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Journal of Fish Diseases 2007, 30, 93–100

A stochastic modelling approach to describing the dynamics of an experimental furunculosis epidemic in Chinook salmon, Oncorhynchus tshawytscha (Walbaum) H Ogut1 and S C Bishop2 1 Faculty of Marine Sciences, Karadeniz Technical University, Surmene, Trabzon, Turkey 2 Roslin Institute (Edinburgh), Roslin, Midlothian, UK

Abstract

A susceptible-infected-removed (SIR) stochastic model was compared to a susceptible-latent-infectious-removed (SLIR) stochastic model in terms of describing and capturing the variation observed in replicated experimental furunculosis epidemics, caused by Aeromonas salmonicida. The epidemics had been created by releasing a single infectious fish into a group of susceptible fish (n ¼ 43) and progress of the epidemic was observed for 10 days. This process was replicated in 70 independent groups. The two stochastic models were run 5000 times and after every run and every 100 runs, daily mean values of each compartment were compared to the observed data. Both models, the SIR model (R2 ¼ 0.91), and the SLIR model (R2 ¼ 0.90) were successful in predicting the number of fish in each category at each time point in the experimental data. Moreover, between-replicate variability in the stochastic model output was similar to betweenreplicate variability in the experimental data. Generally, there was little change in the goodness of fit (R2) after 200 runs in the SIR model whereas 500 runs were necessary to have stable predictions with the SLIR model. In the SIR model, on an individual replicate basis, 80% of 5000 simulated replicates had R2 ¼ 0.7 and above, whereas this ratio was slightly higher (82%) with the SLIR model. In brief, both models were equally effective in predicting the observed data and its variance but the SLIR model was advantageous because it dif-

2007 The Authors. Journal compilation 2007 Blackwell Publishing Ltd

Correspondence Dr H Ogut, Faculty of Marine Sciences, Karadeniz Technical University, Surmene, Trabzon, Turkey 61530 (e-mail: [email protected])

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ferentiated the latent, i.e. infected but not having the ability to discharge pathogen, from the infectious fish. Keywords: dynamics, fish, furunculosis, stochastic model, susceptible-infected-removed, susceptiblelatent-infectious-removed.

Introduction

Models of fish disease dynamics have not been fully explored in terms of their potential for describing disease dynamics in fish hatcheries. Mainly this is due to the fact there have not been adequate data useful in the construction of such models. As suggested by Dye (1992) and Hurd & Kaneene (1993), insight into these models can only be gained by comparing model-produced data to observed data. Simulation models which successfully predict the observed data may be used to evaluate how various factors influence infection dynamics, and identify key steps in the transmission process. This in turn would help lead to the development of control strategies for fish diseases. Simple epidemic models are frequently used in the explanation and description of terrestrial animal disease spread. There are two main assumptions that are often invoked in construction of these types of models: (1) that the population is closed and (2) mixing is homogeneous (Anderson & May 1979; Anderson 1982). However, in modelling of terrestrial animal epidemics, these assumptions are often violated due to the fact that animal populations are dynamic with occurrence of emigration, immigration and territoriality depending on various factors such as resources for foraging and reproduction.



H Ogut and S C Bishop Stochastic modelling of a furunculosis epidemic

Fish in a pond are a closed and homogeneously mixed population. In evaluation of fish diseases in a salmonid hatchery these two assumptions are met. Thus, even very simple susceptible-infectedremoved (SIR) models may explain dynamics of infection transmission in fish epidemics. There are no studies describing the extent and capabilities of stochastic models for capturing the dynamics of fish diseases in an aquatic environment. Ogut, Reno & Simpson (2004) described a simple deterministic SIR model explaining 92% of the variation observed in experimental furunculosis epidemics. There is no other study to our knowledge exploring the potential of these models to aid the understanding and control of aquatic diseases. There is great need to study fish diseases and factors affecting their spread in order to develop effective control measures. For example, Aeromonas salmonicida is one of the most extensively studied fish pathogens. However, it is still problematic in many parts of the world due to the fact that the research has mainly focused on biological aspects of the agent rather than the disease itself, furunculosis (Smith 1997). Mortality, the end point data, is usually collected to compare the virulence of a pathogen under various conditions. The mortality data, alone, are not sufficient to model observed disease epidemics and test various hypotheses related to the risk factors on disease occurrence and spread. However, more comprehensive data describing the dynamics of A. salmonicida infection transmission are now available. Ogut (2001) extensively studied A. salmonicida infection spread in Chinook salmon, Oncorhynchus tshawytscha (Walbaum), and the effects of host density on the spread (Ogut & Reno 2004). From this data, Ogut (2001) also estimated the latent period (3 days), the period starting from the acquisition of the pathogen to the shedding of the bacteria, and infectiousness period (2 days), the period starting from shedding the bacteria to death with clinical signs of furunculosis. These were observed over 100 experimental A. salmonicida epidemics generated by cohabitation, a natural way of inducing epidemics. As described above, this data enabled the development of a deterministic model which successfully predicted natural and disease related mortality, as well as the number of susceptible and infected fish at different time points (Ogut et al. 2004). In this study, two compartmental stochastic models, SIR and susceptible-latent-infectiousremoved (SLIR), were developed and applied to 94

