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Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 432602, 10 pages

http://dx.doi.org/10.1155/2014/432602

## Modeling Transmission Dynamics of *Streptococcus suis* with Stage Structure and Sensitivity Analysis

^{1}Key Lab of Animal Bacteriology of Ministry of Agriculture, College of Veterinary Medicine, Nanjing Agricultural University, Nanjing, Jiangsu 210095, China^{2}China Animal Health and Epidemiology Center, Qingdao, Shandong 266032, China^{3}Department of Mathematics, North University of China, Taiyuan, Shan’xi 030051, China^{4}Qingdao Municipal Center for Disease Control and Prevention, Qingdao, Shandong 266032, China

Received 7 December 2013; Revised 29 January 2014; Accepted 29 January 2014; Published 16 March 2014

Academic Editor: Delfim F. M. Torres

Copyright © 2014 Chaojian Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Streptococcosis is one of the major infectious and contagious bacterial diseases for swine farm in southern China. The influence of various control measures on the outbreaks and transmission of *S. suis* is not currently known. In this study, in order to explore effective control and prevention measures we studied a deterministic dynamic model with stage structure for *S. suis*. The basic reproduction number is identified and global dynamics are completely determined by . It shows that if , the disease-free equilibrium is globally stable and the disease dies out, whereas if , there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. The model simulations well agree with new clinical cases and the basic reproduction number of this model is about 1.1333. Some sensitivity analyses of in terms of the model parameters are given. Our study demonstrates that combination of vaccination and disinfection of the environment are the useful control strategy for *S. suis*.

#### 1. Introduction

Streptococcosis is a zoonosis caused by various pathogenic strains of* Streptococcus suis* (*S. suis*). The disease is primarily associated with* S. suis* capsular type 2 in pigs and humans, commonly manifested clinically as porcine septic streptococcosis and porcine lymphadenopathy with abscess formation [1]. The most common route of infection in pigs is through the respiratory tract, alimentary tract, and damaged skin. Pigs are the most important source of infection, as the pathogen is transmitted mainly via fomites, such as dust in the air or other insect vectors [2–7]. Porcine streptococcosis may cause death in animals and humans [8, 9]. Up to now, at least 832 human cases of porcine streptococcosis have been reported, with a mortality rate of approximately 10% [5, 10]. Despite a worldwide distribution, porcine streptococcosis is found primarily in Asia, with China and Southeast Asia most severely affected [11]. Porcine streptococcosis not only is responsible for serious public health problems, but also poses severe economic losses in the swine industry.

Streptococcosis is widely distributed in China [12]; especially southern China is the most severely affected. In 1963, the disease began spreading in parts of Guangxi and then to most provinces and municipalities in southern China, including Guangdong, Sichuan, and Fujian. In the 1980s, the disease grew in severity, with many areas experiencing fulminating or endemic outbreaks. Since the 1990s, a rapid increase of pig production has led to an increasing number of reports of* S. suis* infection and outbreaks of the disease [13–17]. In southern China,* S. suis* infection has become one of the region’s most important zoonoses.

Most streptococcosis studies are currently focused on the description of its etiology and epidemiology, while the studies on the mechanism of transmission and the risk factors influencing its spread have not been found. In this paper, we constructed dynamical models to discuss the transmission dynamics of the* S. suis* and study the influence of two control measures, as well as the pathogenic* S. suis* shed into the environment from streptococcosis in a porcine herd in southern China.

The paper is organized as follows. In Section 2, we present and interpret the dynamical model that describes the* S. suis* transmission and give the basic reproduction number of the model and some mathematical analysis. And in Section 3, some numerical simulations are showed. Section 4 gives a brief discussion about main results.

#### 2. Mathematical Modeling and Analysis

##### 2.1. Model Formulation

A self-breeding and self-raising swine farm in southern China with good disease diagnosis often done by professional laboratories was employed for this study. There were no introductions of live pigs and the* S. suis* vaccine was not used on the farm. According to the epidemiological characteristics of* S. suis*, the population of this swine farm was divided into five subpopulations compartments: susceptible piglet compartment (which is less than 2 months old), susceptible adult pig compartment (including fattening pigs and sows), recessive infected compartment , recovered compartment , and quarantined clinical compartment at time . Let denote the quantity of the pathogenic bacteria in the environment at time .

