CMES CMES CMES Computer Modeling in Engineering & Sciences 1526-1506 1526-1492 Tech Science Press USA 61635 10.32604/cmes.2025.061635 Article Computational Modeling of Streptococcus Suis Dynamics via Stochastic Delay Differential Equations Computational Modeling of Streptococcus Suis Dynamics via Stochastic Delay Differential Equations Computational Modeling of Streptococcus Suis Dynamics via Stochastic Delay Differential Equations ShafiqueUmar1 RazaAli27ali.raza@uevora.ptali@phs.uchenab.edu.pk BaleanuDumitru3 NasirKhadija4 NaveedMuhammad5 SiddiqueAbu Bakar1 FadhalEmad6efadhal@kfu.edu.sa Department of Mathematics, National College of Business Administration and Economics, Lahore, 54000, Pakistan Center for Research in Mathematics and Applications (CIMA), Institute for Advanced Studies and Research (IIFA), University of Évora, Rua Romão Ramalho, 59, Évora, 7000-671, Portugal Department of Computer Science and Mathematics, Lebanese American University, Beirut, 1102-2081, Lebanon Department of Zoology, University of Sialkot, Sialkot, 51040, Pakistan Department of Mathematics, Air University, Islamabad, 44000, Pakistan Department of Mathematics and Statistics, College of Science, King Faisal University, Al Ahsa, 31982, Saudi Arabia Department of Physical Sciences, The University of Chenab, Gujrat, 50700, Pakistan Corresponding Authors: Ali Raza. Email: ali.raza@uevora.pt or ali@phs.uchenab.edu.pk; Emad Fadhal. Email: efadhal@kfu.edu.sa 2025 11042025 143 1 449 476 29 11 2024 05 2 2025 © 2025 The Authors. 2025 Published by Tech Science Press. This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Streptococcus suis (S. suis) is a major disease impacting pig farming globally. It can also be transferred to humans by eating raw pork. A comprehensive study was recently carried out to determine the indices through multiple geographic regions in China. Methods: The well-posed theorems were employed to conduct a thorough analysis of the model’s feasible features, including positivity, boundedness equilibria, reproduction number, and parameter sensitivity. Stochastic Euler, Runge Kutta, and Euler Maruyama are some of the numerical techniques used to replicate the behavior of the streptococcus suis infection in the pig population. However, the dynamic qualities of the suggested model cannot be restored using these techniques. Results: For the stochastic delay differential equations of the model, the non-standard finite difference approach in the sense of stochasticity is developed to avoid several problems such as negativity, unboundedness, inconsistency, and instability of the findings. Results from traditional stochastic methods either converge conditionally or diverge over time. The stochastic non-negative step size convergence nonstandard finite difference (NSFD) method unconditionally converges to the model’s true states. Conclusions: This study improves our understanding of the dynamics of streptococcus suis infection using versions of stochastic with delay approaches and opens up new avenues for the study of cognitive processes and neuronal analysis. The plotted interaction behaviour and new solution comparison profiles.

 Streptococcus suis disease model stochastic delay differential equations (SDDEs) existence and uniqueness Lyapunov function stability results reproduction number computational methods Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University KFU250259 Introduction

In [1], the authors provide a mathematical model for streptococcus suis infection in the pig population. The technique used to evaluate the model is beneficial for overcoming the disease. In [2], the authors constructed a factual model to regulate model specifications in several antibiotic ramifications various perceptions, and instructions of infection to resist streptococcus suis. In [3], the authors studied separately from recently discovered streptococcus suis species. In [4], the author’s presence of streptococcus suis illness in seedling pigs was linked to pigs that performed averagely and had a sow effect rather than any notable disease traits. In [5], the authors’ enhanced infection was only seen in the upper respiratory tract in this investigation. We used two separate models to assess the variations in streptococcus suis disease. In [6], the authors provide more convincing evidence for the beneficial effects of the drug vs. streptococcus suis disease by elucidating the underlying molecular process. In [7], the authors evaluate the effect of implementing an autogenous vaccination program on the emergence of illnesses linked to streptococcus suis in natural environments as a challenging undertaking. In [8], the authors illustrate that the survival of other streptococcus suis pathotypes in porcine blood is also restricted by antibody-mediated, as evidenced by the fact that the bacterial surface was usually substantially greater following development in standard piglets’ plasma than following incubation in serum obtained before any colostrum adoption. In [9], the authors are shown to be the most effective solvent for a substance called separation by ultrasound-assisted extraction (UAE), and the response surface methodology (RSM) framework accurately represented the anticipated optimization of Emirati. In [10], the authors discovered during the streptococcus suis-2 disease, vimentin increased lung damage, neutrophil counts, and the production of proinflammatory cytokines and chemokines in the lungs of pigs and swine tracheal epithelial cells (STEC). In [11], the authors illustrate how the host-defense peptide cathelicidins are avoided by the Streptococcus suis pepo protease, which affects the pathophysiology of Streptococcus suis. It was discovered that Pepo cleaves the anti-Streptococcus suis cathelicidins, mouse cathelicidin mouse and human cathelicidin. In [12], the authors present the mathematical framework of climate influence on Streptococcus suis infection in pig-human populations generally. In [13], the authors developed a fractional-order mathematical framework relying on fractional derivative concepts. In [14], the authors’ investigation is based on the hypotheses of further studies in this domain, especially utilizing both experimental and real-world data. The model suggested that batch-level isolation might cause a likelihood of Streptococcus suis incidence in the facility. In [15], the authors studied that Streptococcus suis is a human pathogen that is frequently responsible for meningitis in Asian nations that consume pork. In [16], the authors provide an exclusive preventative option accessible to pig breeders as a possibility to medicines for controlling the Streptococcus suis infection. In [17], the authors determine that Streptococcus suis strain extracted from an appropriate pig tonsil is aggressive and possesses multiple mechanisms that encourage niche conflict in pig tonsil. In [18], the authors create a computational model of Streptococcus suis infection in a pig community. The approach employed to analyze the model is useful in conquering the illness. In [19], the authors examine blood cortisol levels as a distress readout metric and buprenorphine therapy as a refining measure in a novel pig Streptococcus suis disease model. In [20], the authors created a scientific simulation to control model parameters in several antibiotic implications, different perspectives on infection, and guidelines for resisting Streptococcus suis. In [21], the author explores the use of Stochastic Differential Equations (SDEs) in applications of sciences and many more. In [22], the authors studied the existence and approximate controllability of the Hilfer fractional neutral stochastic hemivariational inequality with the Rosenblatt process. Stochastic or probabilistic components are included in a mathematical model of Streptococcus suis infection dissemination by numerical simulation and analysis, with an emphasis on accounting for uncertainty in disease transmission. Public health efforts for disease control and prevention are informed by this kind of modeling, which provides insights into how the illness could spread under various circumstances.

The main key point to study is the structure-preserving and dynamical analysis of the Streptococcus suis disease model. The fundamental properties of the model like positivity, boundedness, and local and global stabilities are studied rigorously. The authors used well-known methods like Euler Maruyama, stochastic Euler, and stochastic Runge Kutta for the computational analysis and made a comparison analysis with the proposed method like nonstandard finite difference in the sense of stochastic. The Nonstandard Finite Difference (NSFD) method gives a guarantee of Structure-preserving properties of the model like positivity, boundedness, and dynamical consistency of the solution instead of other standard methods.

The paper is organized as follows: An overview of Streptococcus suis infection-like conditions and a thorough assessment of the literature is provided in Section 1. Building the delayed model and the ensuing mathematical analysis are the focus of Section 2. In Section 3, the local and global levels of the model’s stability, reproduction number, and equilibria are examined. The sensitivity analysis of the model’s parameters is covered in Section 4. The stochastic conceptualization phase is presented in Section 5. The numerical approach of the NSFD technique and numerical simulations and the presentation of the results are the explicit focus of Section 6. Final opinions provide a conclusive overview of the work under Section 7.

 Model Formulation

This section presents the delay model formulation of infection spread by pigs and humans. Four classifications were used to categorize the pig population: susceptible class S𝓹(t), infectious class I𝓹(t), quarantine class Q𝓹(t), and recovered class R𝓹(t). Because Streptococcus Suis may spread from pig to people, the model includes the susceptible human class S𝓱(t), infectious human class I𝓱(t), and recovered class R𝓱(t). System (1)–(7) defines the SIQR-SIR model diagram for people and pigs, as shown in Fig. 1.dS𝓹(t)dt=Λ𝓹−MβS𝓹(t−τ)I𝓹(t−τ)e−bτ−bS𝓹(t)t≥0,τ<t dI𝓹(t)dt=MβS𝓹(t−τ)I𝓹(t−τ)e−bτ−(δ+m+b)I𝓹(t)t≥0,τ<t dQ𝓹(t)dt=δI𝓹(t)−(ε+m+b)Q𝓹(t)t≥0 dR𝓹(t)dt=εQ𝓹(t)−bR𝓹(t)t≥0 dS𝓱(t)dt=Λ𝓱−γS𝓱(t−τ)I𝓹(t−τ)e−μτ−μS𝓱(t)t≥0,τ<t dI𝓱(t)dt=γS𝓱(t−τ)I𝓹(t−τ)e−μτ−(α+μ+β2)I𝓱(t)t≥0,τ<t dR𝓱(t)dt=β2I𝓱(t)−μR𝓱(t)t≥0

SIQR-SIR model diagram for people and pigs [<xref ref-type="bibr" rid="ref-13">13</xref>]

By S𝓹(0)≥0,I𝓹(0)≥0,Q𝓹(0)≥0,R𝓹(0)≥0,S𝓱(0)≥0,I𝓱(0)≥0,R𝓱(0)≥0 initial conditions.

