Nonlinear fractional stochastic delay modeling and computational analysis of herpes simplex virus type II dynamics

Abstract

Herpes simplex virus (HSV) is a widespread infection responsible for painful blisters and ulcers. According to the World Health Organization, approximately 519.5 million people aged 15–49 years (13.3%) worldwide are infected with herpes simplex virus type II (HSV-II), the primary cause of genital herpes. In this study, we develop a nonlinear stochastic fractional delay differential equation (SFDDE) model to describe the transmission dynamics of HSV-II in a human population. The population is divided into susceptible , exposed , asymptomatic , symptomatic , HSV-infected , and recovered compartments. The model’s fundamental properties, including existence, uniqueness, positivity, and boundedness of solutions, are established. Local and global stability analyses are conducted around the HSV-free and HSV-present equilibrium points, and the basic reproduction number is derived using the next-generation matrix method along with sensitivity analysis. Numerical simulations based on a stochastic nonstandard finite difference (NSFD) scheme confirm the theoretical results and demonstrate the stability of the proposed model. These findings highlight the importance of nonlinear fractional stochastic modeling in understanding and controlling HSV-II transmission dynamics.

Introduction

In¹, the authors provided the analysis for transmission of Herpes Simplex Virus II with deterministic model of nonlinear differential equations.

In², the authors analyzed different types of viruses like HSV 1 and HSV 2 with using the world health organization (WHO) reported cases. In³, the authors demonstrated the Aids and HSV-2 infection and also analyses the dynamics and relation between co-infection. In⁴, the authors stated different numerical schemes for the control of HSV disease in human population and make more realistic strategies for their dynamics. In⁵, the authors exhibited the advanced mathematical model construction for the Herpes disease which vastly spread in humans. In⁶, the authors addressed that from 2000 to 2020 five priority areas from a WHO workshop were identified. Data on the author’s country, gender, authorship position, and funding source were gathered through manuscript reviews and online searches. The data was analyzed using IBM SPSS V.26. In⁷, the authors analyzed HSV-1 and HSV-2 infections to find differences based on gender at birth, age, and site of infection. Investigated the influence of comorbidities and pregnancy on infection type. Performed detailed analysis of data to reveal these differences. In⁸, the authors provided a systematic search of MEDLINE, CINAHL, Global Health, and Cochrane databases (2000–2020) was conducted to identify the relevant English-language studies coming from LMICs. Two researchers independently screened and extracted predefined variables, organizing data into Excel. Analysis was carried out using IBM SPSS V.26. In⁹, The authors evaluated immunocompetent adults who were diagnosed with meningitis at a tertiary care hospital in Korea. The study covered the period between 2016 and 2018. Data analysis was done on clinical patterns and outcomes. In¹⁰, the authors carried out a retrospective analysis on 21,210, 49,494, and 32,457 outpatients and inpatients between 1 day to 17 years old. Patients had been subjected to nucleic acid testing for HSV-2, EBV, and CMV. Study duration from January 2018 to December 2023. In¹¹, the authors assessed Cebus apella (C. apella), a New World primate, using a genital infection model for HSV-2. All four animals’ vaginal swabs were used to cultivate HSV-2 for nine to fourteen days after intravaginal inoculation. All the monkeys were initially seronegative for HSV-2. In¹², the authors collected a total of 145 individuals with quantified genital herpes severity, and their HSV genital shedding rate was measured. In¹³, the authors examined the evidence of disease transition within individuals and make control strategies for the HSV-II. In¹⁴, the authors provided the mathematical model with vaccinated parameter for the control of HSV-II. In¹⁵, the authors found HSV-2 UL24 is very potent in suppressing the activation of the IFN-β promoter. It results in drastically decreased levels of IFN-β at both mRNA and protein levels. Results demonstrate the inhibitory function of UL24 in the IFN-β signaling pathway. In¹⁶, the authors founded Chemical and physical stress intermittently reactivate latent viruses in the dorsal root ganglia (DRG). This results in recurrent viral shedding in the genital mucosal epithelium. As a consequence, symptomatic patients suffer episodes of genital herpes. In¹⁷, the authors adapted a dynamic HIV transmission model for South Africa, incorporating HSV-2 to model its synergistic impact on HIV. Assessed the implications of (i) cohort vaccination of 9-year-olds with a prophylactic vaccine to reduce susceptibility to HSV-2, and (ii) therapeutic vaccination of symptomatically HSV-2 infected subjects to reduce HSV shedding. In¹⁸, the authors addressed an optimal control problem applied to a coinfection model of HIV and HSV-2. The model is set up by formulating a system of ordinary differential equations. It delves into the management and control dynamics of this coinfection. In¹⁹, the authors utilized probability trees to model the natural history of genital herpes (GH) caused by HSV-1 and HSV-2 infections among those aged 18–49 years. Insights into infection dynamics and disease outcomes were thereby offered. In²⁰, the authors adhered to the systematic review of the Cochrane and PRISMA guidelines, which considered all publications from both the Chinese and English bibliographic systems. All sources that were considered were published up to March 18th, 2023. In²⁴, the authors studied a mathematical model of tumor development and therapy utilizing virotherapy. In^28,29, the authors conducted numerical studies on a fractional-order nonlinear SIR–SI model for dengue fever epidemics and a fuzzy fractional human liver model using a novel double-parametric approach.