There are some assumptions on the dynamical transmission of* S. suis* in this swine farm. (1) The number of pigs supplemented (the number of newborn piglets) is considered to be approximately constant because the swine farm is a self-breeding and self-raising one. (2) Antibiotics are used not only for treatment in the pig raising process, but also for disease prevention. (3) The recessive infected and quarantined clinical cases have the same recovery rate after antibiotic treatment, and both recover to the recovered compartment. (4) The infection rate is different between adults and piglets, and the infection rate of adults is less than that of piglets. The dynamical transmission of* S. suis* in swine farm is demonstrated in the flowchart.

From Figure 1 we can know that the new infection occurring with piglet and adult pigs is given by and , respectively, where and describe the transmission rate from recessive infected pig and bacteria in the environment to the susceptible piglet. is the ratio of the transmission rate with susceptible adult pig to the transmission rate with susceptible junior pig. The recruitment rate of individuals into this swine farm is given by ; within the piglet and adult pig group, the removed rates are and . represents the rate from susceptible junior pig to susceptible adult pig and describes transfer rate from recessive infected pig to quarantined clinical pig. denotes the removed rate for* S. suis* and and represent the recovery rate from recessive infected compartment and quarantined clinical compartment to recovered compartment, respectively. In the environment compartment, the bacteria shedding rate from the recessive infected pig is , the decaying rate of bacteria is , the disinfection number of the environment is , and the effective disinfection rate of every time is . From the previous assumptions, the model is a system of ordinary differential equations:

All parameters are assumed to be nonnegative in system (1). System (1) always has a disease-free equilibrium , where

Lemma 1. *Every forward solution of system (1) eventually enters
**
and is a positively invariant set for (1).*

*Proof. *Let be the total number of pigs. So from system (1) we can see that
Thus, if , if , and if , which implies that is positively invariant with respect to system (1).

Now we introduce the basic reproduction number for system (1), according to the definition of [18]. We order the infected variables first by disease state; we only need the vector for system (1). Consider the following system: Follow the recipe from van den Driessche and Watmough [18] to obtain Calculate and for derivatives about and generate them into the disease-free equilibrium : So the basic reproduction number for system (1) is

The endemic equilibrium of system (1) is determined by equations

We can obtain

Direct calculation for shows Now we define the following equation: From system (11), we can find that the function is also monotonically decreasing for < . As the function and < when , system (1) has a unique endemic equilibrium .

##### 2.2. Global Stability of the Equilibrium

Because the fourth and fifth equations are independent of other equations for system (1), we only need to consider the following system: From previous analysis, we know that system (13) also has a disease-free equilibrium , , and when and a unique endemic equilibrium when . In the following, we will prove the global stability of the disease-free equilibrium and endemic equilibrium of system (13) by using a Lyapunov function. The Lyapunov function is a powerful tool for the stability analysis of autonomous differential system; it has been used for some epidemiological models with constant inflow and bilinear incidences or nonlinear incidences [19–23].

For the disease-free equilibrium, we have the following conclusion.

Theorem 2. *The disease-free equilibrium of system (13) is globally asymptotically stable when .*

*Proof. *For the disease-free equilibrium , define the Lyapunov function

Then the derivative of along solutions of system (13) is

Therefore, when , and the equality holds if and only if , , and . Thus, the disease-free equilibrium of system (13) is globally asymptotically stable by LaSalle’s Invariance Principle [24]. This completes the proof.

Next, we will also prove global stability of the endemic equilibrium of system (13) by using a Lyapunov function. It is important for us to understand the extinction and persistence of the disease.

