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The Mathematical Model for Streptococcus suis Infection in Pig-Human Population with Humidity Effect

1Department of Mathematics, Faculty of Science, Mahasarakham University, Mahasarakham, 44150, Thailand
2Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen, 40002, Thailand
*Corresponding Author: Din Prathumwan. Email: dinpr@kku.ac.th
Received: 17 July 2021; Accepted: 27 September 2021

Abstract: ﻿In this paper, we developed a mathematical model for Streptococcus suis, which is an epidemic by considering the moisture that affects the infection. The disease is caused by Streptococcus suis infection found in pigs which can be transmitted to humans. The patients of Streptococcus suis were generally found in adults males and the elderly who contacted pigs or who ate uncooked pork. In human cases, the infection can cause a severe illness and death. This disease has an impact to the financial losses in the swine industry. In the development of models for this disease, we have divided the population into 7 related groups which are susceptible pig compartment, infected pig compartment, quarantined pig compartment, recovered pig compartment, susceptible human compartment, infected human compartment, and recovered human compartment. After that, we use this model and a quarantine strategy to analyze the spread of the infection. In addition, the basic reproduction number R0 is determined by using the next-generation matrix which can analyze the stability of the model. The numerical simulations of the proposed model are illustrated to confirm the results from theorems. The results showed that there is an effect from moisture to the disease transmission. When the moisture increases the disease infection also increases.

Keywords: Streptococcus suis; mathematical model; stability analysis; infectious disease; reproduction number

1  Introduction

Nowadays, one of the major problems in the world is zoonotic pathogen such as Rift Valley fever, SARS, pandemic influenza H1N1 2009, Yellow fever, Avian Influenza (H5N1) and (H7N9), West Nile virus [1], the Middle East respiratory syndrome coronavirus (MERSCoV) [2], and novel coronavirus (COVID-19) [3]. Streptococcus suis is an infectious agent which is widely found in pigs around the world. This pathogen is generally found in the upper respiratory tract, genital, and alimentary tracts of the pigs [4,5]. Streptococcus suis is a Gram-positive bacterium and a facultative anaerobic bacterium which is a primary agent of sepsis and meningitis in pigs [6,7]. In pig farms, the disease can transmit from pig to pig rapidly. The reported rate of mortality in pigs from Streptococcus suis is about 20% [8]. It can classify at least 35 serotypes, for examples, serotypes 1 to 31, 33, and 1/2 [9,10]. Serotypes 1 to 9 and 14 are most commonly found in pigs around the world [11,12].

In 1968, the first human who was infected with Streptococcus suis was found in Denmark [4,5]. This situation confirms that Streptococcus suis can transmit from pig to human. Serotype 2 is reported that it is frequently found in human infections [13,14], but in some cases, the infections are caused by other serotypes. Kerdsin et al. reported that serotypes 2 and 14 are involved in human cases that found in Thailand [15,16]. Dutkiewicz et al. presented that disease found in human caused by S. suis serotypes 2, 4, 5, 14, 16, 21 and 24 [12]. S. suis serotype 9 which is the most common found in pigs in Europe was found the first case infection in human in May 2013 locating in northern of Thailand [17]. In addition, Hatrongjit et al. reported the first case of serotype 31 in human [18] in the central region of Thailand. The common symptoms of this disease in human are fever, headache, meningitis, septicemia, arthritis, pneumonia, and hearing loss [19].

To prevent the spread of the disease, the mathematical models become powerful tools for describing the dynamics of the disease. They can forecast the future behaviors of the disease with many assumptions. The solutions of the model can be simulated satisfying the given parameters from the hypothesis. Recently, many mathematical models have been used to describe the behavior of infectious disease [2024]. Rahman et al. proposed the nonlinear SEITR fractional order model of tuberculosis disease by using Atangana-Baleanu derivative which consists of 5 population groups [25]. Arfan et al. [26] presented the fractional order model to predict the dynamic of tumor with drug intervention. The nonlinear fractional order mathematical model for predicting the dynamic of COVID-19 was created by Shah et al. [27]. Almuqrin et al. proposed the fractional model for forecasting the transmission of Ebola virus in bats population by using Atangana-Baleanu fractional derivative [28]. Srivastava et al. improved the mathematical model for studying the dynamic of diabetes patients [29]. Alzaid et al. [30] studied the dynamic of HIV-1 infection by using the mathematical model and showed that the mathematical model is efficient . The fractional order model of HIV with source term was proposed by Shah et al. to consider the behavior of CD4 + T-cells which depending on the concentration of the viral load [31]. The symmetry and asymmetry concepts can be linked to the epidemic model [3234].