The pig model attribute can be expressed as follows: β is the rate of transmission, M is the relative humidity; m is the disease-induced pig death rate; δ is the rate from infectious class to quarantine class in pigs; ε is the pig recovered rate; μ is the human natural death rate, γ is the transmission rate from infected pig to human, α is the disease death rate, and β2 is the human recovery rate.

 Model Analysis

This section examines the delay model feasible region, which carries biological significance as the suggested model takes into account. Consider every parameter and variable in the delay model is non-negative. Next, the model’s equilibria and the fundamental reproduction number are determined. Furthermore, we investigate each equilibrium at locally and globally stable.

Feasible Region

The feasible region of the system (1)–(7) is shown ℒ={(S𝓹,I𝓹,Q𝓹,R𝓹,S𝓱,I𝓱,R𝓱)∈R+7;N≤Λ𝓹+Λ𝓱ℬ}

Theorem 1: The solution of the system (1)–(7) is positive in the feasible region.

Proof: Consider the system (1)–(7), we have

dS𝓹dt|S𝓹=0=Λ𝓹>0, dI𝓹dt|I𝓹=0=MβS𝓹(t)I𝓹(t)e−bτ>0, dQ𝓹dt|Q𝓹=0=δI𝓹(t)>0, dR𝓹dt|R𝓹=0=εQ𝓹(t)>0, dS𝓱dt|S𝓱=0=Λ𝓱>0,dI𝓱dt|I𝓱=0=γS𝓱(t)I𝓹(t)e−μτ>0,dR𝓱dt|R𝓱=0=β2I𝓱(t)>0.

Hence, system (1)–(7) has a positive solution with the initial condition in the feasible region. □

Theorem 2: The solution of the model (1)–(7) is bounded in the feasible region.

Proof: The total number of people and pigs may be written as N(t)=S𝓹(t)+I𝓹(t)+Q𝓹(t)+R𝓹(t)+S𝓱(t)+I𝓱(t)+R𝓱(t) dN(t)dt=dS𝓹dt+dI𝓹dt+dQ𝓹dt+dR𝓹dt+dS𝓱dt+dI𝓱dt+dR𝓱dt dN(t)dt≤Λ𝓹+Λ𝓱−ℬN(t) dN(t)dt+ℬN(t)≤Λ𝓹+Λ𝓱

Which is a linear differential equation N(t)≤Λ𝓹+Λ𝓱ℬ+(N(0)−Λ𝓹+Λ𝓱ℬ)e−ℬt

Using Grown’s inequality

limt→∞SupN(t)≤Λ𝓹+Λ𝓱ℬ as desired. □

 Model Equilibria and Reproduction Number

This section includes two types of model equilibria for Streptococcus Suis Equilibrium.

StreptococcusSuisFreeEquilibrium=SSFE=D0=(S𝓹0,I𝓹0,Q𝓹0,R𝓹0,S𝓱0,I𝓱0,R𝓱0)=(Λ𝓹b,0,0,0,Λ𝓱μ,. 0,0)

Streptococcus Suis Endemic Equilibrium =SSEE=D∗=(S𝓹∗,I𝓹∗,Q𝓹∗,R𝓹∗,S𝓱∗,I𝓱∗,R𝓱∗)

S𝓹∗=(δ+m+b)Mβe−bτ, I𝓹∗=Λ𝓹Mβe−bτ−b(δ+m+b)(δ+m+b)Mβe−bτ, Q𝓹∗=δ(ε+m+b)I𝓹∗, R𝓹∗=εδb(ε+m+b)I𝓹∗, S𝓱∗=Λ𝓱γI𝓹∗e−μτ−μ, I𝓱∗=γΛ𝓱I𝓹∗e−μτ(γI𝓹∗e−μτ−μ)(α+μ+β2), R𝓱∗=β2γΛ𝓱I𝓹∗e−μτμ(γI𝓹∗e−μτ−μ)(α+μ+β2).

The reproduction number is vastly essential in epidemiology. This determines the probability that the illness exists in the community or not. If the reproduction is less than one, disease can be prevented in the community; if the reproduction number is larger than one, disease exists in the community. Use the next-generation approach to calculate the reproduction number. Thus, ℱ is the transmission matrix, while 𝒱 is the transition matrix. ℱ=[MβS𝓹e−bτ0γS𝓱e−μτ0],𝒱=[(δ+m+b)00(α+μ+β2)] ℱ𝒱−1=[MβS𝓹e−bτ0γS𝓱e−μτ0][1(δ+m+b)001(α+μ+β2)] ℱ𝒱−1=[MβS𝓹e−bτ(δ+m+b)0γS𝓱e−μτ(α+μ+β2)0]

The largest eigenvalue of the matrix called the spectral radius or reproduction number at Streptococcus suis free equilibrium, follows as ℛ0=MβΛ𝓹e−bτb(δ+m+b).

 Stability Analysis

We will demonstrate the following well-known result about local and global stability in both model equilibrium points. Consider the function as follows: A=Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t) B=MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t) C=δI𝓹(t)−(ε+m+b)Q𝓹(t) D=εQ𝓹(t)−bR𝓹(t) E=Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t) F=γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t) G=β2I𝓱(t)−μR𝓱(t)

The Jacobian matrix has the following elements:

∂A∂S𝓹=−MβI𝓹e−bτ−b, ∂A∂I𝓹=−MβS𝓹e−bτ, ∂A∂Q𝓹=0, ∂A∂R𝓹=0, ∂A∂S𝓱=0, ∂A∂I𝓱=0, ∂A∂R𝓱=0, ∂B∂S𝓹=MβI𝓹e−nτ, ∂B∂I𝓹=MβS𝓹e−bτ−(δ+m+b), ∂B∂Q𝓹=0, ∂B∂R𝓹=0, ∂B∂S𝓱=0, ∂B∂I𝓱=0, ∂B∂R𝓱=0, ∂C∂S𝓹=0, ∂C∂I𝓹=δ, ∂C∂Q𝓹=−(ε+m+b), ∂C∂R𝓹=0, ∂C∂S𝓱=0, ∂C∂I𝓱=0, ∂C∂R𝓱=0, ∂D∂S𝓹=0, ∂D∂I𝓹=0, ∂D∂Q𝓹=ε, ∂D∂R𝓹=−n, ∂D∂S𝓱=0, ∂D∂I𝓱=0, ∂D∂R𝓱=0, ∂E∂S𝓹=0, ∂E∂I𝓹=−γS𝓱e−μτ, ∂E∂Q𝓹=0, ∂E∂R𝓹=0, ∂E∂S𝓱=−γI𝓹e−μτ, ∂E∂I𝓱=0, ∂E∂R𝓱=0, ∂F∂S𝓹=0, ∂F∂I𝓹=γS𝓱e−μτ, ∂F∂Q𝓹=0, ∂F∂R𝓹=0, ∂F∂S𝓱=γI𝓹e−μτ, ∂F∂I𝓱=−(α+μ+β2), ∂F∂R𝓱=0, ∂G∂S𝓹=0, ∂G∂I𝓹=0, ∂G∂Q𝓹=0, ∂G∂R𝓹=0, ∂G∂S𝓱=0, ∂G∂I𝓱=β2, ∂G∂R𝓱=−μ, J=[−MβI𝓹e−bτ−b−MβS𝓹e−bτ00000MβI𝓹e−bτMβS𝓹e−bτ−(δ+m+b)000000δ−(ε+m+b)000000ε−b0000−γS𝓱e−μτ00−γI𝓹e−μτ−μ000γS𝓱e−μτ00γI𝓹e−μτ−(α+μ+β2)000000β2−μ]

Theorem 3: The Streptococcus Suis Free Equilibrium=SSFE=D0=(S𝓹0,I𝓹0,Q𝓹0,R𝓹0,S𝓱0,I𝓱0,R𝓱0)=(Λ𝓹b,0,0,0,Λ𝓱μ,0,0) is locally asymptotical stable (LAS) if R0<1. Otherwise, the system is unstable at D0 if R0>1.

Proof: For stability at D0=(S𝓹0,I𝓹0,Q𝓹0,R𝓹0,S𝓱0,I𝓱0,R𝓱0), the Jacobian matrix (8) becomes J(D0)=[−b−MβS𝓹0e−bτ000000MβS𝓹0e−bτ−(δ+m+b)000000δ−(ε+m+b)000000ε−b0000−γS𝓱0e−μτ00−μ000γS𝓱0e−μτ000−(α+μ+β2)000000β2−μ] |J(D0)−λ|=|−b−λ−MβS𝓹0e−bτ000000MβS𝓹0e−bτ−(δ+m+b)−λ000000δ−(ε+m+b)−λ000000ε−b−λ0000−γS𝓱0e−μτ00−μ−λ000γS𝓱0e−μτ000−(α+μ+β2)−λ000000β2−μ−λ| λ1=λ5=−b,λ2=λ4=−μ,λ3=−(α+μ+β2),λ6=−(ε+m+b),λ7=MβS𝓹0e−bτ−(δ+m+b) λ7=−(δ+m+b)(1−ℛ0)

Hence the streptococcus Suis free equilibrium of the given system (1)–(7) is stable in the sense of local if ℛ0<1. Else, if ℛ0>1, then, D0 is unstable in the sense of local. □

Theorem 4: The Streptococcus Suis Endemic Equilibrium=SSEE=D∗=(S𝓹∗,I𝓹∗,Q𝓹∗,R𝓹∗,S𝓱∗,I𝓱∗,R𝓱∗) is Locally Asymptotical Stable (LAS) if ℛ0>1.

