Stochastic Modeling of Imperfect Salmonella Vaccines in an Adult Dairy Herd

Abstract

Salmonella is a major cause of bacterial foodborne disease. Human salmonellosis results in significant public health concerns and a considerable economic burden. Dairy cattle are recognized as a key source of several Salmonella serovars that are a threat to human health. To lower the risk of Salmonella infection, reduction of Salmonella prevalence in dairy cattle is important. Vaccination as a control measure has been applied for reduction of preharvest Salmonella prevalence on dairy farms. Salmonella vaccines are usually imperfect (i.e., vaccines may provide a partial protection for susceptible animals, reduce the infectiousness and shedding level, shorten the infectious period of infected animals, and/or curb the number of clinical cases), and evaluation of the potential impacts of imperfect Salmonella vaccines at the farm level is valuable to design effective intervention strategies. The objective of this study was to investigate the impact of imperfect Salmonella vaccines on the stochastic transmission dynamics in an adult dairy herd. To this end, we developed a semi-stochastic and individual-based continuous time Markov chain (CTMC) vaccination model with both direct and indirect transmission, and applied the CTMC vaccination model to Salmonella Cerro transmission in an adult dairy herd. Our results show that vaccines shortening the infectious period are most effective in reducing prevalence, and vaccines decreasing host susceptibility are most effective in reducing the outbreak size. Vaccines with multiple moderate efficacies may have the same effectiveness as vaccines with a single high efficacy in reducing prevalence, time to extinction, and outbreak size. Although the environment component has negligible contributions to the prevalence, time to extinction, and outbreak size for Salmonella Cerro in the herd, the relative importance of environment component was not assessed. This study indicates that an effective vaccination program against Salmonella Cerro spread in the herd can be designed with (1) vaccines with a single high efficacy in reducing either the infectious period or susceptibility of the host, or (2) if such single high efficacy vaccines are not available, vaccines with multiple moderate efficacies may be considered instead. These findings are also of general value for designing vaccination program for Salmonella serotypes in livestock.

Appendix: The Semi-Stochastic Individual-Based Algorithm

Gillespie algorithms are well known and have been applied to many problems where the master equation could not be solved analytically or numerically (Gillespie 2007). Here we described only the core part of the semi-stochastic individual-based algorithm, the time to next event for our imperfect Salmonella vaccination model. To find the time to the next event, we combined the direct and first reaction Gillespie algorithms (Keeling and Rohani 2008). The direct Gillespie algorithm was applied to all events listed in Table 2 except the recovery events. The time to the next event using the direct Gillespie algorithm, t _d , is the solution of the following equation:

$$\begin{aligned} -\log (\mathit{rand}1) =& t_d\bigl(\mu N + \beta S\bigl(I+(1-e_{\beta\sigma} )Y\bigr)/N \\ &{}+(1-e_\lambda)\beta X\bigl(I+(1-e_{\beta\sigma} )Y\bigr)/N \\ &{}+\phi_1(N-S-I-X-Y)+p_2\psi(S+I+R)\bigr) \\ &{}+\bigl(\eta\bigl(S+(1-e_\lambda)X\bigr)/N\bigr)\int _0^{t_d} W(t) \,dt, \end{aligned}$$

(2)

where the rand1 is a random number generated from a uniform distribution within 0–1, and the environment variable W(t) satisfies the first order ordinary differential equation of W(t) (Clancy 2005):

$$ \frac{dW}{dt}=\sigma I+(1-e_{\beta\sigma} )\sigma Y-(\delta+\eta)W. $$

(3)

During the time interval (0,t _d ), the number of animals in each compartment does not change; therefore, the analytical solution W(t) for the above equation can be found for t∈(0,t _d ):

$$ W(t)=W(0)+\frac{\sigma}{\delta+\eta}\bigl(I+(1-e_{\beta\sigma} )Y\bigr) \bigl(1-e^{-(\delta+\eta)t}\bigr). $$

(4)

Substituting Eq. (4) into Eq. (2), the time to the next event t _d was determined.

For the recovery events, the time since infection is important because the infectious period follows a gamma distribution. We applied the individual-based algorithm to consider this more realistic infectious period distribution in our simulation (Keeling and Rohani 2008). When an animal was infected as either unvaccinated (I) or vaccinated (Y), the time of this mth infectious animal recovering from I or Y (to R or Z), τ _m , was predetermined by generating a random number following the gamma distribution, gamrnd(n,1/(nγ)) for I or gamrnd(n,(1−e _γ )/(nγ)) for Y. We then used the first reaction Gillespie algorithm to find the time to the next event for all events:

$$ t_{\mathrm{next}}=\min\{t_d, \tau_{1}, \ldots, \tau_m, \ldots, \tau_{f}\}, $$

(5)

where at the present time the number of all infectious animals is f≤N. If t _next=t _d , a second random number was generated to select which event was going to occur next, and then a third random number was used to find which animal was directly related to the next event. If t _next=τ _m , the status of the mth infectious animal was changed from I to R or Y to Z. Iterations including the above procedures ran until the prescribed time reached.