The first-ever HSV-II model has been developed in this study which takes into account fractional memory effects, stochastic perturbations, and explicit time delays all at the same time. The model is the result of rigorous analytical results and a stochastic NSFD scheme that preserves the structure of the original model. The new insight provided by this integrated framework to long-term and uncertain HSV-II dynamics is more than that of the existing models. By integrating temporal delays, fractional calculus, and stochastic processes, the stochastic delayed methodology helps scientists create more accurate models of complicated systems.

This paper is organized as follows. Section 1 presents a focused review of the literature on Herpes Simplex Virus–II, including its epidemiology and underlying biological mechanisms. Section 2 describes the formulation of the proposed model and provides analytical results, including existence, uniqueness, positivity, boundedness, the basic reproduction number, local and global stability, and sensitivity analysis. Section 3 introduces the stochastic nonstandard finite difference (NSFD) numerical scheme and discusses its qualitative properties. Section 4 presents graphical simulations illustrating the dynamical behavior of the model along with their biological interpretation and real-world significance. Finally, Sect. 5 summarizes the main findings, highlights the contributions of the study, and outlines potential directions for future research.

Model formulation

This paper presents the dynamical system of ordinary differential equations in formulating the dynamics of the human population in the presence of Human Immunodeficiency Virus (HIV) treatment. The present study considered a model that comprises six compartments of the human population like susceptible individuals , exposed individuals , asymptomatic individuals , symptomatic individuals , Herpes Simplex Virus-II Individuals and recovered individuals , respectively.

.

This recruit susceptible into the population at a rate . Individuals in the susceptible sub-compartment move to the exposed sub-compartment at a per capita rate of becoming infectious (we recall that is approximately the length of the latent period). Exposed individuals progress to the symptomatic sub-compartment with probability , and to the asymptomatic sub-compartment with probability . Asymptomatic individuals are usually considered to be infectious, though at a lower rate .

The susceptible individuals are acquiring HSV-II infection with the force of infection where, is the contact rate and is the transmission coefficient for the asymptomatic individuals. Here, denotes the relative transmission rate of asymptomatic individuals compared to symptomatic individuals, accounting for reduced or enhanced infectiousness of asymptomatic cases. If then, the asymptomatic infect susceptible is more likely than infective. If , then asymptomatic and infective have an equal chance to infect the susceptible but if then, the infective has a good chance to infect the susceptible than the asymptomatic. Some of the asymptomatic and symptomatic individuals progress to Herpes simplex virus-II at a rate respectively and others recover naturally through the body’s immune system at a rate respectively. These herpes simplex virus-II individuals get treated at the rate of and move to the subcompartment of recovery. Here also, the recovered individuals might lose their immunity and recover to the susceptible sub-compartment rate. The individual will suffer from death because of diseases when it attains the HSV-II stage at rate. For all these sub compartments the mortality rate of individuals is noted as . All parameters and the variables taken for this model are kept to be positive.