Proof: Letting from (8), we get J(D∗)=[−MβI𝓹∗e−bτ−b−MβS𝓹∗e−bτ00000MβI𝓹∗e−bτMβS𝓹∗e−bτ−(δ+m+b)000000δ−(ε+m+b)000000ε−b0000−γS𝓱∗e−μτ00−γI𝓹∗e−μτ−μ000γS𝓱∗e−μτ00γI𝓹∗e−μτ−(α+μ+β2)000000β2−μ]

For eigenvalue, consider |J−λI|=0 λ1=−μ,λ2=−(α+μ+β2),λ3=−γI𝓹∗e−μτ−μ,λ4=−b,λ5=−(ε+m+b) |−MβI𝓹∗e−bτ−b−λ−MβS𝓹∗e−bτMβI𝓹∗e−bτMβS𝓹∗e−bτ−(δ+m+b)−λ|=0 [−MβI𝓹∗e−bτ−b−λ][MβS𝓹∗e−bτ−(δ+m+b)−λ]+[MβS𝓹∗e−bτ][MβI𝓹∗e−bτ]=0 λ2+[MβI𝓹∗e−bτ+b−MβS𝓹∗e−bτ+(δ+m+b)]λ+[MβS𝓹∗e−bτ][MβI𝓹∗e−bτ]=0 λ2+a1λ+ao=0

a1>0, If MβI𝓹∗e−bτ+b+(δ+m+b)>MβS𝓹∗e−bτ. So, a1,a0>0

So, by the Routh-Hurwitz Criterion for a 2nd-degree polynomial, the coefficient of the characteristic equation is positive with the constraint ℛ0>1. Hence the endemic equilibria (EE) of the given system (1)–(7) are stable in the sense of locally. Else, if ℛ0<1, then Routh Hurwitz’s condition for stability is violated. Thus, EE is unstable in the sense of local. □

 Global Stability Analysis

The stability of the Streptococcus Suis infection model is demonstrated by well-known outcomes in following global sense.

Theorem 5: The Streptococcus Suis Free Equilibrium =SSFE=D0=(S𝓹0,I𝓹0,Q𝓹0,R𝓹0,S𝓱0,I𝓱0,R𝓱0)=(Λ𝓹b,0,0,0,Λ𝓱μ,0,0) is globally asymptotical stable (GAS) if ℛ0<1. Otherwise, the system (1)–(7) is unstable at D0 if ℛ0>1.

Proof: Define the Volterra Lyapunov function A:ℒ→R defined as ℒ=[S𝓹−S𝓹0−S𝓹0logS𝓹S𝓹0]+I𝓹+Q𝓹+R𝓹+S𝓱+I𝓱+R𝓱 dℒdt=[1−S𝓹0S𝓹]dS𝓹dt+dI𝓹dt+dQ𝓹dt+dR𝓹dt+[1−S𝓱0S𝓱]dS𝓱dt+dI𝓱dt+dR𝓱dt dℒdt=[S𝓹−S𝓹0S𝓹][Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t)]+[MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t)]+[δI𝓹(t)−(ε+m+b)Q𝓹(t)]+[εQ𝓹(t)−bR𝓹(t)]+[1−S𝓱0S𝓱][Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t)]+[γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t)]+[β2I𝓱(t)−μR𝓱(t)] dℒdt≤−Λ𝓹(S𝓹−S𝓹0)2S𝓹S𝓹0−(m+b)I𝓹[1−MβS𝓹e−bτ(m+b)]−(m+b)Q𝓹−bR𝓹−Λ𝓱(S𝓱−S𝓱0)2S𝓱S𝓱0−(α+μ)I𝓱(t)[1−γS𝓱(t)e−μτ(α+μ)]−μR𝓱(t)

This implies that dℒdt≤0 if ℛ0<1 and dℒdt=0 if S𝓹=S𝓹0,S𝓱=S𝓱0,I𝓹=Q𝓹=R𝓹=I𝓱=R𝓱=0. Therefore, D0 is globally asymptotically stable. □

Theorem 6: The Streptococcus suis Endemic Equilibrium=SSEE=D∗=(S𝓹∗,I𝓹∗,Q𝓹∗,R𝓹∗,S𝓱∗,I𝓱∗,R𝓱∗) is Globally Asymptotical Stable (GAS) if ℛ0>1.

Proof: Define the Volterra Lyapunov function Z:ℒ→R defined as Z=k1(S𝓹−S𝓹∗−S𝓹∗ln⁡(S𝓹S𝓹∗))+k2(I𝓹−I𝓹∗−I𝓹∗ln⁡(I𝓹I𝓹∗))+k3(Q𝓹−Q𝓹∗−Q𝓹∗ln⁡(Q𝓹Q𝓹∗))+k4(R𝓹−R𝓹∗−R𝓹∗ln⁡(R𝓹R𝓹∗))+k5(S𝓱−S𝓱∗−S𝓱∗ln⁡(S𝓱S𝓱∗))+k6(I𝓱−I𝓱∗−I𝓱∗ln⁡(I𝓱I𝓱∗))+k5(R𝓱−R𝓱∗−R𝓱∗ln⁡(R𝓱R𝓱∗))

Given positive constants ki(i=1,2,3,4,5,6,7), we can express the following equation: dZdt=k1[S𝓹−S𝓹∗S𝓹]dS𝓹dt+k2[I𝓹−I𝓹∗I𝓹]dI𝓹dt+k3[Q𝓹−Q𝓹∗Q𝓹]dQ𝓹dt+k4[R𝓹−R𝓹∗R𝓹]dR𝓹dt+k5[S𝓱−S𝓱∗S𝓱]dS𝓱dt+k6[I𝓱−I𝓱∗I𝓱]dI𝓱dt+k7[R𝓱−R𝓱∗R𝓱]dR𝓱dt dZdt=−k1Λ𝓹(S𝓹−S𝓹∗)2S𝓹S𝓹∗−k2MβS𝓹e−nτ(I𝓹−I𝓹∗)2−k3δI𝓹(Q𝓹−Q𝓹∗)2Q𝓹Q𝓹∗−k4εQ𝓹(R𝓹−R𝓹∗)2R𝓹R𝓹∗−k5Λ𝓱(S𝓱−S𝓱∗)2S𝓱S𝓱∗−k6γS𝓱I𝓹e−μτ(I𝓱−I𝓱∗)2I𝓱I𝓱∗−k7β2I𝓱(R𝓱−R𝓱∗)2R𝓱R𝓱∗

If we choose ki,where(i=1,2,3,4,5,6,7) dZdt=−Λ𝓹(S𝓹−S𝓹∗)2S𝓹S𝓹∗−MβS𝓹e−bτ(I𝓹−I𝓹∗)2−δI𝓹(Q𝓹−Q𝓹∗)2Q𝓹Q𝓹∗−εQ𝓹(R𝓹−R𝓹∗)2R𝓹R𝓹∗−Λ𝓱(S𝓱−S𝓱∗)2S𝓱S𝓱∗−γS𝓱I𝓹e−μτ(I𝓱−I𝓱∗)2I𝓱I𝓱∗−β2I𝓱(R𝓱−R𝓱∗)2R𝓱R𝓱∗

dZdt≤0 for ℛ0>1 and dZdt=0 if and only if S𝓹=S𝓹∗,I𝓹=I𝓹∗,Q𝓹=Q𝓹∗,R𝓹=R𝓹∗,S𝓱=S𝓱∗,I𝓱=I𝓱∗,R𝓱=R𝓱∗. Hence by Lasalle’s invariance principle D∗ is globally asymptotical stable. □

Theorem 7. The Streptococcus suis Free Equilibrium=SSFE=D0=(S𝓹0,I𝓹0,Q𝓹0,R𝓹0,S𝓱0,I𝓱0,R𝓱0)=(Λ𝓹b,0,0,0,Λ𝓱μ,0,0) is globally asymptotical stable (GAS) if ℛ0<1. Otherwise, the system (1)–(7) is unstable at D0 if ℛ0>1.

Proof: Define the Volterra Lyapunov function Φ:ℒ→R defined as Φ′(I𝓹)=1I𝓹dI𝓹dt Φ″(I𝓹)=1I𝓹d2I𝓹dt2−1I𝓹2(dI𝓹dt)2 Φ″(I𝓹)=1I𝓹(MβS𝓹e−bτ−(δ+m+b))2I𝓹−1I𝓹2(MβS𝓹I𝓹e−bτ−(δ+m+b)I𝓹)2 Φ′′(I𝓹)=(MβS𝓹e−bτ−(δ+m+b))2−(MβS𝓹e−bτ−(δ+m+b))2 Φ′′(I𝓹)≤0ifℛ0<1.

