high R² value of 0.967 and a very low RMSE of 0.008 show a near-perfect alignment between the model’s projection and the synthetic data, validating its accuracy in simulating the rapid rise and fall characteristic of the acute infection phase.

Fig. 21.

Time series analyse of acute compartment and chronic compartment

However, the model’s performance for this chronic compartment in Fig. 21b is notably weaker, as shown by the goodness-of-fit metrics. The R² value of 0.867 suggests only a moderate correlation between the model’s predictions and the synthetic data, while the RMSE of 0.011 indicates a less precise fit. This implies that while the fractional-order model captures the general trend, it is less reliable for accurately predicting the specific long-term dynamics of the chronic population.

The model demonstrates strong predictive accuracy for quit treatment dynamics in Figure 22a, as confirmed by the goodness-of-fit plot. The high R² value of 0.990 and low RMSE of 0.007 show excellent agreement between the model’s predictions and the synthetic data, validating its reliability in simulating the rapid surge and decline characteristic of quit treatment needs during an outbreak.

Fig. 22.

Time series analyse of Quit treatment compartment and recovered compartment

The model demonstrates good predictive accuracy for recovery dynamics in Fig. 22b, as evidenced by the goodness-of-fit metrics. The strong R² value of 0.974 and low RMSE of 0.010 indicate a reliable alignment between the model’s predictions and the synthetic data, confirming its effectiveness in capturing the cumulative nature of the recovery process over time.

Goodness-of-fit analysis

Model performance metrics

The goodness-of-fit metrics demonstrate excellent PINN performance across all compartments which shown in Table 4.

Table 4.

Goodness-of-fit metrics for PINN predictions

Compartment	MSE	RMSE		MAE
Susceptible	0.0001	0.0095	0.9954	0.0078
Exposed	0.0001	0.0098	0.9777	0.0078
Acute	0.0001	0.0080	0.9674	0.0068
Chronic	0.0001	0.0108	0.8673	0.0083
Quit Treatment	0.0000	0.0067	0.9904	0.0055
Recovered	0.0001	0.0100	0.9743	0.0070

Most compartments show , indicating the PINN explains over 97% of variance. Susceptible (0.9954) and Quit Treatment (0.9904) show near-perfect fits, while Chronic has the lowest but still strong (0.8673). RMSE values range from 0.0067 to 0.0108, representing minimal prediction errors relative to compartment scales. MAE values show similar consistency (0.0055–0.0083), indicating balanced error distribution.Susceptible shows best overall fit (R² = 0.9954, RMSE = 0.0095). Quit Treatment has lowest errors (RMSE = 0.0067, MAE = 0.0055). Chronic exhibits relatively lower fit ( = 0.8673) but still acceptable performance. The high goodness-of-fit indicates PINN reliably captures HCV dynamics, making it suitable for epidemiological predictions. The consistent low errors across compartments suggest robust generalization capabilities. The slightly lower chronic fit may reflect greater biological complexity or data limitations in this compartment. PINN demonstrates excellent predictive accuracy with average R² of 0.9621 and low error metrics, validating its effectiveness for HCV modeling applications.

Comparative analysis: ODE, fractional derivatives, and PINN approaches

Methodological comparison framework

This study compares three mathematical approaches for modeling Hepatitis C Virus (HCV) dynamics: Ordinary Differential Equations (ODEs) as the baseline, Fractional Differential Equations (FDEs) with orders to capture memory effects, and Physics-Informed Neural Networks (PINNs) combining data-driven learning with physical constraints. The comparison evaluates their performance in predicting disease progression, treatment outcomes, and long-term equilibria.

Compartmental dynamics analysis

Susceptible and Exposed populations

The susceptible population in Fig. 23a shows distinct depletion patterns across methodologies. ODE predicts rapid depletion, while fractional models () show progressively slower depletion with decreasing , indicating memory effects preserve susceptibility longer. PINN follows an intermediate path but maintains higher long-term susceptibility. Faster depletion in ODE models may underestimate long-term transmission risk, while fractional models suggest need for sustained interventions due to persistent susceptible pools.

Fig. 23.

Dynamics of susceptible and exposed populations

Fractional models predict significantly higher and more sustained exposed populations in Fig. 23b, with showing approximately double the ODE values. This represents prolonged incubation periods and heterogeneous progression rates captured by fractional calculus. PINN closely tracks ODE with smoother transitions. Higher exposed populations in fractional models indicate greater undetected transmission potential, emphasizing need for enhanced surveillance and early detection programs.

Acute and chronic infection dynamics

Acute infections in Fig. 24a show broader, delayed peaks in fractional models compared to ODE’s sharp peak. Lower values produce flatter distributions, representing heterogeneous progression rates. PINN captures the general shape with smoothed transitions. Broader peaks imply extended transmission periods requiring longer-duration prevention efforts, while delayed peaks suggest surveillance systems need sustained alertness.

Fig. 24.