A number of research studied the disease from Streptococcus suis on epidemiology and medicine. However, a few pieces of the research proposed and studied the mathematical model for Streptococcus suis. Shen et al. [35] proposed the SIQRW model to explore the outbreaks of S. Suis. Giang et al. proposed the stochastic model and SEI model to predict the behavior of the disease and fitted the model parameters with collected data [9]. The proposed models consider the transmission on pig population only. However, to our knowledge, no research considering the disease transmission between pigs and humans for Streptococcus suis.

In this paper, we propose a mathematical model to describe the disease transmission of Streptococcus suis in humans and pigs. The novelty of this work is the improved mathematical model from one species to two species population. We classify the population of pigs and humans into 4 classes and 3 classes, respectively. This work contains the analysis of the model and numerical simulations to study the dynamic behavior of the disease. We also find the basic reproduction numbers to explain the rapid transmission and study the stability of the solution of the model.

This paper is constructed as follows; In Section 2, we present the mathematical model which expresses the Streptococcus suis transmission. The analysis of the proposed model is shown in Section 3 which includes a basic reproduction number, equilibrium points, and their stability. The numerical simulations are given in Section 4 and follow by a conclusion in Section 5.

2  Model Formulation

In this section, we describe the model formulation of Streptococcus suis disease transmission in humans and pigs. Based on the classical epidemiology model, we propose a new generalized model of Streptococcus suis infection which is SIQR-SIR model. There are some facts about the environmental factors such as temperature and relative humidity [3638]. So, we consider the effect of the moisture in the air for disease transmission in the pig farm.

The investigated population is divided into two subpopulations. These are pig population and human population. Then, both subpopulations of pigs and human are separated into four classes and three classes, respectively. These are a pig susceptible class (Sp), pig infectious class (Ip), pig isolated class (Qp), pig recovery class (Rp), human susceptible class (Sh), human infectious class (Ih), and human recovery class (Rp). We assume that the total population at time t is N(t). We have that

Sp(t)+Ip(t)+Qp(t)+Rp(t)=Np(t), Sh(t)+Ih(t)+Rh(t)=Nh(t),

and

Sp(t)+Ip(t)+Qp(t)+Rp(t)+Sh(t)+Ih(t)+Rh(t)=Np(t)+Nh(t)=N(t),

where Np(t) is the total population of pigs, and Nh(t) is the total population of human at time t.

We can write the dynamic equations for Np(t) and Nh(t) as the following.

dNpdt=dSpdt+dIpdt+dQpdt+dRpdt, dNhdt=dShdt+dIhdt+dRhdt.

The new infection in pigs can be described . where β1 is the transmission coefficient per unit of time per pig in the susceptible class contact with infectious class [39]. The new infection in human can be described by β2ShIp+β3ShIh where β2 is the transmission coefficient per unit of time per person in the susceptible class in contact with pig infectious class, β3 is the transmission coefficient per unit of time per person in the susceptible class contact with the infectious class. In our model, we investigate the relative humidity for considering the infection of this disease in pigs. Then, the new infection in pigs can be expressed by β1MSpIp instead of β1SpIp, where M is the moisture in the air (relative humidity). In addition, we assume that the disease cannot be transmitted from human to pigs. Throughout this paper, we assume that all parameters are positive constants.

The model of disease transmission by Streptococcus suis can be represented by the system of differential equations as follows:

dSpdt=N1bSpβ1MSpIp,(1a)

dIpdt=β1MSpIpaIpbIpδIp,(1b)

dQpdt=δIpaQpbQpϵQp,(1c)

dRpdt=ϵQpbRp,(1d)

dShdt=N2β2ShIpβ3ShIhμSh,(1e)

dIhdt=β2ShIp+β3ShIhαIhγIhμIh,(1f)

dRhdt=γIhμRh,(1g)

whereb is the pig removal rate, a is the pig death rate induced by the disease, δ is the rate from infectious class to isolated class in pigs,e is the transition rate from isolated class to recovery class, μ is the human natural death rate,α is the death rate induced by disease, and γ is the transition rate from infectious class to recovery class. The model form is valid only if ab and αμ.

Eq. (1a) expresses the rate of change of the population of susceptible pig. The number of susceptible pigs increases only via birth N1 described in the first term on the right-hand side. On the other hand, the population of them decreases due to the removal rate of pig (b), transmission of Streptococcus suis from infected pig to susceptible pig (β1) and moisture effect (M).