data arising from replicated experimental epidemics of furunculosis to determine their validity and potential in describing the observed furunculosis epidemic in chinook salmon. The stochastic method gives a full description of the epidemic, and it allows an exploration of the variability in the expected data, unlike deterministic models, providing a unique opportunity to explore the utility of stochastic models for describing aquatic disease outcomes. In reality, for example, not all animals deterministically die at the same day, e.g. the fifth day of acquiring infection in the case of A. salmonicida (Ogut et al. 2004). Some will die a little earlier and some will die a little later, but mortalities will be accumulated around a median or mean in some cases. In the stochastic approach, when latency and mortality start or end are obtained from a distribution rather than an estimated constant number. Therefore, the stochastic models are favoured over deterministic approaches in that the natural variability in each coefficient is considered. Materials and methods

This section describes the data upon which the models were based, the development of the model using parameters estimated from the experimental data, and the subsequent evaluation of the properties of the model. Data description To determine the daily progress of a furunculosis epidemic, seventy 7.5 L tanks, each with approximately 43 susceptible Chinook salmon (1.97 g), were exposed to a single presumably infectious fish, which were obtained by bath-challenging fish with A. salmonicida (105.2 bacteria mL)1) 3 days before they were to be used as donors. Twenty-four hours after inclusion of a single infectious individual, seven tanks were randomly selected, all fish were anaesthetized and checked for A. salmonicida from head kidney, using Coomassie brilliant blue agar (Cipriano & Bertolini 1998). Formation of blue colonies and brown pigment formation were used to differentiate A. salmonicida from other bacteria. If there were few colonies, then they were streaked further and checked again for brown pigment formation, blue colonies on Coomassie brilliant blue agar and growth at 37 C. As kidneys were used for sampling aseptically, no questionable cases were encountered. The same procedure was repeated every



24 h for 10 days. Every day, one control tank that was exposed to phosphate-buffered saline (PBS) was also terminated and all fish were tested bacteriologically. Fish were monitored daily for mortality and prevalence (infection). Every fish (n ¼ 2928) used in the experiment was examined bacteriologically for A. salmonicida. There were no latently infected or dead fish at the beginning of the epidemic. Ogut (2001) gives more information about biological aspects of the experiment. From the experimental furunculosis epidemics, several disease transmission parameters were estimated. First, analysis of more than 100 epidemics (Ogut 2001; Ogut & Reno 2005) (different data from that described above) showed that a susceptible individual fish becoming infected by cohabitation with an infectious fish showed a latency period of approximately 3 days. Fish were then infectious for the next 2 days (releasing large amounts of bacteria) and died after the end of this infectious period. The infectiousness rate (h), i.e. the rate at which latent fish became infectious, and disease related mortality rate (a), i.e. the rate at which infectious individuals die, were estimated as 0.33 of latents per day and 0.5 of infected individuals per day, respectively. From the daily observations over 10 days in the experimental data described above, subclasses of the population falling into each category (susceptible, latent, infected and dead) were determined. From these data, the transmission coefficient of this epidemic was estimated to be 0.0214 (Ogut et al. 2004).