Acute and chronic infection dynamics

Chronic infection in Fig. 24b shows remarkable consistency across all methodologies, converging to similar long-term values despite initial variations. This indicates chronic prevalence represents a stable system property less sensitive to modeling details. Consistent chronic predictions provide reliable burden estimates for healthcare planning, though initial differences affect intervention timing decisions.

Quit treatment and recovery outcomes

Treatment discontinuation in Fig. 25a shows consistent patterns across fractional models but PINN predicts significantly lower quit rates. This suggests PINN learns more optimistic adherence patterns from physical constraints. If fractional models are accurate, they indicate persistent adherence challenges requiring stronger retention strategies. PINN’s optimistic prediction suggests potential for better outcomes through optimized programs.

Fig. 25.

Quit treatment adherence and recovery outcomes

Recovery compartment in Fig. 25b accumulates slower in fractional models, particularly at lower , representing delayed treatment responses and heterogeneous recovery rates. PINN follows ODE closely with reduced oscillation. Slower recovery implies longer timelines to achieve elimination targets and maintains transmission risk during delayed recovery periods.

Quantitative performance assessment

Final state distribution analysis

The Table 5 reveals systematic patterns:

Table 5.

Final state distribution across methodologies

Compartment	ODE				PINN
Susceptible	0.3836	0.2835	0.3105	0.3363	0.4268
Exposed	0.0798	0.1501	0.1245	0.1056	0.0935
Acute	0.0849	0.1567	0.1313	0.1119	0.1043
Chronic	0.0496	0.0553	0.0541	0.0526	0.0695
Quit Treatment	0.1672	0.1693	0.1722	0.1721	0.1217
Recovered	0.1689	0.1576	0.1641	0.1675	0.1629

1. Fractional models show lower susceptible but higher exposed/acute populations than ODE,

2. Chronic infection is relatively stable across methods,

3. PINN predicts highest susceptibility and lowest treatment dropout.

Exposed and acute compartments show the strongest fractional effects, with predicting approximately double the ODE values, indicating these transitional states are most sensitive to memory effects.

Error analysis and model accuracy

The three key observations of Table 6:

• Fractional models show systematic error reduction as , validating mathematical consistency,

• Exposed and acute compartments have highest errors, confirming sensitivity to memory effects,

• PINN has highest average error (0.0932) but performs competitively on exposed/acute compartments.

Table 6.

RMSE comparison with ODE baseline

Compartment				PINN
Susceptible	0.0683	0.0514	0.0342	0.1878
Exposed	0.0962	0.0644	0.0387	0.0632
Acute	0.0741	0.0520	0.0324	0.0554
Chronic	0.0191	0.0134	0.0084	0.0478
Quit Treatment	0.0358	0.0240	0.0144	0.1160
Recovered	0.0281	0.0184	0.0108	0.0899
Average	0.0537	0.0373	0.0233	0.0932

Chronic compartment shows lowest errors across all methods, indicating it’s least sensitive to modeling approach. Fractional models with offer the best balance of biological realism and mathematical accuracy for HCV modeling, while PINNs represent a promising emerging technology requiring further refinement for epidemiological applications.

Discussion on poor chronic compartment performance

The model demonstrates consistently high predictive accuracy ( 0.97) for the Susceptible, Exposed, Acute, Quit Treatment compartments, and Recovered compartment except the chronic compartment, validating the chosen fractional framework for capturing rapid transmission and intervention dynamics. However, the moderate fit for the Chronic compartment (R² = 0.8673) warrants specific analysis, as it likely stems from three interrelated factors:

1. Data and Biological Complexity: HCV chronic infection is characterized by high individual variability in progression rates, viral persistence mechanisms, and treatment response timelines—factors that are homogenized in a deterministic, compartmental model. The synthetic or population-level data used for validation may not fully capture this latent heterogeneity, leading to a structural limitation in fitting the averaged chronic trajectory precisely.

2. Model Structural Limitation: The chronic compartment () acts as an accumulation basin with slower dynamics, influenced by multiple upstream transitions (from Acute via ) and downstream outflows (to Recovery ρ and Quit Treatment δ(1-θ)). Small compounded errors or simplifications in modeling these transition rates—particularly the simplified treatment failure and adherence process—can propagate into larger discrepancies in the chronic population over time, a challenge less pronounced in faster-cycling compartments.

3. PINN Training Dynamics: During Physics-Informed Neural Network training, the loss function naturally prioritizes fitting compartments with sharper temporal gradients (like the acute peak) to minimize the physics loss. The chronic compartment’s gradual, monotonic evolution presents a flatter error landscape, making it less dominant during gradient updates. This implicit bias toward fitting rapidly changing dynamics is a known characteristic of PINNs and may explain the relatively higher residual error in the chronic stage.

This discrepancy does not invalidate the model but rather highlights a critical boundary of the current framework. It underscores the need for future extensions that incorporate within-compartment heterogeneity or stochastic progression rates to better capture the long-term, variable nature of chronic HCV infection.