Eq. (1b) expresses the rate of change of the population of the infected pig. On the right-hand side, the first term represents the number of infected pigs increases due to susceptible pig becoming infected pig with the rate β1M followed by the negative effects of disease death rate (a), removal rate (b), and isolated rate (δ).

Eq. (1c) expresses the rate of change of the population of the isolated pig. On the right-hand side, the first term represents the increase in the population of isolated pig due to the isolation of infected pig with the rate δ followed by the negative effects of disease death rate (a), removal rate (b), and recovered rate (ϵ).

Eq. (1d) expresses the rate of change of the population of the recovered pig. On the right-hand side, the first term represents the increase in the population of the recovered pig due to treatment with the rate ϵ, and the last term represents the removal rate (b) of pigs.

Eq. (1e) expresses the rate of change of the population of susceptible human. The number of susceptible pigs increases only via birth N2 described in the first term on the right-hand side. On the other hand, the population of them decreases due to the transmission of Streptococcus suis from infected pig to susceptible human (β2) and infected human to susceptible human (β3). The last term represents the natural death rate (μ) of human.

Eq. (1f) expresses the rate of change of the population of the infected human. On the right-hand side, the first term and second term represent the number of the infected human increases due to susceptible human becoming infected human by contact with the infected pigs (β2) and the infected human (β3), respectively. The remaining terms are the negative effects of disease death rate (α), recovered rate (γ), and natural death rate (μ).

Eq. (1g) expresses the rate of change of the population of recovered human. On the right-hand side, the first term represents the increase in the population of recovered human due to treatment with the rate (γ), and the last term represents the natural death rate (μ) of human

Let Sp(0)=Sp0,Ip(0)=Ip0,Qp(0)=Qp0,Rp(0)=Rp0,Sh(0)=Sh0,Ih(0)=Ih0,Rh(0)=Rh0. A flowchart of the SIQR-SIR model of pigs and human which describe by the system (1) is shown in Fig 1.

Figure 1: Flowchart of SIQR-SIR (Susceptible-Infectious-Isolation-Recovery-Susceptible-Infectious-Recovery) model for human and pigs

The values of population classes Sp(t),Ip(t),Qp(t),Rp(t),Sh(t),Ih(t),Rh(t) at time tis non-negative numbers.

Let

N(t)=Sp(t)+Ip(t)+Qp(t)+Rp(t)+Sh(t)+Ih(t)+Rh(t),

and c=min{b,μ}. Then we obtain

dNdt=dSpdt+dIpdt+dQpdt+dRpdt+dShdt+dIhdt+dRhdt =N1+N2a(Ip+Qp)b(Sp+Ip+Qp+Rp)αIhμ(Sh+Ih+Rh) N1+N2cN,

and

N(t)N1+N2c+(N(0)N1+N2c)ect.(2)

Therefore, the positive invariant of the system (1) is

Ω={(Sp,Ip,Qp,Rp,Sh,Ih,Rh)R+7:NN1+N2c}.(3)

3  Model Analysis

3.1 Equilibria

The equilibria are obtained by setting all equations of the system (1) to be zero.

N1bSpβ1MSpIp=0, β1MSpIpaIpbIpδIp=0, δIpaQpbQpϵQp=0,

ϵQpbRp=0,(4)

N2β2ShIpβ3ShIhμSh=0, β2ShIp+β3ShIhαIhγIhμIh=0, γIhμRh=0.

By solving the system (4), We get three equilibrium points:

i)   Disease-free equilibrium

E1=(N1b,0,0,0,N2μ,0,0).

ii)   Pig disease free equilibrium

E2=(N1b,0,0,0,Aβ3,β3N2μAβ3A,γμβ3N2μAβ3A), where Aα+γ+μ.

Note that, E2 exists if β3N2μA>0.

iii) The endemic equilibrium

E=(Sp,Ip,Qp,Rp,Sh,Ih,Rh), where Sp=Bβ1M, Ip=β1MN1bBβ1BM, Qp=δIp(a+b+ϵ), Rp=ϵδIpb(a+b+ϵ), Ih=12β3A(X+X2+4Aβ2β3N2Ip), Sh=N2β2Ip+β3Ih+μ, Rh=γIhμ,

with Xβ3N2A(β2Ip+μ) and Ba+b+δ.

Note that, E exists if β1MN1bB>0.