Thus, the system (1)–(7) is globally asymptotically stable at Streptococcus Suis Free Equilibrium. □

Theorem 8. The Streptococcus Suis Endemic Equilibrium=SSEE=D∗=(S𝓹∗,I𝓹∗,Q𝓹∗,R𝓹∗,S𝓱∗,I𝓱∗,R𝓱∗) is Globally Asymptotical Stable (GAS) if ℛ0>1.

Proof: Define the Volterra Lyapunov function V:ℒ→R defined as dVdt=[1−S𝓹∗S𝓹]dS𝓹dt+[1−I𝓹∗I𝓹]dI𝓹dt+[1−Q𝓹∗Q𝓹]dQ𝓹dt+[1−R𝓹∗R𝓹]dR𝓹dt+[1−S𝓱∗S𝓱]dS𝓱dt+[1−I𝓱∗I𝓱]dI𝓱dt+[1−R𝓱∗R𝓱]dR𝓱dt d2Vdt2=S𝓹∗S𝓹2(dS𝓹dt)2+(1−S𝓹∗S𝓹)d2S𝓹dt2+I𝓹∗I𝓹2(dI𝓹dt)2+(1−I𝓹∗I𝓹)d2I𝓹dt2+Q𝓹∗Q𝓹2(dQ𝓹dt)2+(1−Q𝓹∗Q𝓹)d2Q𝓹dt2+R𝓹∗R𝓹2(dR𝓹dt)2+(1−R𝓹∗R𝓹)d2R𝓹dt2+S𝓱∗S𝓱2(dS𝓱dt)2+(1−S𝓱∗S𝓱)d2S𝓱dt2+I𝓱∗I𝓱2(dI𝓱dt)2+(1−I𝓱∗I𝓱)d2I𝓱dt2+R𝓱∗R𝓱2(dR𝓱dt)2+(1−R𝓱∗R𝓱)d2R𝓱dt2 d2Vdt2=((Λ𝓹)2+(MβS𝓹(t)I𝓹(t)e−bτ+bS𝓹(t))2)S𝓹∗S𝓹2−(2(Λ𝓹)(MβS𝓹(t)I𝓹(t)e−bτ+bS𝓹(t)))S𝓹∗S𝓹2+((Λ𝓹)(MβI𝓹(t)e−bτ+b))S𝓹∗S𝓹−((MβI𝓹(t)e−bτ+b)2S𝓹)S𝓹∗S𝓹+((MβI𝓹(t)e−bτ+b)2S𝓹)−((Λ𝓹)(MβI𝓹(t)e−bτ+b))+((MβS𝓹(t)I𝓹(t)e−bτ)2+((δ+m+b)I𝓹(t))2)I𝓹∗I𝓹2− (2(MβS𝓹(t)I𝓹(t)e−bτ)(δ+m+b)I𝓹(t))I𝓹∗I𝓹2+(2(MβS𝓹(t)I𝓹(t)e−bτ)(δ+m+b)I𝓹(t))I𝓹∗I𝓹−((MβS𝓹(t)I𝓹(t)e−bτ)2+(δ+m+b)2I𝓹(t))I𝓹∗I𝓹+((MβS𝓹(t)I𝓹(t)e−bτ)2+(δ+m+b)2I𝓹(t))−(2(MβS𝓹(t)I𝓹(t)e−bτ)(δ+m+b)I𝓹(t))+((δI𝓹(t))2+((ε+m+b)Q𝓹(t))2)Q𝓹∗Q𝓹2−(2(δI𝓹(t))(ε+m+b)Q𝓹(t))Q𝓹∗Q𝓹2+((δI𝓹(t))(ε+m+b))Q𝓹∗Q𝓹 −((ε+m+b)2Q𝓹(t))Q𝓹∗Q𝓹+((ε+m+b)2Q𝓹(t))−((δI𝓹(t))(ε+m+b))+((εQ𝓹(t))2+(bR𝓹(t))2)R𝓹∗R𝓹2−(2(εQ𝓹(t))(bR𝓹(t)))R𝓹∗R𝓹2+((bεQ𝓹(t)))R𝓹∗R𝓹−b(bR𝓹(t))R𝓹∗R𝓹+b(bR𝓹(t))−((bεQ𝓹(t))) +((Λ𝓱)2+(γS𝓱(t)I𝓹(t)e−μτ+μS𝓱(t))2)S𝓱∗S𝓱2−(2(Λ𝓱)(γS𝓱(t)I𝓹(t)e−μτ+μS𝓱(t)))S𝓱∗S𝓱2+((Λ𝓱)(γI𝓹(t)e−μτ+μ))S𝓱∗S𝓱−((γI𝓹(t)e−μτ+μ)2S𝓱(t))S𝓱∗S𝓱+((γI𝓹(t)e−μτ+μ)2S𝓱(t))−((Λ𝓱)(γI𝓹(t)e−μτ+μ))+((γS𝓱(t)I𝓹(t)e−μτ)2+((α+μ+β2)I𝓱(t))2)I𝓱∗I𝓱2− (2(γS𝓱(t)I𝓹(t)e−μτ)((α+μ+β2)I𝓱(t)))I𝓱∗I𝓱2+((γS𝓱(t)I𝓹(t)e−μτ)(α+μ+β2))I𝓱∗I𝓱−((α+μ+β2)2I𝓱(t))I𝓱∗I𝓱+((α+μ+β2)2I𝓱(t))−((γS𝓱(t)I𝓹(t)e−μτ)(α+μ+β2))+((β2I𝓱(t))2+((μ)R𝓱(t))2)R𝓱∗R𝓱2−(2(β2I𝓱(t))((μ)R𝓱(t)))R𝓱∗R𝓱2+((μβ2)I𝓱(t))R𝓱∗R𝓱−((μ)2R𝓱(t))R𝓱∗R𝓱+((μ)2R𝓱(t))−((μβ2)I𝓱(t))

For simplification, we choose d2Vdt2=ψ1−ψ2 ψ1=((Λ𝓹)2+(MβS𝓹(t)I𝓹(t)e−bτ+bS𝓹(t))2)S𝓹∗S𝓹2+((Λ𝓹)(MβI𝓹(t)e−bτ+b))S𝓹∗S𝓹+((MβI𝓹(t)e−bτ+b)2S𝓹)+((MβS𝓹(t)I𝓹(t)e−bτ)2+((δ+m+b)I𝓹(t))2)I𝓹∗I𝓹2+(2(MβS𝓹(t)I𝓹(t)e−bτ)(δ+m+b)I𝓹(t))I𝓹∗I𝓹+((MβS𝓹(t)I𝓹(t)e−bτ)2+(δ+m+b)2I𝓹(t))+((δI𝓹(t))2+((ε+m+b)Q𝓹(t))2)Q𝓹∗Q𝓹2+((δI𝓹(t))(ε+m+b))Q𝓹∗Q𝓹+((ε+m+b)2Q𝓹(t))+((εQ𝓹(t))2+(bR𝓹(t))2)R𝓹∗R𝓹2+((bεQ𝓹(t)))R𝓹∗R𝓹+b(bR𝓹(t))+((Λ𝓱)2+(γS𝓱(t)I𝓹(t)e−μτ+μS𝓱(t))2)S𝓱∗S𝓱2+((Λ𝓱)(γI𝓹(t)e−μτ+μ))S𝓱∗S𝓱+((γI𝓹(t)e−μτ+μ)2S𝓱(t))+((γS𝓱(t)I𝓹(t)e−μτ)2+((α+μ+β2)I𝓱(t))2)I𝓱∗I𝓱2+((γS𝓱(t)I𝓹(t)e−μτ)(α+μ+β2))I𝓱∗I𝓱+((α+μ+β2)2I𝓱(t))+((β2I𝓱(t))2+((μ)R𝓱(t))2)R𝓱∗R𝓱2+((μβ2)I𝓱(t))R𝓱∗R𝓱+((μ)2R𝓱(t)). ψ2=(2(Λ𝓹)(MβS𝓹(t)I𝓹(t)e−bτ+bS𝓹(t)))S𝓹∗S𝓹2+((MβI𝓹(t)e−bτ+b)2S𝓹)S𝓹∗S𝓹+((Λ𝓹)(MβI𝓹(t)e−bτ+b))+(2(MβS𝓹(t)I𝓹(t)e−bτ)(δ+m+b)I𝓹(t))I𝓹∗I𝓹2+((MβS𝓹(t)I𝓹(t)e−bτ)2+(δ+m+b)2I𝓹(t))I𝓹∗I𝓹+(2(MβS𝓹(t)I𝓹(t)e−bτ)(δ+m+b)I𝓹(t))+(2(δI𝓹(t))(ε+m+b)Q𝓹(t))Q𝓹∗Q𝓹2+((ε+m+b)2Q𝓹(t))Q𝓹∗Q𝓹+((δI𝓹(t))(ε+m+b))+(2(εQ𝓹(t))(bR𝓹(t)))R𝓹∗R𝓹2+b(bR𝓹(t))R𝓹∗R𝓹+((bεQ𝓹(t)))+(2(Λ𝓱)(γS𝓱(t)I𝓹(t)e−μτ+μS𝓱(t)))S𝓱∗S𝓱2+((γI𝓹(t)e−μτ+μ)2S𝓱(t))S𝓱∗S𝓱+((Λ𝓱)(γI𝓹(t)e−μτ+μ))+(2(γS𝓱(t)I𝓹(t)e−μτ)((α+μ+β2)I𝓱(t)))I𝓱∗I𝓱2+((α+μ+β2)2I𝓱(t))I𝓱∗I𝓱+((γS𝓱(t)I𝓹(t)e−μτ)(α+μ+β2))+(2(β2I𝓱(t))((μ)R𝓱(t)))R𝓱∗R𝓱2+((μ)2R𝓱(t))R𝓱∗R𝓱+((μβ2)I𝓱(t)).

