One of the key reasons for building simulation models is so that we can explore the effectiveness of different strategies for controlling disease in populations. Ideally, we want to find the combination of interventions that will provide the greatest reductions in the prevalence of infected individuals and prevent new susceptible individuals from becoming infected with the least overall costs. It is important we are confident that our simulation model is accurately capturing the population demographics and disease transmission dynamics before starting to impose interventions on the system otherwise the results will likely not make biological sense. In this chapter, we will first learn about the different options that can be used to control disease and how these can be built into our simulation model.
By the end of this module, participants should be able to:
In order to eradicate disease from a population, we ultimately need to empty any compartments that contain individuals who are infected with the pathogen (including exposed, infectious, and carrier individuals as well as any other vectors or reservoirs in other species). This is done primarily by using strategies that reduce the flow of susceptible individuals into the infected compartments and increase the flow of infected individuals out of their infected compartments. Remember that the flow of individuals is determined both by the flow rate and the number of individuals in the compartment. The more we can reduce the R0 value below 1, the faster we can eradicate disease from the population.
We can break our control interventions into three general categories based on how they affect different aspects of the disease transmission dynamics.
These strategies work on the principle of reducing the pool of susceptible individuals who could potentially become exposed and move into the I compartment. We primarily do this by vaccinating (V) animals or ensuring good colostrum intake so that animals have good maternal antibodies (MA). In most models, animals will have maternal antibodies from being born to a R dam, but in some systems, vaccinating dams can also give birth to animals that will have waning immunity to the pathogen. Not all vaccinations are 100% effective and so it is important to multiply the rate of vaccination (i.e. how many individuals on average vaccinated each time step) by the proportion of animals that will be protected (i.e the vaccine efficacy Ve). Depending on the duration of immunity from vaccination, animals may become susceptible again over time if your intervention programme does not re-booster animals at the appropriate frequency.
Another important point to think about if you are estimating costs for vaccination is because you can rarely in practice distinguish between S, I, and R animals, you will likely end up vaccinating the entire population. However, the model will only show the S individuals moving into the V compartment.
One other possible strategy for reducing the number of S is to purposefully infect them with the pathogen so that they develop natural immunity. Classic examples are chicken pox in humans and BVD in cattle (where farmers advocate leaving PI animals in the herd as natural vaccinators). There are obviously some ethical issues around this and it is usually better if we can focus on control strategies that prevent individuals from becoming infected in the first place.
Remember that the β term includes elements for both the contact rate and the probability of transmission. Practical ways we can reduce the contact rate between susceptible and infected individuals would be, for example, to encourage susceptible herds to remain completely closed to movements or to quarantine infected herds so they cannot make contacts with susceptible herds. This will generally not change the overall structure of your model, but simply reduce the magnitude of β. If you were able to completely prevent contact between management groups in a herd, you could just simply knock out the infection pressure from the other management group in your transmission term. If contact between groups is reduced, but not completely eliminated, you would have to create separate β terms for each I compartment and just scale down the β from infection pressure caused by other management groups.
The cornerstone of this strategy is being able to first identify infected individuals so that you can actually target them with interventions such as culling or treatment. The diagnostic test could be based on clinical signs or some other laboratory based test. If the diagnostic test is not 100% sensitive, you will need to adjust the rate of test and removal by the test sensitivity. Also remember that if you are budgeting out the cost of testing, you will have to include all S and R individuals who will likely be tested as well. Adding in treatments that cause individuals to recover faster will end up modifying the magnitude of γ.
One of the most common ways of assessing the efficacy of interventions in compartmental modelling studies is to compare the values for Ro before and after intervention. If we have simply adjusted the value of a parameter, then we can just use our existing equation for Ro. If we have made changes to the structure of the model by adding new compartments or arrows, then we will need to write a new set of model equations and derive a new equation for Ro. We can also re-plot our epidemic curves to look at how the spreading pattern of the disease has changed as the result of intervention. If you are considering controlling disease in an endemic situation, just remember that you will need to run you old model first to get values for how many individuals you would expect to be in each compartment at endemic stability. Then you will use those values as your starting values for the new model with control interventions.
Another cool thing we can do with the Ro equation is to ask questions like “What would my rate of test and culling have to be in order to control disease? To do this, we simply make the Ro value in the equation equal to 1 (our threshold for controlling disease), substitute in the values for all of the other parameters besides our test-and-cull rate, and then solve for the test-and-cull-rate.