Model development The stochastic SIR model


The stochastic model developed here is composed of three compartments: susceptible (S ), infected (I ) and dead (R ). There are three parameters in the model: the transmission coefficient, disease related mortality rate and natural mortality rate. The transmission coefficient is the rate at which new infections occur, i.e. susceptible fish become infected. It is the sum of the factors related to host, pathogen and environment which determine the success of a contact between an infectious and a susceptible fish. Highly infectious strains of microparasites may be expected to have larger transmission coefficient values and vice versa. Recovery rate is not considered in the models described here since a short-term acute epidemic is modelled. 95

In this model, the distinction between the latency stage and infectious stage was ignored. In reality, a susceptible fish becoming infected with A. salmonicida goes into a latency stage which takes approximately 3 days (Ogut 2001), and disease-related mortality is not observed prior to the fifth day after becoming infected. Thus, in this model a lag period, i.e. a period in which no disease related mortality was observed, for the disease related mortality was applied (3 0.5 days). The stochastic SLIR model The same approach described above was used for this model. This model is more realistic than the simple SIR model in that the SLIR model distinguishes between latently infected fish and infectious fish, with fish entering the infectious stage prior to dying. In the experimental data, latent and infectious individuals were not separated due to experimental difficulties. Therefore, in the data fitting, the expected compartments, L and I in the SLIR model, were combined and compared with the number of infected individuals in the observed data. A lag period (3 0.5 days) for disease related mortality was also used in this method. The mean length of the latent period is 3 days, therefore, the rate at which latently infected fish become infectious is 0.33 fish day)1. Two types of events in the SIR model and three types of events in the SLIR model are shown in Table 1. The stochastic setting of the compartmental model is described below for the SLIR model. There are two components in the stochastic models: the event type (Table 1) and the inter-event time (Renshaw 1991; MacKenzie & Bishop 2001a,b). The time periods between events may be considered random, with probability of specific events defined by the components of the model as described by Renshaw (1991). The inter-event time at time t, in a population of S susceptible animals, L latent and I infectious animals at time t in a furunculosis epidemic has a mean: 1 ðbSI þ hL þ aI Þ

ð1Þ

where b is transmission coefficient, h is rate of infectiousness and a is disease related mortality of the infectious class. Note that the equations given here describe S, I and L at time t. The inter-event time is drawn from an exponential distribution of )ln(r)*mean inter-event time,



Table 1 Possible events and their rates during a furunculosis epidemic in the SLIR and the SIR models. S, L, I and R are susceptible, latent, infected and dead classes, respectively. b, h and a are transmission coefficient, infectiousness rate and mortality rate of infected, respectively. Thus, (S, L) represents the number of individuals in the S and L compartments, (S ) 1,L + 1) represents removing one individual from the susceptible class to the latent class

smaller than the sum of the probabilities given in equations (3) and (4), then one latent individual was moved to the infectious class. Lastly, if r was larger than sum of the equations (3) and (4), then the case was a disease related mortality. Model exploration

Event and model type The SLIR model Latency Infectiousness Dead The SIR model Infected Dead

Symbolic representation of the event

Rate

(S,L) fi (S ) 1,L + 1) (L,I) fi (L ) 1,I + 1) (I,R) fi (I ) 1,R + 1)

bSI hL aI

(S,I) fi (S ) 1,I + 1) (I,R) fi (I ) 1,R + 1)

bSI aI

where r is a random number between 0 and 1. Event type probabilities are calculated as described below; First, RATE is calculated; RATE ¼ I ðbS þ aÞ þ hL

ð2Þ

This is the overall rate at which all events are occurring in the population. Then, the probabilities of different event types are calculated separately. Because the population has homogeneous mixing, the contact rate is set to 1.0. The probability that the next event is a new infection (removal of a susceptible individual to the infected class) is bSI RATE

ð3Þ

We explored the predictive capabilities of the model, comparing the number of fish predicted to be in each compartment at each time point with the observed experimental data. Additionally, we explored the variability inherent in a stochastic model, determining the replication necessary to achieve reliable predictions. Specifically, simulations were run 5000 times. After 100 runs, the daily mean of expected values for each compartment were calculated from the total number of runs carried out to that point, and all the compartments was compared to the daily observed epidemic values by weighted least squares, where the weighting factor was the standard deviation of the numbers of animals within each category, on each day. The residual sum of squares was calculated for each simulation. Additionally, the goodness-of-fit (R2) values were calculated, both from the individual simulations and from the average outcomes after every 100 simulations for a total of 5000 runs. Lastly, between-replicate variability of epidemics obtained from the simulations was compared with between-replicated variability observed in the experimental data. Results

The probability that the next event is a new infectious individual moved from the latent class is hL ð4Þ RATE and the probability of the removal of an infectious individual due to disease related mortality is aL RATE


ð5Þ

After calculating all of these probabilities at time t, they were compared with a random number (r) generated between 0 and 1. If r was smaller than the probability obtained by the equation (3), one susceptible individual was removed to the latentclass as a new infection case. If r was larger than the probability obtained by the equation (3) and r was 96