The system of nonlinear stochastic fractional delay differential equations is described as follows:

1

.

2

.

3

.

4

.

5

.

6

.

under the initials;

Where fluctuation and is the Brownian motion with time delay .

Preliminaries

The basic definitions are as follows;

Definition 1

For a function , the Caputo fractional derivative of order is;

.

Definition 2

For the function , the expression describes the equivalent fractional integral with order .

.

Where is the gamma function displayed.

Definition 3

A function satisfies a Lipshitz condition in the variable y on set if a constant exists with

,

whenever are in is Lipshitz constant.

Existence and uniqueness

In this part, the existence and uniqueness of the model (1–6) are established with . For this,

7

.

8

.

9

.

10

.

11

.

12

.

The functions listed in system (7–12) under the integral are

13

.

14

.

15

16

17

18

Let and define the operator on the Banach space with the supremum norm by.

where . Existence and uniqueness follow if  is Lipschitz in on a bounded region and the contraction condition holds for some , where is the Lipschitz constant.

Furthermore, it is assumed that exist as positive constants and that are non-negative limiting functions.

Such that

.

Theorem 1

Show that Lipschitz conditions fulfill by kernel for each model (1–6) with , when .

Proof

For a detailed proof, see Appendix A, where Eqs. 36 are derived and used in the proof.

Theorem 2

Prove that: The system of Eqs. (25–30) has a definite uniform function. if there exists at least a such that . For the model system with there exists at least a solution if for .

Proof

For a detailed proof, see Appendix B, where Eqs. (60)-(81) are derived and used in the proof.

Theorem 3

The system (1–6) demonstrates the uniqueness with if .

Proof

For a detailed proof, see Appendix C, where Eqs. 82 are derived and used in the proof.

Positivity and boundedness

Theorem 4

Assume the initial history functions satisfy and all parameters are nonnegative. For , the solution of system (1–6) remains nonnegative for all , i.e.,

Proof

Using the Caputo fractional formulation, each state variable admits the equivalent Volterra integral form. For example,

and similarly, for , where and denotes the right-hand side of (1). Now suppose (for contradiction) that some component becomes negative. Let be the first time such that at least one component hits zero and then attempts to cross into the negative region. At time , that component equals zero, while all other components remain nonnegative by minimality of , and the delayed terms are also nonnegative because the history is nonnegative.

On each boundary face, the corresponding right-hand side is nonnegative under nonnegative states. In particular, evaluating the drift terms on the boundary yields:

• if , then ;

• if , then ;

• if , then ;

• if , then ;

• if , then ;

• if , then .

Therefore, at the first hitting time , the Caputo integral representation implies the solution cannot decrease past zero (fractional comparison), which contradicts the assumption that a component crosses into negative values. Hence all components remain nonnegative for all .

Lemma 1

(Fractional Grönwall Inequality). Let be a nonnegative, continuous function on and assume that

.

where are constants. Then

where denotes the Mittag-Leffler function.

Theorem 5

The system (1–6) under the nonnegative initial conditions is bounded for all when , , and the solution lies in the feasible region

.

Proof

Let By summing Eqs. (1–6) and using the nonnegativity of all state variables and parameters, we obtain

.

The associated comparison Eq. 

19

has the explicit solution

20

By the fractional comparison principle and Lemma 1, it follows that

.

Since as , we conclude that

21

Therefore, all solutions of system (1–6) remain bounded and lie in the feasible region .

Model equilibria and reproduction number

This part will provide the model equilibria for system (1–6). Therefore,

22

.HSV-Free Equilibrium

23

.HSV-Present Equilibrium

.

By using Next generation method to get reproduction number for system (1–6) is as follows:

24

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The basic reproduction number plays a threshold role in determining the existence and stability of the equilibrium points. When , the HSV-free equilibrium is locally (and globally) asymptotically stable, indicating that HSV-II cannot invade the population and the infection dies out. When , the HSV-free equilibrium becomes unstable and a unique HSV-present equilibrium emerges, corresponding to persistent disease transmission. Thus, governs the qualitative transition between disease extinction and persistence in the proposed model.