We can also use the value of R0 to estimate the proportion of animals in the population that must be vaccinated to achieve adequate herd immunity.
The larger the value of R0 and the less effective the vaccine, the greater the proportion of individuals that must be vaccinated.
Read through the description of swine influenza pathogenesis below.
In a classic swine influenza epidemic, the disease can spread through an entire swine herd in 1 - 3 days with morbidity reaching 100%. Pigs begin excreting virus particles less than 24 hours after infection. The predominant clinical signs are fever, anorexia, coughing, dyspnea, and mucous discharge. Recovery occurs within 3 – 7 days, but the virus can persist in carrier animals for 1 and 3 months. Mortality from the primary and secondary infections attributable to the virus ranges from 1 – 4%. There are no good estimates in the literature for the proportion of animals that become carriers, however, one study was able to isolate viral DNA from about 30% of swine in the Midwest US and another study isolated virus from 25% of samples that were submitted to pathology to identify causes of respiratory disease. In this worst case scenario, this means that approximately 27.5% of animals become carriers for influenza. The average age at replacement for sow production units is 36 months. Antibodies produced in natural infections have been documented at 28 months post infection. Although no other studies have examined time points beyond 28 months, it is reasonable to assume that recovered animals will remain resistant to the same influenza strain until they are culled at 36 months.
| Parameter | Value | Justification |
|---|---|---|
N |
2000 sows | Baseline size of swine production unit |
μ |
0.000926 an/day | Average age at replacement is 36 months, converted to an average of 1080 days |
β |
0.000469 | Estimated by calibration |
γ |
0.2 an/day | Animals show clinical recovery in 3–7 days; an average of 5 days was used |
σ |
0.0167 an/day | Animals remain carriers for 1–3 months; an average of 60 days was used |
R₀ |
1.923 |
Calculated using R₀ = N / X*, where X* is the equilibrium proportion of susceptible animals
|
In a simple endemic model, R₀ = N / X*, where N is the total population size and X*
is the equilibrium prevalence of susceptible animals. Based on a serological survey of swine influenza in finisher units,
the average prevalence of susceptible animals across 12 herds with known influenza exposure was approximately 52%.
Background information
Swine influenza is a highly contagious orthomyxovirus of the type A group that causes acute respiratory disease in swine. Although human cases have been reported, the primary host range is swine and the virus does not readily spread through human populations. Swine influenza can be found in operations on every continent but the majority of cases in the United States are localized to the Midwest. The disease has serious economic consequences because it can delay finishing pigs from reaching slaughter weight and reduce fertility in sow production units. In a survey of finishing units, swine influenza was consistently ranked among the top three infectious disease concerns and was cited as being a significant challenge for control by 18 of 19 companies (Holtkamp, 2007). Public health concerns have also emerged over the zoonotic potential of swine influenza. Pigs may act an intermediate host and mixing vessel to allow genetic reassortment between human and avian strains (Scholtissek, 1998). Thus, the prompt detection and eradication of influenza outbreaks will be key to preventing new pandemics and increasing the global productivity of the swine industry in the face of rising demands.
Basic Endemic Model
In a classic influenza epidemic, the disease can spread through an entire swine herd in 1 – 3 days with morbidity reaching 100%. Pigs begin excreting virus particles less than 24 hours after initial infection. The predominant clinical signs are fever, anorexia, coughing, dyspnea, and mucous discharge. Recovery occurs within 3 – 7 days, but the virus can persist in carrier animals for 1 and 3 months (Straw et al). Mortality from the primary and secondary infections attributable to the virus ranges from 1 – 4% (Merck, 2005). There are no readily available estimates in the literature for the proportion of animals that become carriers, however, one study was able to isolate viral DNA from about 30% of swine in the Midwest (Hinshaw et al) and another study isolated virus from 25% of samples that were submitted to pathology to identify causes of respiratory disease (Foni, 2003). In this worst case scenario, this would suggest that approximately 27.5% of animals become carriers for influenza. The average age at replacement for sow production units is 36 months (Pla, 2003).