3.2 Basic Reproduction Number

The basic reproduction number is the expected number of secondary cases produced by a single infection in a completely susceptible population. To compute the basic reproduction number and to study the local stability of the equilibrium, we use the next generation matrix method [40,41]. We define x=f(x)v(x), where x=[Ip,Qp,Ih]Twheref(x) is the matrix of new infection terms, and v(x) is the matrix of transfer terms into compartment and out of compartment as:

f=[β1MSpIp0β2ShIp+β3ShIh],v=[aIp+bIp+δIpδIp+bQp+ϵQpαIh+γIh+μIh].

The corresponding Jacobian matrices are

F=[β1MSp00000β2Sh0β3Sh],V=[B00δa+b+ϵ000A].

Then, we obtain

FV1=[β1MSpB00000β2ShB0β3ShA]|E1 =[β1MN1bB00000β2N2μB0β3N2μA]

We obtain three eigenvalues of FV1as:

λ1=0,λ2=β3N2μA,λ3=β1MN1bB.

Hence, the spectral radii are R01=β3N2μA,and R02=β1MN1bB.

3.3 The Stability of Disease-Free Equilibrium (E1)

Theorem 1. The disease-free equilibrium point (E1) is locally asymptotically stable if R01<1 and R02<1.

Proof. The model system (1) has the Jacobian matrix at the point E1 as:

J(E1)=[bβ1MN1b000000β1MN1bB000000δ(a+b+ϵ)000000ϵb0000β2N2μ00μβ3N2μ00β2N2μ000β3N2μA000000γμ]

The corresponding eigenvalues of J(E1) are

λ1=λ2=b,λ3=λ4=μ,λ5=(a+b+ϵ), λ6=β1MN1bBB=B(R021), λ7=β3N2μA=A(R011).

Note that λ6 is negative if R02<1 and λ7 is negative if R01<1.

Therefore, the disease-free equilibrium (E1) of the model system (1) is locally asymptotically stable due to all negative real part eigenvalues if R01<1 and R02<1. □

3.4 The Stability of Pig Disease Free Equilibrium (E2)

Theorem 2. The pig disease free equilibrium (E2) is locally asymptotically stable if R01>1 and R02<1.

Proof. The pig disease-free equilibrium

E2=(N1b,0,0,0,Aβ3,β3N2μAβ3A,γμβ3N2μAβ3A)

exists and is positive if R01>1.

The model system (1) has the Jacobian matrix at the point E2 as:

J(E2)=[bβ1MN1b000000β1MN1bB000000δ(a+b+ϵ)000000ϵb0000β2N2μR0100μR01A00β2N2μR0100μ(R011)0000000γμ]

The corresponding eigenvalues of J(E2) are

λ1=λ2=b,λ3=μ,λ4=(a+b+ϵ), λ5=β1MN1bB=B(R021), λ6=μR01+μ2R0124μA(R011)2, λ7=μR01μ2R0124μA(R011)2.

Note that λ5 is negative if R02<1 while λ6 and λ7 have negative real part if R01>1.

Therefore, the pig disease free equilibrium (E2) of the system (1) is locally asymptotically stable if R01>1 and R02<1.

3.5 The Stability of Endemic Equilibrium (E)

Theorem 3. The endemic equilibrium (E) is locally asymptotically stable if R01<1 and R02>1.

Proof. The endemic equilibrium

E=(Sp,Ip,Qp,Rp,Sh,Ih,Rh),

where

Sp=N1bR02 Ip=b(R021)β1M Qp=δIp(a+b+ϵ) Rp=ϵδIpb(a+b+ϵ) Ih=12β3A(X+X2+4Aβ2β3N2Ip) Sh=N2β2Ip+β3Ih+μ Rh=γIhμ

exists and is positive if R02>1.

The model system (1) has the Jacobian matrix at the point E as:

J(E2)=[bΨ1Ψ200000Ψ1Ψ2B000000δ(a+b+ϵ)000000ϵb0000β2Sh00Ψ3μβ3Sh00β2Sh00Ψ3β3ShA000000γμ]

where

Ψ1=β1MN1(R021)BR02,Ψ2=Mβ1N1bR02,Ψ3=β2N1(R021)BR02+β3Ih.

The corresponding eigenvalues of J(E) are

λ1=b,λ2=μ,λ3=(a+b+ϵ), λ4=[β1M(IpSp)+B+b]+Y12, λ5=[β1M(IpSp)+B+b]Y12, λ6=[β2Ip+β3(IhSh)+A+μ]+Y22, λ7=[β2Ip+β3(IhSh)+A+μ]Y22,

where

Y1=[β1M(IpSp)+B+b]24[β1M(BIpbSp)+Bb],

and

Y2=[β2Ip+β3(IhSh)+A+μ]24[β2AIp+β3AIh+μ(Aβ3Sh)].

