Development of a model of the spread of pilchard kill events in southern Australian waters. We will construct a 1-D SIR (susceptible, infected, removed) model of the spread of the 1995 and 1998/9 pilchard mass mortality events.
2 The Aims, Limitations and Structure of the Modelling Project
We will begin by examining such models as valuable illustrations of the processes underlying the spread of epidemics. There is the initial development of the epidemic, there are gaps in the evidence of the epidemic progression (which may or may not be valid), and there are physiological and.
3 Available Data and the Characterisation and
But complexity detracts from the comprehensibility of the model and can make it dependent on parameters whose value cannot be estimated independently. In the following chapters we will explore model development from simple standard epidemic models to a form adapted to describe these pilchard herpesvirus epidemics.
Parameterisation of the Epidemic's Spread in Space and Time
Spread of the Epidemic
We have to describe the location of the front in relation to the distance the epidemic has actually advanced along the coast. And this corresponds to the speed of the westward wave, allowing for a delay at Cape Leeuwin.
Duration of Infection in a Population
Further away from the origin, mortality lasted less than 10 days, and given that the epidemic takes about a week to cross 250 km and pilchards can be detected some time after death, this means that the local duration of infection is very short. There is one exception, outside central Western Australia mortality persisted for 14 days, but this duration is due to a single event occurring 10 days after other mortality events.
Distance
- Constraints on Viral Transmission Rate
- Pilchard Population Structure and Density
- The Virus
- Length of the Infection in a Single Individual
- Survivorship and Mortality
- Conclusions
However, these local aggregations do not appear to affect the rate of spread of the epidemic. We do not expect breeding-related behavior to explain the differences in transmission from the epidemic front.
4 Local Epidemic Models
- Continuous Turnover
- Fixed Infection Phase Period Lengths
- Effect of Schools on Transmission
- Transmission at the School Level
- Transmission between Schools
However, for the replication of the mortality pattern inflicted by pilchard herpesvirus, these models exhibit a fundamental flaw. When infection starts in a swarm, a smaller number of infected individuals will be introduced to one of the schools.
![Figure 4.1 Distribution of mortality, normalised to maximum, following infection under the SIR, SEIR and [S][E][I][R] models and mortality of prawns caused by White Spot Syndrome Virus](https://thumb-ap.123doks.com/thumbv2/5docco/10389008.0/34.894.151.646.126.387/figure-distribution-mortality-normalised-following-infection-mortality-syndrome.webp)
The Model Structure
Therefore, factors such as birth and non-disease death of the pilchards were not included in the model. The model then acts as a conveyor belt, inputs for the period are continuously placed into the initial variable subcomponent of the school (eg Ez1). This is especially true in the initial and final stages of the spread of infection between the schools.
The value of β at the swarm population level (and higher levels) emerges from this value and the interaction of the schools.
Model Results
A good measure of the length of the epidemic is the number of days in which 50% of the population is infected. This is not the days from the start of the epidemic, but rather the number of days around the peak of the epidemic. Another interesting, although less compact, measure of the effect of schooling on the models is apparent β.
Not surprisingly, the time required to cause 50% infection of the pilchards is highly sensitive to the infection rate parameter in both the SEIR and [S][E][I][R] models.
E Ipy
Discussion
We developed a simple dynamic model to investigate the effect of the interaction within a school of schooling fish. We apply models of the effect on transfer of both the local interactions within the school and of the interaction between the schools of a school. These conclusions do not seem to depend on the structure of the model used.
The rate of transmission at small and larger levels is likely to be independent of sphere size.
6 Modelling Large Scale Dispersion in a Spatially Structured Dynamic Model
The Models
Details may differ due to the role of the E phase and the fixed period length versus continuous turnover in delaying the feedback between the occurrence of an infection and the initiation of further infection by that infected individual. Our overall conclusion is that in the case of the pilchard epidemic we can dispense with the direct modeling of school behaviour. But they do not describe the dynamic nature of the spread of the epidemic, especially at its origin.
They cannot be used to describe the non-linear characteristics of an epidemic and must be re-evaluated for each different model structure.
Model Implementation
The population class holds the spatial and infection stage (S, E, I or R) and substage structure of the model. Diffusion is divided by the time step size and the square of the space step size to create a proportional exchange between adjacent boxes. The flux varies with the inverse square of the spatial scale and so when it is reduced, flux can increase rapidly.
The length of the I (infectious) and E (latent) periods, in days, along with an update resolution (per day) are used to calculate the number of I and E subphases in the infection periods.
Analytical Solution of the Model's Wave Speeds
This option detects the time when the I wave reaches the target landmarks, which are spaced at 0.1 spatial resolution intervals. In the simple SIR (susceptible infected, removed) model described by Murray (1993) to examine the spread of the Black Death, the following (1-dimensional) epidemic rate is derived:. In the next chapter, a method is developed that enables the analytical determination of the solutions of the model of the length of the fixed latent period.
The solution of the model is very insensitive to the turnover of the infectious phase.