Local stability

The local stability analysis is carried out within the framework of Caputo fractional-order systems. For fractional differential equations of order , an equilibrium point is locally asymptotically stable if all eigenvalues of the corresponding Jacobian matrix satisfy . This criterion generalizes the classical ODE stability condition and accounts for the memory effects inherent in fractional-order models. Using this fractional stability criterion, we analyze the HSV-free and HSV-present equilibrium points of system (1–6).

Theorem 6

For the Caputo fractional-order system (1–6) with and , the HSV-free equilibrium  is locally asymptotically stable if . Conversely, becomes unstable when .

Proof

Linearizing system (1–6) around the HSV-free equilibrium  yields the corresponding Jacobian matrix . For a Caputo fractional-order system of order , an equilibrium point is locally asymptotically stable if all eigenvalues of the Jacobian matrix satisfy the fractional-order stability condition

.

Applying this criterion to , it follows that all eigenvalues satisfy the above condition when the basic reproduction number , ensuring local asymptotic stability of the HSV-free equilibrium. Conversely, when , at least one eigenvalue violates the fractional stability condition, and the HSV-free equilibrium becomes unstable. Hence, under the fractional-order framework, the stability of  is completely governed by the threshold parameter .

Theorem 7

For the Caputo fractional-order system (1–6) with and , the HSV-present equilibrium  exists when and is locally asymptotically stable provided that all eigenvalues of the Jacobian matrix evaluated at  satisfy

.

Proof

After linearizing Jacobian matrix of the system (1–6), the 6th order polynomial follows as

.

Here.

.

.

.

.

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Since the 6th order polynomial with and are positive coefficient with , , . So, by the Routh-Hurwitz Criterion satisfies when Hence of the given system (1–6) is LAS.

Global stability

In this part, we analyzed the stability of the model (1–6) at both equilibrium points like HSV-Free Equilibrium and HSV-Present Equilibrium point in the sense of global.

Theorem 8

The system (1–6) is GAS at when with .

Proof

Define the Lyapunov function ,

.

.

.

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This implies that when , and only at . Therefore, by the fractional-order extension of LaSalle’s invariance principle (for Caputo systems), the HSV-free equilibrium is globally asymptotically stable for when .

Theorem 9

The system (1–6) is GAS at when with ,

Proof

Let the Lyapunov function ,

.

.

.

If we choose

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Therefore, since and if and only if , it follows from the fractional LaSalle invariance principle for Caputo fractional systems that the HSV-present equilibrium is globally asymptotically stable for when .

Sensitivity analysis

,

,

,

,

,

,

To discuss the graphs for the Herpes simplex virus, which is in the middle of understanding HSV transmission dynamics. Figure 1 shows the sensitivity indices of the reproduction number () concerning different model parameters. Positive sensitivity indices for the parameters mean that they are directly related to , and an increase in these parameters increases the reproduction number and enhances the spread of the disease. On the other hand, negative sensitivity indices for the parameters indicate an inverse relationship, where an increase in these parameters decreases and reduces the transmission of the disease. Table 1 confirms these dynamics to be biologically relevant because the positively sensitive parameters likely would reflect factors like transmission rates or population interactions, while negative ones might reflect intervention-related factors such as recovery rate or vaccine effects. Understanding those relationships enables targeted control strategies since focus could be placed on the reduction of positively sensitive parameters along with enhancing the negatively sensitive ones that could help manage the HSV-II spread effectively.

Fig. 1.

Sensitivity indices of reproduction number .

Table 1.

Values and signs of parameter’s sensitivity:.

Parameters	Signs	Values
	Positive
	Positive
	Positive
	Positive
	Positive
	Negative
	Negative
	Negative
	Negative
	Negative

Numerical methods

We propose a generalized stochastic fractional technique for the solution of the stochastic fractional-order system (1–6) like a NSFD scheme (see²¹. In all cases, denotes the temporal step size.