Conclusion

This study presents a comprehensive mathematical framework for analyzing Hepatitis C Virus (HCV) transmission by integrating a six-compartment fractional-order model with a Physics-Informed Neural Network (PINN) solver. Using the Caputo–Fabrizio derivative with an exponential kernel, the model effectively captures the non-Markovian memory effects inherent in HCV infection dynamics, including viral kinetics, immune response, and Direct-Acting Antiviral therapy adherence.

Our analysis established the mathematical well-posedness of the fractional-order system through positivity, boundedness, existence-uniqueness proofs using fixed-point theory, and Ulam–Hyers stability. The basic reproduction number was derived, confirming that HCV remains in an epidemic phase under current parameterization, with acute transmission (β₂) identified as the dominant driver, contributing over 50% of new infections. Sensitivity analyses via Partial Rank Correlation Coefficients (PRCC) and the Morris method consistently highlighted β₂ as the most influential parameter, emphasizing that interventions targeting the acute infection stage would yield the highest impact on reducing transmission.

Fractional-order simulations revealed that lower fractional orders (e.g., ), which represent stronger memory effects, significantly delay and dampen epidemic peaks across all compartments, particularly in the acute and exposed populations. This suggests that public health strategies that establish long-term behavioral or immunological “memory”—such as sustained education campaigns, stable treatment access, and repeat screening programs—can effectively modulate outbreak intensity and timing.

The PINN implementation demonstrated high predictive accuracy for most compartments (, low RMSE), validating its capability as a mesh-free, physics-constrained solver. However, the model exhibited moderate performance for the chronic compartment (), likely due to the inherent biological heterogeneity and slower dynamics of chronic HCV progression, which are less effectively captured by deterministic compartmental structures.

Limitations and future directions

Despite these advances, several limitations must be acknowledged. First, the model relies on synthetic and literature-derived parameters; future work should incorporate real-world clinical and epidemiological data from Zimbabwe or similar settings for validation and calibration. Second, the deterministic, homogeneous compartmental structure does not account for individual-level variability in viral load, immune response, or adherence behavior—factors that could be addressed through stochastic or agent-based extensions. Third, while the Caputo–Fabrizio kernel is suitable for rapidly saturating memory, alternative operators (e.g., Atangana–Baleanu with Mittag-Leffler kernel) could be explored to capture different types of memory decay. Finally, the current framework does not include spatial heterogeneity, age structure, or network-based contact patterns, which are relevant for designing targeted interventions.

Future research should focus on: (i) integrating multi-source clinical data to refine parameter estimation and validate compartment trajectories; (ii) extending the model to incorporate within-host dynamics and treatment pharmacokinetics; (iii) developing adaptive PINN architectures that better handle slower, cumulative compartments like the chronic stage; and (iv) embedding cost-effectiveness analysis to evaluate intervention strategies under resource constraints.

Public health implications

The findings offer two actionable policy insights. First, the predominance of acute-phase transmission (β₂) underscores the need for enhanced early detection and rapid treatment initiation in high-risk populations to interrupt transmission chains. Second, the memory-modulated dynamics suggest that sustained, long-term public health investments—rather than short-term campaigns—are crucial for achieving durable epidemic control. Together, these results advocate for a dual-strategy approach: prioritizing acute-phase interventions while maintaining consistent, system-wide efforts to build lasting population-level immunity and awareness.

In summary, this hybrid fractional-order PINN framework bridges mechanistic modeling with machine learning, providing a robust, interpretable tool for understanding HCV dynamics and optimizing elimination strategies. By capturing the temporal memory of biological and behavioral processes, the model offers a more realistic representation of epidemic progression, ultimately supporting evidence-based decision-making for HCV control in resource-limited settings.

Abbreviation

HBV

Hepatitis B Virus

HCV

Hepatitis C Virus

IFE

Infection Free Equilibrium

EEP

Endemic Equilibrium Point.

PINN

Physics Informed Neural Network

RMSE:

Root Mean Square Error

MSE

Mean Square Error

MAE

Mean Absolute Error

IC

Initial Condition

WHO

World Health Organization

DAA therapy

Direct-Acting Antiviral Therapy

CF

Caputo-Fabrizio

PRCC

Partial Rank Correlation Coefficient

ML

Machine Learning

Appendix

Definition

Definition 12.1

[112, 113] Let and , . The equality defines the derivative of order of the function ϒ:

Definition 12.2

[114] The fractional integral is related to the new fractional integral derivative with a power-law kernel is defined as follows:

Coefficient of Routh Herwitz criteria

Author contributions

Vetrivel Muthupandi: Writing – original draft, Validation, Methodology, Conceptualization, Resources. Arul Joseph Gnanaprakasam: Writing – review & editing, Validation, Methodology, Supervision, Conceptualization, Investigation, Resources, Visualization. Salah Boulaaras: Writing – review & editing, Validation, Methodology, Supervision, Conceptualization, Investigation, Resources, Visualization.

Data availability

All data supporting the findings of this study are publicly available in the Coalition for Global Hepatitis Elimination repository at https://www.globalhep.org/data-profiles/countries/zimbabwe and published article [98] as cited in the manuscript.

Declarations

Ethics approval and consent to participant

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.