It can see that ψ1>ψ2,d2Vdt2>0 ψ1<ψ2,d2Vdt2<0 ψ1=ψ2,d2Vdt2=0. ◻

 Sensitivity Analysis

This section examined the streptococcus suis model’s sensitivity. Sensity analysis is a study of how various factors related to input uncertainty may be attributed to the inconsistency of a mathematical model’s output outcomes. We calculate the sensitivity of the reproduction number concerning the model’s parameter. This technique provided the most sensitive measure for the reproduction number, which helped the infection spread. The basic format for sensitivity is as follows: D𝓹R=𝓹R×∂R∂𝓹where R depict the reproduction number while the 𝓹 present the parameter of the reproduction.

𝒰Λ𝓹=Λ𝓹ℛ0×∂ℛ0∂Λ𝓹=1>0, 𝒰M=Mℛ0×∂ℛ0∂M=1>0, 𝒰β=βℛ0×∂ℛ0∂β=1>0, 𝒰δ=δℛ0×∂ℛ0∂δ=−1(δ+m+b)<0, 𝒰b=bℛ0×∂ℛ0∂b=−(δ+m+2b)(δ+m+b)<0, 𝒰m=mℛ0×∂ℛ0∂m=−1(δ+m+b)<0.

The values of sensitivity and signs of the model’s parameters are presented in Table 1.

Sensitivity indices
Parameter Sensitivity indices Signs
Λ𝓹 1 Positive
M 1 Positive
β 1 Positive
δ −0.714285 Negative
b −0.714285 Negative
m −1.357 Negative

The most significant contributing aspect to the viral transmission phenomenon is human morality (b), as seen in Fig. 2, which has a negative connection with the fundamental reproduction number (ℛ0). With an increase in the pig mortality rate, the fundamental reproduction number ratio loses value. It suggests that as the number of afflicted pigs rises, so does the systemic infection level. This means that more research on the pig’s natural mortality rate analysis can be done, and it will become clearer why fewer pigs need to be infected. In pigs who have it, the most recent infectious virus is prevalent. “δ” represents the rate from infectious class to quarantine class in pigs, also “m” is the disease-induced pig death rate has negative effects on the reproduction number. On the other hand, there is a positive correlation between reproduction number and recruitment rate “Λ𝓹”, the relative humidity rate “M” and transmission rate “β”. The positive relationship shows that when the value of the parameter rises, so does the reproduction number. Consequently, it implies that decreasing the value of “Λ𝓹”, “M”, and “β”, can reduce the possibility of losing transmitted yield.

Analysis of the sensitive indices of reproduction number Stochastic Formulation Phase 1

The Stochastic delayed differential equations (SDDEs) of the streptococcus suis model (1)–(7) may be represented by the vector 𝒜=[S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t),S𝓱(t),I𝓱(t),R𝓱(t)]T. We wish to compute the variance E∗[Δ𝒜(Δ𝒜)T] and the expectation E∗[Δ𝒜]. Table 2 lists the probable changes together with the associated transition probability.Expectetions=E∗[Δ𝒜]=∑i=114Pi(Δ𝒜)i=[Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t)MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t)δI𝓹(t)−(ε+m+b)Q𝓹(t)εQ𝓹(t)−bR𝓹(t)Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t)γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t)β2I𝓱(t)−μR𝓱(t)]Δt Variance=∑i=114Pi(Δ𝒜)i[(Δ𝒜)i]T. =[P1+P2+P3−P200000−P2P2+P4+P5−P400000−P4P4+P6+P7−P600000−P6P6+P80000000P9+P10+P11000000−P10P10+P12+P13−P1300000−P13P13+P14]Δt Drift=G(𝒜,t)=E∗[Δ𝒜]Δt=[Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t)MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t)δI𝓹(t)−(ε+m+b)Q𝓹(t)εQ𝓹(t)−bR𝓹(t)Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t)γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t)β2I𝓱(t)−μR𝓱(t)]Δt Diffusion=H(𝒜,t)=E∗[Δ𝒜(Δ𝒜)T]Δt= [P1+P2+P3−P200000−P2P2+P4+P5−P400000−P4P4+P6+P7−P600000−P6P6+P80000000P9+P10+P11000000−P10P10+P12+P13−P1300000−P13P13+P14]

Potential modifications to the model’s procedure
Transition Probabilities
(ΔA)1=[1000000]T P1=(Λ𝓹)Δt
(ΔA)2=[−1100000]T P2=(MβS𝓹(t)I𝓹(t)e−bτ)Δt
(ΔA)3=[−1000000]T P3=(bS𝓹(t))Δt
(ΔA)4=[0−110000]T P4=(δI𝓹(t))Δt
(ΔA)5=[0−100000]T P5=((m+b)I𝓹(t))Δt
(ΔA)6=[00−11000]T P6=(εQ𝓹(t))Δt
(ΔA)7=[00−10000]T P7=((m+b)Q𝓹(t))Δt
(ΔA)8=[000−1000]T P8=(bR𝓹(t))Δt
(ΔA)9=[0000100]T P9=(Λ𝓱)Δt
(ΔA)10=[0000−110]T P10=(γS𝓱(t)I𝓹(t)e−μτ)Δt
(ΔA)11=[0000−100]T P11=(μS𝓱(t))Δt
(ΔA)12=[00000−10]T P12=((α+μ)I𝓱(t))Δt
(ΔA)13=[00000−11]T P13=(β2I𝓱(t))Δt
(ΔA)14=[000000−1]T P14=(μR𝓱(t))Δt

Therefore, d𝒜(t)=G(𝒜,t)+H(𝒜,t)dB(t). d[S𝓹I𝓹Q𝓹R𝓹S𝓱I𝓹R𝓱]=[Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t)MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t)δI𝓹(t)−(ε+m+b)Q𝓹(t)εQ𝓹(t)−bR𝓹(t)Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t)γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t)β2I𝓱(t)−μR𝓱(t)]dt+ [P1+P2+P3−P200000−P2P2+P4+P5−P400000−P4P4+P6+P7−P600000−P6P6+P80000000P9+P10+P11000000−P10P10+P12+P13−P1300000−P13P13+P14]dB(t)

By studying the relevant academic literature, the Euler Maruyama approach is employed to simulate the results of Eq. (10). Following is an outline of the data that is shown in Table 2. 𝒜n+1=𝒜n+G(𝒜n,t)Δt+H(𝒜n,t)dB(t). [S𝓹n+1I𝓹n+1Q𝓹n+1R𝓹n+1S𝓱n+1I𝓱n+1R𝓱n+1]=[S𝓹nI𝓹nQ𝓹nR𝓹nS𝓱nI𝓱nR𝓱n]+[Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t)MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t)δI𝓹(t)−(ε+m+b)Q𝓹(t)εQ𝓹(t)−bR𝓹(t)Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t)γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t)β2I𝓱(t)−μR𝓱(t)]Δt+ [[P1+P2+P3−P200000−P2P2+P4+P5−P400000−P4P4+P6+P7−P600000−P6P6+P80000000P9+P10+P11000000−P10P10+P12+P13−P1300000−P13P13+P14]Δt]ΔBnwhere the discretization parameter is denoted by Δt.

Stochastic Formulation Phase 2

Create an uncertainty parameter for the dynamical system (1)–(7) by including Brownian motion. dS𝓹(t)dt=Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t)+σ1S𝓹(t)dB(t)dtt≥0,τ<t dI𝓹(t)dt=MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t)+σ2I𝓹(t)dB(t)dtt≥0,τ<t dQ𝓹(t)dt=δI𝓹(t)−(ε+m+b)Q𝓹(t)+σ3Q𝓹(t)dB(t)dtt≥0 dR𝓹(t)dt=εQ𝓹(t)−bR𝓹(t)+σ4R𝓹(t)dB(t)dtt≥0 dS𝓱(t)dt=Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t)+σ5S𝓱(t)dB(t)dtt≥0,τ<t dI𝓱(t)dt=γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t)+σ6I𝓱(t)dB(t)dtt≥0,τ<t dR𝓱(t)dt=β2I𝓱(t)−μR𝓱(t)+σ7R𝓱(t)dB(t)dtt≥0where σi=1,2,3,4,5,6,7 denote the randomness of each compartment and B (t) indicates the Brownian motion.

 Fundamental Properties of the Stochastic Model

This part covers the analysis of the positivity and boundedness properties of the system (11)–(17).

