A stochastic model for transmission, extinction and outbreak of Escherichia coli O157:H7 in cattle as affected by ambient temperature and cleaning practices

Abstract

Many infectious agents transmitting through a contaminated environment are able to persist in the environment depending on the temperature and sanitation determined rates of their replication and clearance, respectively. There is a need to elucidate the effect of these factors on the infection transmission dynamics in terms of infection outbreaks and extinction while accounting for the random nature of the process. Also, it is important to distinguish between the true and apparent extinction, where the former means pathogen extinction in both the host and the environment while the latter means extinction only in the host population. This study proposes a stochastic-differential equation model as an approximation to a Markov jump process model, using Escherichia coli O157:H7 in cattle as a model system. In the model, the host population infection dynamics are described using the standard susceptible-infected-susceptible framework, and the E. coli O157:H7 population in the environment is represented by an additional variable. The backward Kolmogorov equations that determine the probability distribution and the expectation of the first passage time are provided in a general setting. The outbreak and apparent extinction of infection are investigated by numerically solving the Kolmogorov equations for the probability density function of the associated process and the expectation of the associated stopping time. The results provide insight into E. coli O157:H7 transmission and apparent extinction, and suggest ways for controlling the spread of infection in a cattle herd. Specifically, this study highlights the importance of ambient temperature and sanitation, especially during summer.

Appendices

Appendix A: Proof of Theorem 3.1

System (2.2) can be written as $d{{\varvec{x}}}/dt=f({{\varvec{x}}})$, where ${{\varvec{x}}}=(I,E)^T$ and ${\varvec{f}}=(f_1,f_2)^T$ is continuously differentiable in region $R$. Every positive semi-orbit of this system lies in $R$, a closed and bounded subset of $\mathbb R ^2$. In addition, there are a finite number of equilibria in $R$. We apply the powerful Poincaré–Bendixson theory for two-dimensional autonomous systems of differential equations, in particular, a result known as the Poincaré–Bendixson trichotomy (Smith and Waltman 1995). This result states that for system $d{{\varvec{x}}}/dt=f({{\varvec{x}}})$ with the preceding properties that the omega-limit set of a positive semi-orbit beginning in $R$ must approach one of the following limiting sets: an equilibrium, a periodic orbit, or a cycle graph (equilibria connected by solution trajectories). We show that the omega-limit set for every solution of (2.2) beginning in $R$ is the DFE in the case $T\le 1$ or the endemic equilibrium in the case $T>1$.

Proof of Theorem 3.1

For the case $T\le 1$, there is only one equilibrium in $R$, namely, the DFE. For $T<1$, the DFE is locally asymptotically stable and for $T=1$, the DFE is neutrally stable (one zero eigenvalue and one negative eigenvalue). Since every periodic orbit must contain at least one equilibrium in the interior of its orbit, there cannot exist any periodic orbits for $T\le 1$ in $R$. Moreover, there does not exist any cycle graphs (homoclinic orbit) when $T\le 1$ because the origin is not a saddle point. Thus, by the Poincaré–Bendixson trichotomy, solutions of (2.2) must approach the DFE. The DFE is globally asymptotically stable when $T\le 1$.

For the case $T>1$, system (2.2) has two equilibria in $R$, namely, the DFE and the endemic equilibrium. First, we rule out periodic orbits encircling the endemic equilibrium by applying the Dulac threshold (Smith and Waltman 1995). No periodic orbit can touch the boundary $E=0$ or $I=0$ because it is repelling and the origin is an equilibrium. Given the Dulac function $B=1/(IE)$, the expression $\partial (Bf_1)/\partial I+\partial (Bf_2)/\partial E<0$ is of one sign in the interior of $R$ which by Dulac’s threshold implies there does not exist any periodic orbits in $R$ (Smith and Waltman 1995). Second, we rule out cycle graphs. When $T>1$, the DFE is a saddle point with stable manifold outside of $R$ and unstable manifold in $R$. In addition, the endemic equilibrium is locally asymptotically stable. Hence, there cannot be any homoclinic orbits or heteroclinic orbits and consequently no cycle graphs. Solutions cannot approach the DFE because it is repelling in $R$. Hence, by the Poincaré–Bendixson trichotomy, solutions of (2.2) must approach the endemic equilibrium. The endemic equilibrium is globally asymptotically stable in the interior of $R$ when $T>1$.

Appendix B: Estimation of the maximal carrying capacity $K_{E}$ of free-living pathogen in the environment

According to Jiang et al. (2002), the carrying capacity of the free-living pathogen in fecal debris is $10^{5}$ CFU per gram of feces. Notice that a host will on average produce 30 kg of feces per day. So, if there are $N$ hosts, the maximal carrying capacity of the free-living pathogen in fecal debris will be about $30,000\times N\times 10^5=3\times 10^{9}\times N$ CFU. In reality, the maximum carrying capacity for the system is not known because the total amount of fecal debris in the cattle environment is unknown. Our assumption underlying the maximum carrying capacity implies that at most there could be $30,000\times N$ grams of feces in the cattle environment on any given day and since each gram of feces has the carrying capacity of $10^5$ CFU/g the approximated maximum carrying capacity is $3\times 10^{9}\times N$ CFU. We tested the robustness of the model results and conclusion to this assumed maximal carrying capacity by running numerical analysis with different values of the maximal capacity. While the proportion of pathogens in the environment over time relative to the maximal carrying capacity obviously changed for different assumed values of this parameter, the qualitative behavior was identical, thus confirming the robustness of the model to the assumption.

Appendix C: Estimation of the pathogen growth rate $g_{E}$

Gautam et al. (2011) show that the relationship between the growth rate $Z$ and the temperature $T$ can be expressed as

$$\begin{aligned} Z = 24\,(0.0032 \, T-0.016). \end{aligned}$$

(C.1)

Taking the expectation of both sides of (C.1), we have

$$\begin{aligned} \mathbb E [Z] = 0.0768 \, \mathbb E [T]-0.384. \end{aligned}$$

(C.2)

Plugging the average temperature into (C.2) gives us the estimates of the average growth rate in (5.5).

Appendix D: Estimation of the pathogen cleaning rate $r_{E}$

Four scenarios of cleaning, including cleaning every day, twice per week, once per week and no cleaning, are considered to reflect this wide variation in cleaning frequencies across the cattle industry. The cleaning rate for each of these scenarios was calculated using the formula described in Vosough Ahmadi et al. (2007) as

$$\begin{aligned} C=I \exp (-q t), \end{aligned}$$

where $C$ represents the number of CFU of E. coli O157:H7 per milliliter of bedding materials at time $t$ (in days), $I$ is the intercept or initial number of bacteria, and $q$ is the rate of bacterial depletion. Vosough Ahmadi et al. (2007) obtained a death rate of $0.46$ CFU per day for cleaning practice of two times per week by fitting an exponential distribution to the data reported by Davis et al. (2005). To account for cleaning of water, $0.17$ CFU per day is added to the estimated death rate. Using this approach we estimate the overall death rate for different cleaning frequencies as given in Table 1.

Table 1 The estimated value of cleaning rate $r_{E}$ (CFU per day)

See Tables 1 and 2.

Table 2 Definition of parameters used in stochastic ${SIS}_\mathrm{E}$ model for E. coli O157:H7 transmission infection in a cattle herd