| Symbol | Definition |
|---|---|
N |
Number of sows in herd (constant) |
μ |
Rate of culling / introduction of new sows |
S |
Number of susceptible sows |
β |
Force of infection |
I |
Number of infected sows |
σ |
Rate of carriers becoming recovered / resistant |
R |
Number of recovered / resistant sows |
θ |
Partitioned outcomes from infection:
|
C |
Number of carrier sows |
D |
Number of animals that die from influenza |
Model constraints
Antibodies produced in natural infections have been documented at 28 months post infection (Derosiers, 2004). Although no other studies have examined time points beyond 28 months, it is reasonable to assume that recovered animals will remain resistant to the same influenza strain until they are culled at 36 months. Piglets were excluded from the model as a source of infection since maternal antibodies should protect them for at least 4 weeks and piglets are typically weaned and removed from the sow unit at 3 weeks. The assumption was made that all animals entering the herd are susceptible to the endemic influenza strain and that the rate of entry of new animals is adjusted to maintain a constant herd size. Influenza viruses are known to undergo antigenic variation which may or may not render animals susceptible to subsequently introduced strains (Jahnke, 1996). This model was based on only 1 endemic strain in the herd and no introduction of new influenza sub-types over the duration of control measures. Carrier animals were assumed to be equally as infectious as the clinically infected animals, representing the worst case scenario.
Estimating model parameters
| Parameter | Value | Justification |
|---|---|---|
μ |
0.000926 /an/day | Average age at replacement is 36 months, converted to an average of 1080 days |
β |
0.0000469 |
Calculated from R₀ = 1.923 and herd size 2000 sows
|
θ |
0.2 /an/day | Animals show clinical recovery in 3–7 days; an average of 5 days was used |
σ |
0.0167 /an/day | Animals remain carriers for 1–3 months; an average of 60 days was used |
R₀ |
1.923 |
In a simple endemic model, R₀ = N / X*, where N is total population size and X*
is the equilibrium prevalence of susceptible animals
|
According to a serological survey of swine influenza in finisher units, the average prevalence of susceptible animals across 12 herds with known influenza exposure was approximately 52% (Poljak et al, 2008).
Endemic Models with Control Measures
The existence of a carrier state is the primary reason why swine influenza persists in domestic production systems. Assuming equal infectivity, the contribution of clinically infected individuals to Ro is 0.467 whereas the contribution of carrier individuals is 1.456. There are no effective treatments to stop carrier shedding so control efforts have largely focused on herd depopulation, mass vaccination, and preventing infected animals from entering the herd. The purpose of this project is to establish if test and removal would be more effective for controlling endemic influenza than vaccination.
Estimating additional model parameters for control measures
| V = number of vaccinated individuals | ||
| κ = rate of return to susceptibility in vaccinated sows | 0.00556 /an/day | Wyeth recommends revaccinating every 6 months, so if animals were vaccinated once upon entry to the herd, they would be susceptible again at about 180 days. |
| 0.8 μ 0.02 θ | 0.0007408 /an/day 0.004 /an/day | The efficacy of the vaccine is approximately 80%, so these individuals will enter the herd vaccinated. |
| 0.2 μ 0.005 θ | 0.0001852 /an/day 0.001 /an/day | Vaccination will not take in approximately 20% of swine, so these animals will enter the herd susceptible. |
| δ = rate of test and removal of carrier sows | 0.03053 /an/day | Calculated back from R₀ = 1 and the test-and-removal R₀ equation. |
Vaccinate incoming susceptible animals
For the vaccination strategy, all susceptible animals entering the herd will be vaccinated with Wyeth’s Suvaxyn SIV vaccine (Wyeth, 2007). In a challenge study using an H3N2 strain, the vaccine was able to prevent viral shedding in 80% of animals 3 days post challenge so presumably the remaining 20% will enter the herd susceptible. The animals that recovered or became carriers following the initial epidemic are presumed to be immune to swine influenza for their remaining duration in the herd. Setting Ro to equal 1, the maximum susceptible population can be no more than 1,040 sows out of the 2000 sow herd or 52%. For a serologically naïve herd, this means that vaccination should theoretically be successful in controlling a disease outbreak, even if the efficacy of the vaccine is considerably less than 80% in the field.
Test and remove carriers
Extrapolating from the equation p= 1 – 1/Ro, you would have to remove at least 48% of clinically infected individuals to control an outbreak. At first glance, this appears reasonable. However, due to the variability in clinical signs, the explosive nature of the disease where animals may only exhibit signs for 3 days, and the difficulties in monitoring individual animals in the herd, a more feasible option may be to test and eliminate carriers. Viral isolation is considered the gold standard for diagnosis of active infections, but may take 1-2 weeks and cost up to $30 per test (Jahnke 2000, ISU 2008). To ensure Ro is less than 1, you would have to remove the carrier from the herd within 33 days (δ= 0.03053). This is adequate time to perform viral isolation tests for identify carriers. If the carrier sow is near term or recently farrowed, this would also allow enough time to complete a 21 day lactation to minimize the economic loss associated with prematurely culling sows. If the carrier were removed within 2 weeks (δ= 0.07143), Ro would decrease to 0.756 and if the carrier were removed within 1 week (δ= 0.1429), which represents the limit of the viral isolation test, Ro would drop to 0.627.