Let us consider the vector as follows: U(t)=(S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t))andV(t)=(S𝓱(t),I𝓱(t),R𝓱(t))and norm |U(t)|=S𝓹2(t)+I𝓹2(t)+Q𝓹2(t)+R𝓹2(t)

And |V(t)|=S𝓱2(t)+I𝓱2(t)+R𝓱2(t)

Furthermore, let M13,1(R4x(0,∞):R+) and M23,1(R3x(0,∞):R+) represent the collection of all non-negative functions V1(U,t) and V2(U,t) that are defined on R4x(0,∞) consequently. Additionally, the function is twice differentiable in U and V and once differentiable in it. We have established the differentiable operator T1 and T2 that is linked to seven-dimensional stochastic delay differential equations (SDDEs). dU(t)=H1(U,t)dt+k1(U,t)dB(t) dV(t)=H2(V,t)dt+k2(V,t)dB(t)

As T1=∂∂t+∑i=14H1i(U,t)∂∂Ui+12∑i,j=14k1T(U,t)k1(U,t)∂2∂Ui∂Uj

And T2=∂∂t+∑i=13H2i(V,t)∂∂Vi+12∑i,j=13k2T(V,t)k2(V,t)∂2∂Vi∂Vj

If T1 and T2 acts on function U∗,V∗∈M13,1(R4x(0,∞):R+) then we denote T1U∗(U,t)=Ut∗(U,t)+UU∗(U,t)M1(U,t)+12Trace(k1T(U,t)UUU∗(U,t)k1(U,t)) T2V∗(V,t)=Vt∗(V,t)+VV∗(V,t)M2(V,t)+12Trace(k2T(V,t)VVV∗(V,t)k2(V,t))where T is Transportation.

Theorem 9 Shows that for the system (11)–(17) and any given initial conditions (S𝓹(0),I𝓹(0),Q𝓹(0),R𝓹(0))∈R+4, and (S𝓱(0),I𝓱(0),R𝓱(0))∈R+3 there are unique solutions (S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t)) and (S𝓱(t),I𝓱(t),R𝓱(t)) t≥0. Furthermore, these solutions will always remain in R+7 with a probability of one.

Proof. Given that all model parameters satisfy the local Lipschitz limitations. Thus, according to Ito’s formula, the provided model has a positive solution locally on the interval [0,τe], and the time of explosion is represented by τe. The model can be proven to have a global solution when τe is equal to infinity.

Let n0=0 be a sufficiently big value such that (S𝓹(0),I𝓹(0),Q𝓹(0),R𝓹(0)) and (S𝓱(0),I𝓱(0),R𝓱(0)) are all inside the interval {1n0,n0}. Let’s define a series for every non-negative integer n as follows: τn=inf{t∈[0,τe]:S𝓹(t)∈(1n,n),orI𝓹(t)∈(1n,n),orQ𝓹(t)∈(1n,n),orR𝓹(t)∈(1n,n),orS𝓱(t)∈(1n,n)orI𝓱(t)∈(1n,n),orR𝓱(t)∈(1n,n)}where we set infφ=∞(φisemptyset). Since τn is non-decreasing as n→∞, τ∞=limn→∞τn

The inequality states that τ∞ is less than or equal to τe. Now, we aim to demonstrate that τ∞ is equal to infinity, as intended.

If this condition fails to be satisfied, then there exist values T > 0 and b1∈(0,1) that satisfy the statement. U{τn≤T}≥b1∀n≥n1

Define a C4−function f:R+4→R+ by f(S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t))=(S𝓹−1−ln⁡S𝓹)+(I𝓹−1−ln⁡I𝓹)+(Q𝓹−1−ln⁡Q𝓹)+(R𝓹−1−ln⁡R𝓹)

Define a C3−function g:R+3→R+ by g(S𝓱(t),I𝓱(t),R𝓱(t))=(S𝓱−1−ln⁡S𝓱)+(I𝓱−1−ln⁡I𝓱)+(R𝓱−1−ln⁡R𝓱)

By using Ito’s formula (25), we calculate df(S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t))=(1−1S𝓹)dS𝓹+(1−1I𝓹)dI𝓹+(1−1Q𝓹)dQ𝓹+(1−1R𝓹)dR𝓹+σ12+σ22+σ32+σ422dt.

df(S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t))=(1−1S𝓹)((Λ𝓹−MβS𝓹(t)I𝓹(t)e−bτ−bS𝓹(t))dt+σ1S𝓹(t)dB(t))+(1−1I𝓹)((MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t))dt+σ2I𝓹(t)dB(t))+(1−1Q𝓹)((δI𝓹(t)−(ε+m+b)Q𝓹(t))dt+σ3Q𝓹(t)dB(t))+(1−1R𝓹)((εQ𝓹(t)−bR𝓹(t))dt+σ4R𝓹(t)dB(t)) df(S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t))=(Λ𝓹+4b+2m+δ+ε+σ12+σ22+σ32+σ422)dt+σ1S𝓹(t)d(B(t)+σ2I𝓹(t)d(B(t)+σ3Q𝓹(t)d(B(t)+σ4R𝓹(t)d(B(t)

For simplify, we assume M1=(Λ𝓹+4b+2m+δ+ε+σ12+σ22+σ32+σ422). Then Eq. (27) could be written as df(S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t))≤M1dt+(σ1S𝓹(t)+σ2I𝓹(t)+σ3Q𝓹(t)+σ4R𝓹(t))d(B(t)).where M1 is a positive constant, after that integrating from 0 to τn∧τ, we get ∫0τn∧τdf(S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t))≤∫0τn∧τM1dt+∫0τn∧τ(σ1S𝓹(t)+σ2I𝓹(t)+σ3Q𝓹(t)σ4R𝓹(t))d(B(t))where τn∧τ=min(τn,T), the taking the expectations lead to EU∗(S𝓹(τn∧τ),I𝓹(τn∧τ),Q𝓹(τn∧τ),R𝓹(τn∧τ))≤U∗(S𝓹(0),I𝓹(0),Q𝓹(0),R𝓹(0))+M1T.

Set Ωn={τn≤T} for n>n1 and from (18), we have X(Ωn≥b).

For each element a1 in the set Ωn, there exist certain indices 𝑖 such that Ui(τn,a1) is equal to either n or 1n, where 𝑖 takes on the values 1, 2, 3, 4.

Hence,U∗((S𝓹(τn,a1),I𝓹(τn,a1),Q𝓹(τn,a1),R𝓹(τn,a1)))islessthanmin{n−1−ln⁡n,1n−1 −ln⁡1n}.

Next, we obtain U∗(S𝓹(0),I𝓹(0),Q𝓹(0),R𝓹(0))+M1T≥E(IΩm(a1)U∗(S𝓹(τn),I𝓹(τn),Q𝓹(τn),R𝓹(τn)))≥min{n−1−ln⁡n,1n−1−ln⁡1n}

The indicator function is denoted as IΩn(a1) within the set Ωn. As n approaches infinity, we get there to the contradiction that infinity is equal to the value of U∗(S𝓹(0),I𝓹(0),Q𝓹(0),R𝓹(0))+M1T, which is finite.

As desired.

Again, by using Ito’s formula (26), we calculate dg(S𝓱(t),I𝓱(t),R𝓱(t))=(1−1S𝓱)dS𝓱+(1−1I𝓱)dI𝓱+(1−1R𝓱)dR𝓱+σ52+σ62+σ722dt dg(S𝓱(t),I𝓱(t),R𝓱(t))=(1−1S𝓱)((Λ𝓱−γS𝓱(t)I𝓹(t)e−μτ−μS𝓱(t))dt+σ5S𝓱(t)dB(t))+(1−1I𝓱)((γS𝓱(t)I𝓹(t)e−μτ−(α+μ+β2)I𝓱(t))dt+σ6I𝓱(t)dB(t))+(1−1R𝓱)((β2I𝓱(t)−μR𝓱(t))dt+σ7R𝓱(t)dB(t)) dg(S𝓱(t),I𝓱(t),R𝓱(t))=(Λ𝓱+3μ+α+β2+σ52+σ622)dt+σ5S𝓱(t)dB(t)+σ6I𝓱(t)dB(t)+σ7R𝓱(t)dB(t)

For simplify, we assume M2=(Λ𝓱+3μ+α+β2+σ52+σ62+σ722), Then Eq. (32) could be written as dg(S𝓱(t),I𝓱(t),R𝓱(t))≤M2dt+(σ5S𝓱(t)+σ6I𝓱(t)+σ7R𝓱(t))d(B(t))where M2 is a positive constant, after that integrating from 0 to τn∧τ, we get ∫0τn∧τdg(S𝓱(t),I𝓱(t),R𝓱(t))≤∫0τn∧τM2dt+∫0τn∧τ(σ5S𝓱(t)+σ6I𝓱(t)+σ7R𝓱(t))d(B(t))where τn∧τ=min(τn,T), the taking the expectations lead to EV∗(S𝓱(τn∧τ),I𝓱(τn∧τ),R𝓱(τn∧τ))≤V∗(S𝓱(0),I𝓱(0),R𝓱(0))+M2T

Hence, V∗((S𝓱(τn,V1),I𝓱(τn,V1),R𝓱(τn,V1))) is less than min{n−1−ln⁡n,1n−1−ln⁡1n}.

Next, we obtain V∗(S𝓱(0),I𝓱(0),R𝓱(0))+M2T≥E(I𝓱Ωm(v1)V∗(S𝓱(τn),I𝓱(τn),R𝓱(τn)))≥min{n−1−ln⁡n,1n−1−ln⁡1n}

The indicator function is denoted as I𝓱Ωn(v1) within the set Ωn. As n approaches infinity, we get there to the contradiction that infinity is equal to the value of V∗(S𝓱(0),I𝓱(0),R𝓱(0))+M2T, which is finite.

As desired. □

Theorem 10. If the spectral radius v and the variance σ22<MβΛ𝓹e−bτb(δ+m+b), then the number of infected pig population in the system (11)–(17) will exponentially approach zero.