Theorem 3. *The unique endemic equilibrium of system (13) is globally asymptotically stable when .*

*Proof. *System (1) can be transformed into the following form:

Define the Lyapunov function
Then the derivative of along solutions of system (16) is
where

For the endemic equilibrium , we have the following equation:
Hence, we can obtain

The equality holds only for , , and . By LaSalle’s Invariance Principle [24], the endemic equilibrium is globally asymptotically stable. This completes the proof.

So far all our analyses focus on the mathematical models and their dynamic behavior, such as the basic reproduction number and the global stability of the disease-free equilibrium and endemic equilibrium. In the next section we will present some numerical simulations about the actual data and give some sensitivity analyses of the basic production number on parameters.

#### 3. Numerical Simulations and Sensitivity Analysis

The number of new births, sales, and deaths for live pigs, inventory at the beginning of the quarter, quarantined clinical cases, and deaths from* S. suis* and disinfection numbers from the past three years are recorded in Table 1.

##### 3.1. Parameter Estimation

In order to carry out the numerical simulations, we need to estimate the model parameters. Some parameter values can be calculated by using the data in Table 1, some parameter values need to be assumed, and other parameter values need the parameter estimation. For the recruitment rate of individuals into this swine farm , which is the average of new birth, the value is . For the transfer rate from susceptible piglet to susceptible adult pig , we know that susceptible piglet can survive in piglet compartment for about two months, and the unit time of this model is a quarter, so the transfer rate is . With the removed rate of this swine farm, we assume the removed rates of susceptible piglet compartment and susceptible adult pig compartment are the same, which is the average of the proportion for sales and death to the breeding stock of this swine farm, which is . For the removed rate for quarantined clinical , we can calculate , so the recovered rate from quarantined clinical compartment to recovered compartment is . Due to the fact that antibiotics are used for treatment in the pig raising process and disease prevention and we do not know the transfer rate from recessive infected compartment to recovered compartment, we assume that and . We can obtain the disinfection time of the environment from the data in Table 1, which is . With the effective disinfection rate of every time , we assume . According to the survival of* S. suis* in the environment mentioned previously, combined with the local climate characteristics, and assuming that the survival time of is 10 days under natural conditions, we obtain . For other parameters , , , and , we need some parameter estimation. Hence, the parameter values are listed in Table 2.

For the initial values, we also need the fitting data. can be directly obtained, but and are assumed. Since the rate of piglet births is assumed to be continuous, we assume that the initial value of is the final size with the disease-free state. So satisfies the following equation: , with its final size given by ; therefore . Hence, we can also obtain . For the initial value of , we use the data fitting to obtain .

##### 3.2. Numerical Simulations

Using system (1), we simulate newly quarantined clinical cases in this swine farm from the first quarter of 2011 to the second quarter of 2013. The numerical simulation of newly quarantined clinical cases is shown in Figure 2(a), and Figure 2(b) shows the cumulative quarantined clinical cases, which indicate that our model provides a good match to the reported data. According to the predictions of the model (Figure 2(a)), if no further effective prevention and control measures are taken, the disease will not vanish and the clinical cases of streptococcosis will increase during the period after 2013. With the simulated parameter values, we obtain that the estimated value for the basic reproduction number of this swine farm is , which predicts that* S. suis* will persist in this farm.

In an epidemic model, the basic reproduction number is calculated and shown to be a threshold for the dynamics of the disease. The main purpose is to control the disease by making the basic reproduction number less than 1, so we must know how the basic reproduction number depends on the model parameter values. In the following result, we will show some sensitivity analyses of the basic production number .

As shown in Figure 3(a), in order to make , with all other parameters held constant, when the disinfection times reach 18, the basic reproduction number is still larger than 1. In the real world, it is very difficult to disinfect this whole farm 18 times per quarter for costs. Figure 3(b) shows that the effective disinfection rate must reach 1 which can make , when other parameters remain constant. Figures 3(c) and 3(d) illustrate that increases with the proportion parameter and the recruitment rate , when or , which can make . However, in practice it is also very difficult to reach these two measures.

Finally, the influence of different parameters on the incidence of disease was studied based on the sensitivity analysis of the basic reproduction number as mentioned above.