Non-standard finite difference (NSFD) scheme

Consider the stochastic fractional delayed system (1–6) with Caputo derivative of order . Using the integral representation of the Caputo derivative, a general equation of the form.

can be written in a time-stepping form where the deterministic increment is scaled by . Following the nonstandard finite difference (NSFD) rules of Mickens, we discretize the gain terms explicitly and treat the loss terms implicitly through a denominator in order to preserve positivity and boundedness for any step size. For the stochastic part, we apply the Euler-Maruyama approximation by replacing the Brownian increment with , which yields multiplicative noise terms of the form . Implementing these principles componentwise for , and evaluating the delayed mortality survival factor consistently, leads to the stochastic fractional NSFD scheme given by Eqs. (25)-(30).

25

26

27

28

29

30

Here is the time step and are independent Brownian increments with mean zero and variance . Furthermore, we investigate the properties of fractional NSFD of model (79–84).

Theorem 10

(Positivity). The system (79–84) satisfies the positivity under the initial conditions.

Proof

The model (79–84) is free from negative term. Thus, the initial condition also positive, as desired.

Theorem 11

(Boundedness). The system (79–84) satisfies the boundedness under the initial conditions for , such that ,for each .

Proof

The sum of model (79–84) for the bondedness is as follows;

.

The proof relies on mathematical induction, with serving as the end point of sequence of identities and inequalities.

Graphical simulation

In this part, we use graphical simulations for fractional model (1–6). To that end, the model parameters are fixed as given in Table 2 (see¹. Moreover, in all simulations, are taken as independent Wiener increments, and the noise intensities were fixed at . To investigate the combined effect of memory and delay, simulations for different fractional orders were conducted. while changing the delay parameter over various representative values. This allows comparison of disease dynamics under interchangeable enumerate all combinations and point out how explicit delays change the persistence effects induced by fractional memory. The numerical findings confirm that increasing generally delays peaks and prolongs persistence while decreasing enhances long-term memory effects.

Table 2.

Values of parameters.

Parameters	Values	Source [1]
		Assumed
		Assumed
		Estimated
		Estimated
		Fitted
		Fitted

Discussion

The graphical discussion for Herpes Simplex-II Virus (HSV-II) addresses the dynamics of how disease relates to various fractional order values, affecting the different compartments in the model with time delay. Figure 2 represents the time profile of the susceptible population, as it is affected by changes in fractional orders. A decrease in fractional order implies a significant memory effect on the system that gradually reduces the rate of decline in the susceptible population, reflecting persistence over time for those at risk due to fractional time delay effects. In Fig. 3, the fractional order heavily affects the rate at which exposed individuals transition to the asymptomatic or symptomatic phases. As the fractional order increases, the exposed population drops more sharply, implying more rapid disease progression. The graph for asymptomatic individuals (Fig. 4) exhibits the variability of the peak and persistence of this compartment with fractional order. The lower fractional orders lead to prolonged peaks that suggest a delay in the recovery or progression due to the time-delay effect in the system. In Fig. 5 symptomatic individuals show more significant sensitivity to changes in fractional orders. The delay parameter increases the oscillatory behavior and seems to indicate fluctuations of disease prevalence over time when memory effects and delays interact. Figure 6 for the viral load shows how fractional dynamics affect the prevalence of the virus. Lower fractional orders lead to sustained viral activity, which reflects the combined effects of delayed immune response and the persistence mechanisms inherent in the disease.

Fig. 2.

Impact of on at .

Fig. 3.

Impact of on at .

Fig. 4.

Impact of on at .

Fig. 5.

Impact of on at .

Fig. 6.

Impact of on at .