Proof: Let’s examine the initial data (S𝓹(0),I𝓹(0),Q𝓹(0),R𝓹(0),S𝓱(0),I𝓱(0),R𝓱(0))∈R+7 and the system (11)–(17) has a solution (S𝓹(t),I𝓹(t),Q𝓹(t),R𝓹(t),S𝓱(t),I𝓱(t),R𝓱(t)) if it satisfies the stochastic delayed differential equation, where σ represents randomness and c represents drift. dI𝓹(t)=(MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t))dt+cσ2I𝓹(t)dB(t)

Applying Ito’s lemma to the function f(I𝓹) = ln(I𝓹), we obtain dln(I𝓹(t))=g′(I𝓹(t))dP+12g′′(I𝓹)I𝓹2σ22dt. dln⁡(I𝓹(t))=1I𝓹(t)dI𝓹+12(−1I𝓹2)I𝓹2σ22dt. dln⁡(I𝓹(t))=1I𝓹(t)dI𝓹−12σ22dt. dln⁡(I𝓹(t))=1I𝓹(t)[(MβS𝓹(t)I𝓹(t)e−bτ−(δ+m+b)I𝓹(t))dt+cσ2I𝓹(t)dB(t)]−12σ22dt. dln⁡(I𝓹(t))=(MβS𝓹(t)e−bτ−(δ+m+b))dt+cσ2dB(t)−12σ22dt. ln⁡(I𝓹(t))=ln⁡I𝓹(0)+(MβS𝓹(t)e−bτ−(δ+m+b)−12σ22)dt+∫0tcσ2dB(t),

Notice that, N(t)=∫0tcσ2dB(t) with N(0)=0.

If σ22>MβΛ𝓹e−bτb(δ+m+b), ln⁡(I𝓹(t))>(MβΛ𝓹e−bτb(δ+m+b)−(δ+m+b)−12MβΛ𝓹e−bτb(δ+m+b))t+N(t)+ln⁡I𝓹(0), ln⁡I𝓹(t)t>(MβΛ𝓹e−bτ2b(δ+m+b)−(δ+m+b))+N(t)t+ln⁡I𝓹(0)t,

limt→∞ln⁡I𝓹(t)t>(MβΛ𝓹e−bτ2b(δ+m+b)−(δ+m+b))>0, with limt→∞N(t)t=0,

If σ22<MβΛ𝓹e−bτb(δ+m+b), then ln⁡(I𝓹(t))<(MβΛ𝓹e−bτb(δ+m+b)−(δ+m+b)−12σ22)t+N(t)+ln⁡I𝓹(0), ln⁡I𝓹(t)t<(δ+m+b)(MβΛ𝓹e−bτb(δ+m+b)2−1−12σ22)+N(t)t+ln⁡I𝓹(0)t,

limt→∞supln⁡I𝓹(t)t<(δ+m+b)(ℛ0S−1), when ℛ0S<1, we get limt→∞supln⁡I𝓹(t)t≤0,

limt→∞I𝓹(t)=0, as desired. ℛoS=ℛod−σ222(δ+m+b)<1. ◻

 Stochastic Nonstandard Finite Difference Scheme

The NSFD method is used in this analysis to address the stochastic delay differential equations regulating Streptococcus suis dynamics. The discrete approximations are very carefully selected to keep stability and be able to precisely model the turbulent behavior of the system. This ensures that the theoretical analysis is kept consistent and that the numerical solutions stay physically significant. For (11)–(17), the stochastic non-standard finite difference scheme has the following equation: S𝓹n+1=S𝓹n+h(Λ𝓹+σ1S𝓹nΔBn)1+h(MβI𝓹ne−bτ+b) I𝓹n+1=I𝓹n+h(MβS𝓹nI𝓹ne−bτ+σ2I𝓹nΔBn)1+h(δ+m+b) Q𝓹n+1=Q𝓹n+h(δI𝓹n+σ3Q𝓹nΔBn)1+h(ε+m+b) R𝓹n+1=R𝓹n+h(εQ𝓹n+σ4R𝓹nΔBn)1+hb S𝓱n+1=S𝓱n+h(Λ𝓱+σ5S𝓱nΔBn)1+h(γI𝓹ne−μτ+μ) I𝓱n+1=I𝓱n+h(γS𝓱nI𝓹ne−μτ+σ6I𝓱nΔBn)1+h(α+μ+β2) R𝓱n+1=R𝓱n+h(β2I𝓱n+σ7R𝓱nΔBn)1+hμwhere h represents a discretization parameter and n is a non-negative integer.

Stability Analysis

Assuming ΔBn=0, the system (37)–(43) consists of functions A,B,C,D,andE.

A=S𝓹+h(Λ𝓹)1+h(MβI𝓹e−bτ+b), B=I𝓹+h(MβS𝓹I𝓹e−bτ)1+h(δ+m+b), C=Q𝓹+h(δI𝓹)1+h(ε+m+b), D=R𝓹+h(εQ𝓹)1+hb, E=S𝓱+h(Λ𝓱)1+h(γI𝓹e−μτ+μ),

F=I𝓱+h(γS𝓱I𝓹e−μτ)1+h(α+μ+β2), G=R𝓱+h(β2I𝓱)1+hμ.

The Jacobian matrix consists of the following elements:

∂A∂S𝓹=11+h(MβI𝓹e−bτ+b), ∂A∂I𝓹=−MβI𝓹e−bτ(hΛ𝓹)(1+h(MβI𝓹e−bτ+b))2, ∂A∂Q𝓹=0, ∂A∂R𝓹=0, ∂A∂S𝓱=0, ∂A∂I𝓱=0, ∂A∂R𝓱=0

∂B∂S𝓹=h(MβI𝓹e−bτ)1+h(δ+m+b), ∂B∂I𝓹=1+h(MβS𝓹e−bτ)1+h(δ+m+b), ∂B∂Q𝓹=0, ∂B∂R𝓹=0, ∂B∂S𝓱=0, ∂B∂I𝓱=0, ∂B∂R𝓱=0

∂C∂S𝓹=0, ∂C∂I𝓹=h(δ)1+h(ε+m+b), ∂C∂Q𝓹=11+h(ε+m+b), ∂C∂R𝓹=0, ∂C∂S𝓱=0, ∂C∂I𝓱=0, ∂C∂R𝓱=0

∂D∂S𝓹=0, ∂D∂I𝓹=0, ∂D∂Q𝓹=h(ε)1+hb, ∂D∂R𝓹=11+hb, ∂D∂S𝓱=0, ∂D∂I𝓱=0, ∂D∂R𝓱=0

∂E∂S𝓹=0, ∂E∂I𝓹=−γhe−μτ(hΛ𝓱)(1+h(γI𝓹e−μτ+μ))2, ∂E∂Q𝓹=0, ∂E∂R𝓹=0, ∂E∂S𝓱=11+h(γI𝓹e−μτ+μ), ∂E∂I𝓱=0, ∂E∂R𝓱=0

∂F∂S𝓹=0, ∂F∂I𝓹=h(γS𝓱e−μτ)1+h(α+μ+β2), ∂F∂Q𝓹=0, ∂F∂R𝓹=0, ∂F∂S𝓱=h(γI𝓹e−μτ)1+h(α+μ+β2), ∂F∂I𝓱=11+h(α+μ+β2), ∂F∂R𝓱=0

∂G∂S𝓹=0, ∂G∂I𝓹=0, ∂G∂Q𝓹=0, ∂G∂R𝓹=0, ∂G∂S𝓱=0, ∂G∂I𝓱=h(β2)1+hμ, ∂G∂R𝓱=11+hμ

Theorem 11. For all values of n≥0, the eigenvalues of the Jacobian matrix at the streptococcus suis-free equilibrium for the system (37)–(43) are located within the unit circle if the value of ℛ0<1.

Proof. The Jacobian matrix at the streptococcus suis-free equilibrium, denoted as (S𝓹0,I𝓹0,Q𝓹0,R𝓹0,S𝓱0,I𝓱0,R𝓱0), can be expressed as (Λ𝓹b,0,0,0,Λ𝓱μ,0,0). J(D0)=[11+hb00000001+h(MβS𝓹0e−bτ)1+h(δ+m+b)000000h(δ)1+h(ε+m+b)11+h(ε+m+b)000000h(ε)1+hb11+hb0000−γhe−μτ(hΛ𝓱)(1+hμ)20011+hμ000h(γS𝓱0e−μτ)1+h(α+μ+β2)00011+h(α+μ+β2)000000h(β2)1+hμ11+hμ] |J(D0)−λ|=|11+hb−λ00000001+h(MβS𝓹0e−bτ)1+h(δ+m+b)−λ000000h(δ)1+h(ε+m+b)11+h(ε+m+b)−λ000000h(ε)1+hb11+hb−λ0000−γhe−μτ(hΛ𝓱)(1+hμ)20011+hμ−λ000h(γS𝓱0e−μτ)1+h(α+μ+β2)00011+h(α+μ+β2)−λ000000h(β2)1+hμ11+hμ−λ|

Therefore,

λ1=λ4=11+hb<1, λ2=1+h(MβS𝓹0e−bτ)1+h(δ+m+b)<1, λ3=11+h(ε+m+b)<1, λ5=λ7=11+hμ<1, λ6=11+h(α+μ+β2).