7 A Tractable Deterministic Multiple Diffusion Coefficient Model With Latent Period For Virus Epidemics In
These analytical solutions allow us to quickly assess the sensitivity of the wave velocity of the model to parameterization and thus focus on a more detailed analysis of other characteristics, such as the duration of mortality at a given point. The fixed latent and fixed infectious period length version of the model appears to have very similar solutions. Thus, we can use the analytical method as a tool to investigate both models in the relevant region of the parameter space.
Pilchards
Introduction
The Model
This then becomes manageable, but at the cost of realism because the negative exponential.. distribution favors small values of the variation far too much to match the reality of a latent period that is almost constant. The model then becomes the .. deterministic equivalent of one in which the latent period is a random variable with fairly realistic distribution. Finally, we make its deviation extremely small while holding the mean to obtain, for all practical purposes, behavior indistinguishable from that of the above model.
The system is the deterministic equivalent of a model in which the latent period is random with distribution that of the sum of n independent variations each of which has a negative exponential distribution with parameter σ.
Travelling Wave Solutions
In other words, the latent period has a gamma distribution with mean nσ and variance nσ2. We do not assume that the latent phase physically consists of a chain of different, although similar, subphases. Recall now that the model is the deterministic equivalent of a model in which the latent period is a random variable with a gamma-type probability density actually given specifically by .
Thus, from (7.43) we can say that if the latent period is a Gamma variable with mean band variance v, then so will the rate of the waveform solutions.
8 Parameter Sensitivities of the Speeds and Duration of Model Mortality Waves
- Wave Speed
- Duration of Mortality Waves
- A Maximum Limit on Latent Period Length
- Levels of Infection and Mortality
- Conclusions
Wave speed is largely independent of the rate of turnover of infected individuals (death or recovery). The last parameter to which the model wave velocity is highly sensitive is the length of the latent period (Fig. 8.3). All versions of the model are sensitive to the square root of the diffusion coefficient.
The epidemic wave speed of the model also depends on the square root of the diffusion coefficient.
9 The Initialisation and Stabilisation of Wave Speeds
- Wave Speed and Stability under Different Latent Periods
- The Initial Formation of the Epidemic
- The Initial Formation of Epidemic Waves
- Conclusions
The duration of mortality at the origin can be increased if the length of the infectious period is shortened (Fig. 9.4). We look at the original nature of the wave that occurs within a short distance from the origin. The wave develops smoothly without extension of the epidemic around the origin compared to later waveforms.
Such a long latent period is not in accordance with the speed of the next epidemic wave.
10 Non-Linear Transmission and the Epidemic Spread
This relationship between the infection rate and I has particular significance for the origin of epidemics. However, with the initial penetration of the infection wave, I is small and I2 is very small. This means that the leading front of the infection in the quadratic model is an area of low production of infection.
The difference between the linear and quadratic models lies largely in their implications for the origin of the epidemic.
11 Discussion of the Modelling, Impacts and Management of Epidemics
- General Discussions on the Modelling
- What Controls Epidemic Behaviour?
- Differences Between and within Epidemics
- Longer Term Impacts of the Epidemics
- Management Options
The latent period is a major factor in the pattern of persistent recurrent mortality during the initial phase of the epidemic. This process would only lead to changes in the speeds of the two arms. Two parameters, the rate of viral transmission and the latent period of infection, can be changed resulting in changes in the wave speed.
Population recovery after the epidemics depends on the production of sardines (Wada and Jacobson 1998).
12 Benefits, Future Developments and Conclusions of Research
Benefits
Future Developments
Conclusions
We used observations, especially the initial behavior of the epidemic to select the model structure; however. 7 We will construct a simple fishery recovery model to investigate the period required for stocks to become vulnerable to renewed mortality. 10 We will produce a final report detailing the final form of the model produced and including analysis of the model's structure, parameters and outputs.
Three scientific papers have been produced on the modeling, which exceeds plans to expand the results.
A study of the comparison of diffusion with increase in the amount of matter and its application to a. 1986) Influence of nonlinear incidence rates on the behavior of SIRS epidemiological models. Virus dynamics: a model of the effects of size, shape, movement and abundance of unicellular planktonic organisms and other particles. Sardinops sagax (Clupeidae), within a large south-eastern Australian bay. 1970) Migration of the sardine Sardinops ocellata in Southern Africa.
Modeling the behavior of the northern anchovy, Engraulis mordax, as a schooling predator exploiting patchy prey.
ACKNOWLEDGMENTS
Interaction between schools and the spread of infection at the local level The main model describing the spread of infection on a continental scale The analytical method for determining the rate of spread of the epidemic A model describing post-epidemic recovery of the population. A collected dataset describing the spread of the 1995 and 1998/9 epidemics. A review of epidemiological models (submitted to the JPSWG). Dr. Brian Jones of Fisheries Western Australia, Perth Dr. Sergui Sokolov of CSIRO Marine Research.
They include members of the Pilchard Joint Scientific Working Group, researchers from CSIRO and national fisheries laboratories, and members of the public who have reported mass mortality events to national authorities.
![Figure 5.2 Parameter sensitivity of the period required for 50% infection in the SEIR and [S][E][I][R]](https://thumb-ap.123doks.com/thumbv2/5docco/10389008.0/42.894.149.771.241.600/figure-parameter-sensitivity-period-required-50-infection-seir.webp)