Finally, Fig. 7 illustrates the recovery dynamics and how fractional orders make the return to the recovered state delayed. Lower orders elongate the recovery period due to strong memory and time-delay effects, while higher orders exhibit recovery rates. This means that by changing the fractional order, one can simulate the different behaviors of diseases such as prolonged exposure phases, oscillatory symptomatic peaks, and delayed recovery. Such insights are crucial for long-term disease control and the planning of effective interventions. By adjusting the fractional order, one can simulate varying disease behaviors, such as prolonged exposure phases, oscillatory symptomatic peaks, and delayed recovery. These insights are important for understanding long-term disease control and planning effective interventions. In the context of HSV-II, fractional-order memory reflects the cumulative influence of past infection history, including immune persistence, latency, and recurrent reactivation, on current transmission dynamics. Lower values of the fractional order indicate stronger dependence on historical states, resulting in slower progression and prolonged persistence. This effect differs from the discrete delay , which represents a specific biological waiting time, such as incubation or delayed response.

Fig. 7.

Impact of on at .

In this model, explicit delay would mainly reflect biological delays of waiting processes, such as incubation and host response delays. Stochastic perturbations capture random variability in transmission and immune dynamics. These components introduce uncertainty and heterogeneity that cannot be modeled by a deterministic model in isolation.

Conclusion

In this study, the dynamics of Herpes Simplex Virus–II (HSV-II) were investigated through a stochastic fractional delay differential model incorporating essential epidemiological factors. The model guarantees the existence, uniqueness, positivity, and boundedness of solutions. Two equilibrium states were identified: HSV-Free Equilibrium (HSV-FE) and HSV-Present Equilibrium (HSV-PE), with the basic reproduction number calculated via the Next-Generation Matrix method. Both local and global stability analyses were performed, along with a sensitivity analysis to examine the influence of key parameters on disease transmission. Numerical simulations based on the Stochastic Non-Standard Finite Difference (NSFD) scheme were conducted for various fractional orders , demonstrating improved positivity, boundedness, and stability compared to standard fractional-order modeling methods. The theoretical findings were further validated through graphical simulations, providing deeper insights into HSV-II dynamics and supporting informed public health decision-making. Overall, the combination of advanced mathematical modeling and robust computational simulations provides a valuable framework for understanding complex biological systems and predicting disease dynamics. The suggested framework might be expanded in future work by availing probabilistic safety and control-theoretic concepts, e.g., risk-aware analysis and adaptive control strategies, which have been proven to work well in complicated engineered systems[^25–27]. These types of methods could even further improve the robustness and real-time usability of HSV-II modeling when the uncertainty is present.

Appendix A

Proof: First, we analyze the Lipschitz’s condition for the function. For this, we take help with the following and :

Since, . Lipschitz’s condition is satisfied. Next, for consider and .

For, . Lipschitz condition is satisfied.

Next, for analyzing for and .

For, . Lipschitz condition is satisfied.

Next, for analyzing for and .

For, . Lipschitz condition is satisfied.

Next, for analyzing for and .

For, . Lipschitz condition is satisfied.

Next, for analyzing for and .

For, . Lipschitz condition is satisfied.

Next, there is constant in (13–18).

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Remaining variation is as follows:

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Therefore, we have

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Let,

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Similarly,

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As required.

Appendix B

Proof

Consider the are bounded. Then.

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Since, will converge because the system (31–36) exists and consistent. For this, consider changes as . Thus,

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The result of Lipschitz condition for and the triangle inequality, and for , fulfills the Lipschitz condition.

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with repetition of (55),

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Next, at , one acquires

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Assuming as the limit.

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By applying the hypothesis , we get

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Similarly,

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As desired.

Appendix c

Proof

Examine how the sets represent the solutions to (1–6).

.

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After simplifying,

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By applying the hypothesis , we have from (66) yield.

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Similarly,

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Hence proved.

Author contributions

Ali Raza contributed to the conceptualization, methodology, software, validation, formal analysis, investigation, resources, data curation, writing of the original draft, review and editing, visualization, supervision, project administration, and funding acquisition. Mansoor Alsulami contributed to methodology, Visualization, validation, and writing – review and editing. Marek Lampart contributed to resources, supervision, validation,writing – review and editing,and project administration. Umar Shafique contributed to data curation, software, methodology,validation and writing – review and editing. All authors reviewed and approved the final version of the manuscript. All authors read and approved a manuscript with the given study.

Data availability

The datasets analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.