Both stochastic models were successful in predicting the mean number of fish in each category in the experimental data at each time point, although the simple SIR model was slightly better (statistically different; student’s t-test, P < 0.05) than the simple SLIR model (R2 ¼ 0.91 vs. R2 ¼ 0.90; Figs 1–3). Moreover, the SIR was also successful in representing variability in the observed data (Fig. 4), as the between-replicate variation in the numbers of infected fish at each time point was similar in the simulated and experimental data. The mode of the weighted sum of least squares, calculated on an individual replicate basis, occurred at R2 ¼ 0.9 (Fig. 5) in both models. Of 5000 simulated epidemics with the SLIR model, 82% of them had R2 ¼ 0.7 and above, whereas 80% of the simulated epidemics had R2 ¼ 0.7 and above with the simple SIR model.



Figure 1 Observed number of fish (with SE bars) in the susceptible compartment in an experimental furunculosis epidemic (h), with the predicted number of susceptible fish from the SIR stochastic model (——) and from the SLIR stochastic model (- - - - -) (n ¼ 44, initial I ¼ 1, b ¼ 0.0214, a ¼ 0.29 and c ¼ 0.00015 with a latency period of 3 days).

Figure 4 Variation in the observed number of infected animals (SD) and the expected number of infected animals (SD) obtained from the SIR stochastic model.

3000

Frequency

2500 2000 1500 1000 500 0 0

Figure 2 Observed number of fish (with SE bars) in the infected compartment (combining latent and infectious classes) in an experimental furunculosis epidemic (h), with the predicted number of susceptible fish from the SIR stochastic model (——) and from the SLIR stochastic model (- - - - -) (n ¼ 44, initial I ¼ 1, b ¼ 0.0214, a ¼ 0.29 and c ¼ 0.00015 with a latency period of 3 days).


Figure 3 Observed number of fish (with SE bars) in the diseasedependent mortality compartment in an experimental furunculosis epidemic (h), with the predicted number of susceptible fish from the SIR stochastic model (——) and from the SLIR stochastic model (- - - - -) (n ¼ 44, initial I ¼ 1, b ¼ 0.0214, a ¼ 0.29 and c ¼ 0.00015 with a latency period of 3 days).

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Goodness of fit

1

Figure 5 Frequency of goodness of fit (R2) of six 5000 runs. The R2 was rounded to the closest first decimal place (e.g. 0.23 to 0.2 and 0.59 to 0.6).

The predictive value of the data consisting of daily means of all simulation runs showed a slight dependency on the number of simulations (Fig. 6). There was no improvement in the goodness-of-fit (R2), when compared with the experimental data, after 200 runs in the SIR and 500 in the SLIR model. Both the stochastic models explained much of the observed data. In particular, in the SLIR stochastic model we distinguished latent animals from infectious animals within the infected category. The number of latent animals increased until 6 days post-exposure (dpe), and then slightly decreased to the end of the experiment. The number of latent individuals was always predicted to be higher than the infectious individuals during the experiment, i.e. for the time period simulated by the model.



Figure 6 Change of goodness of Fit (R2) as number of simulations increased using the SIR model (solid line) and the SLIR model (hatched line). 5000 simulations were carried out and after every 100 runs the mean values of each compartment were compared with the compartment of the observed experimental epidemic.

Discussion


The stochastic models used here described the observed epidemic well. A large proportion of independently simulated epidemics (>80%) closely resembled the observed epidemics, as judged by the goodness-of-fit criterion. The stochastic SIR model fitted the latter slightly better than the stochastic SLIR due to the fact that the stochastic SIR model better took the large variation at day 5 into account. An analysis of the variability of stochastically simulated outcomes showed that 200–500 simulation runs are required to obtain reliable and stable answers, given the conditions in these models. Furthermore, the variability in outcomes is similar to the between-replicate variability seen in the experimental data, indicating that the stochastic models are indeed capturing the patterns seen in real epidemics. There is a great body of literature on stochastic and deterministic models of terrestrial animal disease dynamics. There is also great amount of information comparing the properties of stochastic models to their counterpart deterministic models, starting with the classical work of Bailey (1975). There is, however, no information on models of diseases occurring in animals of aquatic systems, which offer some unique capabilities for application of both deterministic and stochastic epidemic models. A fish stocked hatchery tank reflects the perfect conditions for the theory of SIR models based on two assumptions: (1) the population is mixing homogeneously and (2) the population is closed. In recent years, stochastic models have been used more frequently for describing animal disease 98