Next, we consider

β1M(IpSp)+B+b=β1M(b(R021)β1MN1bR02)+B+b =b(R021)β1MN1bR02+B+b =bR02 >0,

and

β1M(BIpbSp)+Bb=β1M(Bb(R021)β1MbN1bR02)+Bb =bB(R021)bβ1MN1bR02+Bb =Bb(R021).

We obtain that λ4 and λ5 has negative real part if R02>1.

Then, let us consider

β2Ip+β3(IhSh)+A+μ =β2Ip+β3(IhN2β2Ip+β3Ih+μ)+A+μ =β32Ih2+[2β2β3Ip+2μβ3+β3A]Ih+β22Ip2+(A+2μ)β2Ip+μ(A+μ)β3N2β2Ip+β3Ih+μ =[β32+β3A]Ih2+[2β2β3Ip+β3(2μ+AN2)+A(β2Ip+μ)]Ihβ2Ip+β3Ih+μ +β2Ip(β2Ip+A+2μN2)+μ(A+μ)β3N2β2Ip+β3Ih+μ.

Since Ih=12β3A(X+X2+4Aβ2β3N2Ip), it follows that

β3AIh2[β3N2A(β2Ip+μ)]Ihβ2N2Ip=0.

Then, we get

β2Ip+β3(IhSh)+A+μ =β32Ih2+[2β2β3Ip+β3(2μ+A)]Ih+β2Ip(β2Ip+A+2μ)+μ2+μA(1R01)β2Ip+β3Ih+μ.

If R01<1, Ip and Ih are positive i.e., R02>1, then we obtain

β2Ip+β3(IhSh)+A+μ>0.

Thus, the eigenvalues λ5 and λ6 have a negative real part if R01<1 and R02>1.

Therefore, the endemic equilibrium (E) of the model system (1) is locally asymptotically stable if R01<1 and R02>1. □

We can draw the bifurcation diagram to describe the basic reproduction numbers and stability as shown in Fig. 2.

Figure 2: The bifurcation region

4  Numerical Examples and Discussion

The numerical results of the system (1) are computed by using MATLAB with the given initial values:

Sp(0)=4057,Ip(0)=1000,Qp(0)=0,Rp(0)=0,Sh(0)=50000,Ih(0)=1000,Rh(0)=0.

The numerical results of the system (1) with the parameter values as shown in Tab. 1. Note that the parameters and initial values are obtained from data in [35] and [42].

The solution trajectories tend to the disease-free equilibrium (E1) which satisfy Theorem 1 with the remaining parameter values μ=0.9,α=0.9,γ=0.9,M=0.9,a=0.9,β2=0.1, and δ=0.9as shown in Fig. 3. The calculated reproduction numbers of this case are R01=0.9568<1 and R02=0.5625<1.

Figure 3: The simulation results of the system (1), (a) The plots of asymptotic population density of pig susceptible population (Sp), pig infectious population (Ip), pig isolated population (Qp), and pig recovery population (Rp) with respect to time t tends to the equilibrium (E1), and (b) The plots of asymptotic population density of human susceptible population (Sh), human infectious population (Ih), and human recovery population (Rh) with respect to time t tends to the equilibrium (E1)

The results showed that the number of pig infectious cases dramatically decreased in the first quarter. After that, the decreasing of pig infectious cases slowly decreased. Then, it tended to the equilibrium value. Fig. 3a, the number of pig susceptible population increased after infectious cases were recovered. In addition, the isolated population increased when the number of infectious cases increased to control the spread of the disease. Then, it decreased after the number of infectious cases decreased. In human case (Fig. 3b), the number of infectious populations increased in the first quarter, then the number increased and approached to the equilibrium number. Therefore, the parameters of this case provided that the disease died out.

The solution trajectories tend to the disease-free equilibrium (E2) which satisfy Theorem 2 with the remaining parameter values μ=0.6,α=0.5,γ=0.4,M=0.3,a=0.7,β2=0.1, and δ=0.8 as shown in Fig. 4. The calculated reproduction numbers of this case are R01=2.5833>1and R02=0.2125<1.

