Computational Modeling of Streptococcus Suis Dynamics via Stochastic Delay Differential Equations

1  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.

2  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(1)

dI𝓹(t)dt=MβS𝓹(t−τ)I𝓹(t−τ)e−bτ−(δ+m+b)I𝓹(t)t≥0,τ<t(2)

dQ𝓹(t)dt=δI𝓹(t)−(ε+m+b)Q𝓹(t)t≥0(3)

dR𝓹(t)dt=εQ𝓹(t)−bR𝓹(t)t≥0(4)

dS𝓱(t)dt=Λ𝓱−γS𝓱(t−τ)I𝓹(t−τ)e−μτ−μS𝓱(t)t≥0,τ<t(5)

dI𝓱(t)dt=γS𝓱(t−τ)I𝓹(t−τ)e−μτ−(α+μ+β2)I𝓱(t)t≥0,τ<t(6)

dR𝓱(t)dt=β2I𝓱(t)−μR𝓱(t)t≥0(7)

Figure 1: SIQR-SIR model diagram for people and pigs [13]

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.

3  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.

3.1 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. □

3.2 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).

3.3 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. □

3.4 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