Using the definition of ℛ0, we can show that if ℛ0<1, then λ2<1, and D0 is L.A.S. on the contrary, it is obviously to verify that λ2>1, if ℛ0>1, which shows that D0 is unstable. □

Theorem 12. For all values of n≥0, the eigenvalues of the Jacobian matrix at the streptococcus suis-endemic equilibrium for the system (37)–(43) are located within the unit circle if the value of ℛ0>1.

Proof. The Jacobian matrix at the streptococcus suis- endemic equilibrium, denoted as (S𝓹∗,I𝓹∗,Q𝓹∗,R𝓹∗,S𝓱∗,I𝓱∗,R𝓱∗). J(D∗)=[11+h(MβI𝓹∗e−bτ+b)−MβI𝓹∗e−bτ(hΛ𝓹)(1+h(MβI𝓹∗e−bτ+b))200000h(MβI𝓹∗e−bτ)1+h(δ+m+b)1+h(MβS𝓹∗e−bτ)1+h(δ+m+b)000000h(δ)1+h(ε+m+b)11+h(ε+m+b)000000h(ε)1+hb11+hb0000−γhe−μτ(hΛ𝓱)(1+h(γI𝓹∗e−μτ+μ))20011+h(γI𝓹∗e−μτ+μ)000h(γS𝓱∗e−μτ)1+h(α+μ+β2)00h(γI𝓹∗e−μτ)1+h(α+μ+β2)11+h(α+μ+β2)000000h(β2)1+hμ11+hμ]

So, the eigenvalues of the Jacobian at D∗ as follows:

λ1=11+hμ<1, λ2=11+h(α+μ+β2),λ3=11+h(γI𝓹∗e−μτ+μ),λ4=11+hb,λ5=11+h(ε+m+b) provided that R0>1. |11+h(MβI𝓹∗e−bτ+b)−λ−MβI𝓹∗e−bτ(hΛ𝓹)(1+h(MβI𝓹∗e−bτ+b))2h(MβI𝓹∗e−bτ)1+h(δ+m+b)1+h(MβS𝓹∗e−bτ)1+h(δ+m+b)−λ|=0 A1=TrceofJD∗=11+h(MβI𝓹∗e−bτ+b)+1+h(MβS𝓹∗e−bτ)1+h(δ+m+b) A2=DeterminentofJD∗=((11+h(MβI𝓹∗e−bτ+b))(1+h(MβS𝓹∗e−bτ)1+h(δ+m+b)))+(MβI𝓹∗e−bτ(hΛ𝓹)(1+h(MβI𝓹∗e−bτ+b))2)(h(MβI𝓹∗e−bτ)1+h(δ+m+b))

Lemma. For the quadratic equation λ2−A1λ+A2=0, |λi|<1, i=1,2. if and only if the following conditions are satisfied:

(i) 1+A1+A2>0.

(ii) 1−A1+A2>0.

(iii) A2<1. □

 Comparison Section

This section examines the characteristics of the graphs representing the number of infected pig population using the Euler Maruyama, stochastic Euler, and stochastic Runge Kutta schemes, in comparison to the NSFD scheme, across various step sizes and parameters values (See Table 3).

Values of parameter
Parameters Values Source [13]
Λ𝓹 0.5 Fitted
Λ𝓱 5 Fitted
β 2.3 (SSEE) 1.3 (SSFE) Estimated
β2 0.1 Estimated
μ 0.01 Estimated
M 1 Fitted
γ 0.1 Fitted
ε 0.3 Fitted
b 0.5 Estimated
m 0.3 Fitted
δ 0.6 Estimated
α 0.1 Estimated
Discussion

Fig. 3a,b provides a comparison between the infected class of the Stochastic NSFD and the Euler Maryama Method. Fig. 3a shows convergence for both approaches at ℎ = 0.01. When the step size was raised to ℎ = 1.0, the Euler Maryama Method diverged whereas the Stochastic NSFD Method remained convergent, as shown in Fig. 3b. Similarly, Fig. 3c,d compares the infected class of the Stochastic NSFD and the Stochastic Euler Method. At ℎ = 0.01, both techniques converged in Fig. 3c. However, when the step size was increased to ℎ = 1.0, the Stochastic Euler Method diverged, while the Stochastic NSFD method-maintained convergence, as shown in Fig. 3d. Similarly, Fig. 3e,f compares the infected class of the Stochastic NSFD and Stochastic RK Method. At ℎ = 0.01, both methods converged, as shown in Fig. 3e. However, when the step size increased to ℎ = 2.0, the Stochastic RK Method diverged, while the Stochastic NSFD method continued to converge, as shown in Fig. 3f. Fig. 4a shows how delay affects the model’s susceptible class at different 𝜏 values (0.1, 0.2, 0.3, 0.4, 0.5). Fig. 4b shows the effect of delay on the infected class of the model at various values 𝜏 = 0.1, 0.2, 0.3, 0.4, and 0.5, indicating a gradual decline in disease from the infected class over time. Finally, Fig. 5 shows the behavior of delay on the reproduction number of the model.

Comparison graph of computational methods at the Streptococcus Suis endemic equilibrium of the model (a) The comparison behavior of the infected pig population through Euler Maruyama and stochastic NSFD methods at <inline-formula id="ieqn-367"><mml:math id="mml-ieqn-367"><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn></mml:math></inline-formula> (convergent) (b) The comparison behavior of the infected pig population through Euler Maruyama and stochastic NSFD methods at <inline-formula id="ieqn-368"><mml:math id="mml-ieqn-368"><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> (divergent) (c) The comparison behavior of the infected pig population through stochastic Euler and stochastic NSFD methods at <inline-formula id="ieqn-369"><mml:math id="mml-ieqn-369"><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn></mml:math></inline-formula> (convergent) (d) The comparison behavior of the infected pig population through stochastic Euler and stochastic NSFD methods at <inline-formula id="ieqn-370"><mml:math id="mml-ieqn-370"><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math></inline-formula> (divergent) (e) The comparison behavior of the infected pig population through stochastic Runge Kutta and stochastic NSFD methods at <inline-formula id="ieqn-371"><mml:math id="mml-ieqn-371"><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.01</mml:mn></mml:math></inline-formula> (convergent) (f) The comparison behavior of the infected pig population through stochastic Runge Kutta and stochastic NSFD methods at <inline-formula id="ieqn-372"><mml:math id="mml-ieqn-372"><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math></inline-formula> (divergent) Time-Plot with the time delay on susceptible and infected population. (a) The effect of different values of delay on susceptible pig population (b) The effect of different values of delay on infected pig population Time plot of the effect of time delay <inline-formula id="ieqn-373"><mml:math id="mml-ieqn-373"><mml:mrow><mml:mo>(</mml:mo><mml:mi>τ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></inline-formula> with reproduction number <inline-formula id="ieqn-374"><mml:math id="mml-ieqn-374"><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mrow><mml:mi>ℛ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> Conclusion

This paper provides a comprehensive assessment of the mathematical analysis, including trustworthy delay techniques, of the delayed model for streptococcus suis infection. Subpopulations are classified by the model into four categories: susceptible class S𝓹(t), infectious class I𝓹(t), quarantine class Q𝓹(t), and recovered class R𝓹(t). Because Streptococcus Suis may spread from pig to people, the model includes the susceptible human class S𝓱(t), infectious human class I𝓱(t), and recovered class R𝓱(t). The model’s dynamic analysis examines positivity, boundedness, equilibria, and the threshold parameter. The sensitivity of the parameters is revealed by the outcomes of the model. The linearization of the model is based on existing concepts such as the Routh-Hurwitz criteria and the Jacobian. The focus of the research is on the use of Lassalle’s invariance principle and Lyapunov’s theory to ensure the global stability of the model. It is discovered that the Stochastic delayed NSFD method is the most accurate, successful, and efficient method. In these models, stability is necessary to avoid unpredictable behavior and incorrect results in terms of stability, optimism, and staying within normal bounds even with enormous time increments. Stochastic delayed NFSD performs exceptionally well. Other methods such as Stochastic delayed NFSD Euler Maryama, Stochastic Euler, and Stochastic RK-4 are considered valuable tools in our toolbox, however, at high time scales, they break down, leading to a loss of stability and consistency. After a great deal of testing and comparison, Stochastic delayed NFSD has emerged as the champion in stability and reliability, passing important tests like “local stability” and the Routh-Hurwitz criterion in the study of accurate predictions. This model can be taken forward by adding some dynamism in terms of space to real-world data for parameter estimation, thereby making it more applicable. It could also be taken up in terms of other diseases so that the applicability of the model further validates its relevance.

To fully capture the real-world complexities, there is a need to develop a model without the assumption of disease transmissibility and parameter estimation. Moreover, validation based on data is needed to enhance the reliability of the model.

Not applicable.

 Funding Statement

This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia [KFU250259].

 Author Contributions

Umar Shafique, Ali Raza, Dumitru Baleanu, Khadija Nasir, Muhammad Naveed, Abu Bakar Siddique and Emad Fadhal reviewed the results and approved the final version of the manuscript. All authors reviewed the results and approved the final version of the manuscript.

 Availability of Data and Materials

All the data used and analyzed is available in the manuscript.

 Ethics Approval

Not applicable.

 Conflicts of Interest

The authors declare no conflicts of interest to report regarding the present study.

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