The results showed that the number of pig infectious cases dramatically decreased in the first quarter. After that, the decreasing of pig infectious cases slowly decreased. Then, it tended to the equilibrium value. The number of pig susceptible population increased after infectious cases decreased. In human case (Fig. 4b), the number of infectious populations dramatically increased in the first quarter, then it decreased and approached to the equilibrium number. Therefore, the result of this case indicated that the disease in pig died out while it still appeared in human.

Figure 4: The simulation results of the system (1), (a) The plots of asymptotic population density of pig susceptible population (Sp),pig infectious population (Ip), pig isolated population (Qp), and pig recovery population (Rp)with respect to time ttends to the equilibrium (E2),and (b) The plots of asymptotic population density of human susceptible population (Sh), human infectious population (Ih), and human recovery population (Rh) with respect to time ttends to the equilibrium (E2)

The numerical results of the system (1) with the remaining parameter values μ=0.5,α=0.6,γ=0.1,M=0.9,a=0.3,β2=0.01,and δ=0.1 with R01=0.9568<1 and R02=1.2473>1. The solution trajectories tend to the endemic equilibrium (E) which satisfy Theorem 3 as shown in Fig. 5. The number of pig susceptible population decreased in the first five quarters then it increased and approached to the equilibrium. The number of pig infectious population increase in the first quarter then it tends to equilibrium after five quarters. In human case, the number of human infectious population increased sharply in the first two quarters. After that, it decreased and tend to equilibrium value. The number of recovery population increased in the first quarter then it decreased and approached to equilibrium value. The result of this case showed that the disease still appeared in both human and pigs.

To consider the effect of moisture, the numerical results of the system (1) with the remaining parameter values μ=0.5,α=0.6,γ=0.1,a=0.01,β2=0.01, and δ=0.01. The solutions trajectories are computed with various values of relative humidity M as 0.1,0.3,0.5,0.7, and 0.9 is shown in Fig. 6.

In Fig. 6, the result indicates that when the relative humidity (M) increases the pig infectious population density also increases. Particularly, the trajectories of pig infectious population decrease and approach to constant values when M=0.1,0.3, and 0.5. On the other hand, the number of pig infectious population increase and go to equilibrium value when M=0.7 and 0.9. Moreover, when M=0.1 and 0.3 the trajectories approach zero. This means that the disease will die out when there is less moisture in the air. Therefore, the result confirms that there is an effect of the moisture in the air to the infection of Streptococcus suis as mentioned in the literature.

Figure 5: The simulation results of the system (1), (a) The plots of asymptotic population density of pig susceptible population (Sp), pig infectious population (Ip), pig isolated population (Qp), and pig recovery population (Rp) with respect to timet tends to the equilibrium (E), and (b) The plots of asymptotic population density of human susceptible population (Sh), human infectious population (Ih), and human recovery population (Rh) with respect to time ttends to the equilibrium (E)

Figure 6: The simulation results of the pig infectious population (Ip) for various values of relative humidity Mwith respect to time t

5  Conclusions

We have proposed the mathematical model for predicting the disease transmission of Streptococcus suis between pigs and humans, which is developed from the SIQR model that considered only in the pig population. The pigs-human epidemiology model of Streptococcus suis transmission which is SIQR-SIR model is investigated. The model combined with susceptible (Sp), infection (Ip), isolation (Qp), and recovery (Rp) for pig population and susceptible (Sh), infectious (Ih), and recovery (Rh) for human population. Moreover, we studied the effect of moisture in the air on disease transmission. The equilibria of the model were analyzed. Then, the next-generation matrix was used to find the basic reproduction numbers R01, and R02. We obtained the conditions of the equilibrium points E1,E2, and E which are locally asymptotically stable in Theorem 1, Theorem 2, and Theorem 3, respectively. The results show that if R01<1 and R02<1, the equilibrium points E1 is locally asymptotically stable while E2 is locally asymptotically stable if R01>1 and R02<1. Finally, the endemic equilibrium (E) is locally asymptotically stable if R01<1 and R02>1. The bifurcation diagram is also presented. The numerical examples confirmed the results of Theorems. The results indicated that moisture affects disease transmission. When the air has higher moisture the transmission of the pathogen is also high. Moreover, the model can be improved by considering infection of newborn pigs from infectious adult female pigs.

Acknowledgement: This research was financially supported by the young researcher development project of Khon Kaen University. Appreciation is extended to Faculty of Science, Khon Kaen University, Khon Kaen, Thailand.

Funding Statement: This research was financially supported by the young researcher development project of Khon Kaen University.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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