From the two figures of Figure 4, we can find that it is very difficult to control the disease only by changing parameters and , because the final size of the new clinical cases is larger than 25 in 2019. In order to effectively control the disease, it is necessary to use other control measures for this swine farm. So in the next section, we will give the dynamic model with vaccination for* S. suis*.

##### 3.3. The Analysis of the Dynamic Model with Vaccination

Since disinfection cannot effectively control the disease, it is necessary to use other measures to effectively control the disease, such as vaccination. In the real world, breeding pigs and fattening pigs are vaccinated. So we assume that vaccinated pigs are susceptible adult pigs and their removed rate is the same as the adult susceptible ones. For the susceptible adult pig, is the vaccination rate and is the losing vaccination rate. Therefore, system (1) can be transformed into the following system after the vaccination measure is applied:

Define the basic reproduction number of system (22) by [18]. We have where

Generally, the immune validity duration is about 4-5 months, taking the averaging 4.5 months, so we can calculate the losing vaccination rate to be . In the following result, we will show sensitivity analysis of the basic production number of system (22) and give the variations of new clinical cases of system (22) for different values of the vaccination rate since the second quarter of 2013.

Figure 5(a) shows that increasing will decrease the basic reproduction number . So as to make the basic reproduction number of system (22) less than 1, the vaccination rate of the susceptible adult pig only reaches about . The second figure of Figure 5 shows the variations of new clinical cases of system (22) for different values of the vaccination rate since the second quarter of 2013; we know that when the vaccination rate reaches , the final size of the new clinical cases is less than 10 in the future; it means that* S. suis* infection of this farm can be effectively controlled within 3-4 years. Hence, vaccination is a very important factor for the* S. suis* transmission; it plays an important role in the persistence of* S. suis* in this swine farm. Compared with Figures 4(b) and 5(b), we conclude that combination of vaccination and disinfection for* S. suis* are more effective than only the disinfection for the environment.

#### 4. Conclusion and Discussion

In southern China,* S. suis* has not only become a serious public health problem, but also caused severe economic losses in the swine production. So both the government and the swine industry have been seeking forceful methods to reduce the outbreaks and the spread of* S. suis*. However, no people have studied the transmission mechanism of this disease, and there are also no given effective control measures for the disease. In this paper, we established a deterministic dynamic transmission model with stage structure for* S. suis*. It is found that the model has two nonnegative equilibria, the disease-free equilibrium and the endemic equilibrium. The disease-free equilibrium exists without any condition whereas the endemic equilibrium exists provided . Through the analysis of the model it has been found that the global asymptotic behavior of system (1) is completely determined by the size of the basic reproduction number ; that is, the disease-free equilibrium is globally asymptotically stable if while an endemic equilibrium exists uniquely and is globally asymptotically stable if .

The model simulations (see Figure 2(a)) agreed with the new clinical cases, and we gave an estimate of the basic reproduction number to be about 1.1333, which implies that* S. suis* of this farm cannot be controlled with the current strategies. The trend of future* S. suis* incidence on the farm was predicted using system (1) and it was found that if the current control measures remained unchanged, the number of* S. suis* cases would increase. By some sensitivity analyses of the basic reproduction number on parameters, we find that disinfection is the effective control measure with* S. suis*, but it can not eradicate it for this farm. From Section 3.3 we know that vaccination is a very important factor for* S. suis*; it plays an important role in the persistence of* S. suis*. Hence, we conclude that combination of disinfection and vaccination control measures will become more reasonable and effective for* S. suis* in this swine farm.

However, there are some limitations and shortcomings in our model. Firstly, while* S. suis* in the environment as defined in the study is a quantity value, it is only an assumption, without actual laboratory data. Secondly, for the lack of the cost of vaccination and elimination for* S. suis*, we can only give some macrocontrol measures for* S. suis* of this farm; specific control measures cannot be given and the optimal control strategy cannot also be taken into account. So we need to continue research in the future.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

This project was funded by the National Key Technologies R D Program of China (2004BA519A60, 2006BAK02A03-1).

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