dynamics. Bouma, De Jong & Kimman (1995) estimated host population size dependent transmission of presudorabies virus. Innocent, Morrison, Brownlie & Gettingby (1997) tested effects of different management practices on disease incidence in a closed dairy herd, and Sta¨rk, Pfeiffer & Morris (2000) determined disease spread between farms stochastically. MacKenzie & Bishop (2001b)) developed a stochastic model of hypothetical epidemics caused by microparasites in a pig farm and later applied the same model to quantify the impact of selection for disease resistance by adapting host genotype in the model (MacKenzie & Bishop 2001b). However, the weakness of many of these models is that they do not have robust data against which to parameterise or test their model. In this current study we use replicated experimental furunculosis epidemics (more than 70 replicates), the epidemics producing similar results each time, to demonstrate the utility of the stochastic model. Such data that can be directly used in simple SIR models are rare. Two stochastic models proved to be efficient in predicting observed data, and both approaches have merits. The SIR model fits the data well with fewer parameters, which is advantageous from a modelfitting perspective. On the other hand, the SLIR stochastic model has an advantage over the SIR model in that it separates latent and infectious individuals and treats them differently. This aspect, easily obtained by the modelling approach, is very important, since in reality only infectious individuals contribute to the occurrence of new cases. During the latency stage there is no release of the bacteria. Moreover, one cannot expect disease related mortality during this stage. As a result, significant improvement in the predictive capability of the SIR model was gained by adding a lag period for the absence of disease related mortality for the first 3 days of the epidemic. A further unique attribute of the data used in this model is the replication, and this allowed comparison of experimental and predicted variability in outcomes. The between replicate variability in expected data resembled the variability in the experimental data. It can be concluded that the stochastic model does capture much of the variation seen in experimental situations. More importantly, it demonstrates how potentially misleading unreplicated experimental data can be. The stochastic model applied here may be considered more realistic than the deterministic




model described by Ogut et al. (2004) for several reasons, although statistical fit is relatively similar. Firstly, the type of event and time between events are considered random (Renshaw 1991; MacKenzie & Bishop 2001a) rather than fixed as in the deterministic models, reflecting our understanding of biology. Secondly, the results of a stochastic model have inherent variability reflecting the observed experimental data, unlike deterministic models which simply predict expected outcomes. Thirdly, as described above, the stochastic model effectively describes between replicate variability in experimental data unlike deterministic models which ignore this variation. Furthermore, though not subject of this paper, differences between epidemics that could be attributed to individual differences in susceptibility, or any of the other infection transmission parameters, may easily be simulated in a stochastic model. An important source of such variability may be differences in the degree of infectiousness of infectious fish used to initiate an epidemic in a group of susceptible fish. In this study, we demonstrated the probable change in the number of latent and infectious individuals during the progress of a furunculosis epidemic. In the deterministic model (Ogut et al. 2004) and the stochastic simple SIR model, which did not differentiate between latent and infectious individuals, the number of infected animals peaked at the seventh day and gradually decreased from that point on. Similarly, the current stochastic SLIR model suggested that the number of latent animals gradually increased until the sixth day and then gradually decreased parallel to the decrease in the number of infectious individuals. The number of latent individuals was always higher than infectious individuals during the course of the epidemic, as expected. Separation of the latent compartment from the infectious compartment could increase predictive capabilities of these models when key risk factors are integrated. As a result, more effective control measures could be developed. In terms of expected outcomes, one can see deterministic trends in the stochastic model, suggesting that the deterministic model could be considered as an approximation of a more realistic stochastic counterpart (Ball & Neal 2002). Kurtz (1970, 1971) showed that the approximation is good if compartments are sufficiently large. In another study, West & Thompson (1997) suggested that deterministic approximation is not appropriate for an underlying stochastic process 99

when there are few infected individuals during the initial phases of an epidemic. Therefore, differences in the results of the deterministic and the stochastic models may arise when there are few initially infected individuals, and this is precisely the situation that is most likely to closely resemble real-life situations. In summary, stochastic models were successful in representing the observed mean values and observed variability in data from experimental epidemics of furunculosis. Additionally, unlike the deterministic model described in Ogut et al. (2004) and the SIR stochastic model, the SLIR model allowed an evaluation of the latent and infectious compartment separately. Now that the general ability of the simple SLIR model has been demonstrated, the next step is to adapt it to include key factors such as temperature, density or other factors relevant to disease control. Acknowledgements

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