Since viral isolation is considered the gold standard for swine influenza diagnosis, it is difficult to estimate its sensitivity and therefore the true impact of false negatives on disease control. Setting the Ro equation from the basic endemic model equal to 1 and substituting Xθ for 0.275θ, the maximum proportion of animals that can become carriers after an acute infection is 0.1, which implies that the viral isolation test would have to identify at least 36.4% of the total carrier animals in the herd. One would hope that as a gold standard diagnostic test, the sensitivity is significantly greater than 36%.
References
Desrosiers R, Boutin R, Broes A. Persistence of antibodies after natural infection with swine influenza virus and epidemiology of the infection in a herd previously considered influenza-negative. J Swine Health Prod. 2004;12(2):78-81.
Foni E, Chiapponi C, Fratta E, et al. Detection of swine infleunza virus by RT-PCR and standard methods. Proceedings International Symposium on Emerging and Re-emerging Pig Diseases. Rome June 29th, 2003. p 270
Hinshaw VS, Bean WJ, Webster RG, and Easterday C. The prevalence of influenza viruses in swine and the antigenic and genetic relatedness of influenza viruses from man and swine. Virology. 1978 Jan;84(1):51-62.
Holtkamp D, Rotto H, and Garcia R. Economic cost of major health challenges in large US swine production systems – part 2. Swine News. 2007;30(4). Available from here.
Iowa State University. Fee Schedule – Veterinary Diagnostic and Production Animal Medicine. Accessed 15 Oct 2008. Available from here.
Janke BH. Significance of antigenic variation in swine influenza virus for serodiagnosis. Proc Sw Dis Conf Pract. 1997:7-14.
Janke BH. Diagnosis of swine influenza. Swine Health Prod. 2000;8(2):79-84.
Merck Veterinary Manual ninth edition. (2005) Respiratory infections in swine. National Publishing Inc, Philadelphia, PA.
Pla LM, Pomar C, and Pomar J. A Markov decision sow model representing the productive lifespan of herd sows. Agricultural Systems. 2003; 76(1): 253-272.
Poljak Z, Friendship RM, Carman S, et al. Investigation of exposure to swine influenza viruses in Ontario (Cananda) finisher herds in 2004 and 2005. Prev Vet Med. 2008; 83; 24-40.
Straw BE, Zimmerman JJ, and D’Allaire S. Diseases of swine. Available from here.
Scholtissek C, Hinshaw VS, and Olsen CW. Influenza in pigs and their role as the intermediate host. In: Textbook of Influenza (Ed. Nicholson KG, Webster RG, Hay AJ) Blackwell Science, Oxford. 137-145.
Wyeth Animal Health. Introducing Suvaxyn SIV – H3N2 challenge study. 2007. Available from: http://www.wyethah.ca/swine.asp?pageid=siv0306
In many cases, your simulation model will be used to justify the cost-effectiveness of different management recommendation and so it is important to have a good understanding of animal health economics so that you can build the appropriate components into your model.
One of the main limitations with the simple economics models like those presented in the Animal Health Economics study guide it that they ignores both the within-flock and between-flock transmission dynamics for situations where we are evaluating the impacts of infectious disease. For example, if we wanted to look at the economics of vaccinating sheep flocks for footrot, we would ideally also like to know how long it would take the flock to clear footrot or even if the interventions would be enough to control footrot in the flock. Fortunately, we can easily integrate economic models with disease simulation models to help us get a better estimate of the costs and benefits of intervention.
The key to doing this lies in understanding what production or health impacts the disease will cause in affected individuals since this is what we will need to track in our models. For example, if one of the effects of disease is mortality, then we will need to create an output storage vector that tracks the number of individuals that die every day (i.e. the 0.025 γI individuals that die per day in the swine influenza example below). Then we can compare things like the annual production outputs of farms before and after implementing a control measure.
If the disease has impacts on things like production levels (impacting total yield) or product quality (impacting price per unit yield), we will also need to build in mechanism to track offtake from the system. In mastitis models, for example, we could estimate an average daily milk yield for animals that are unaffected or affected with mastitis. We would then create an output storage vector to record the total daily milk yield for the farm, which would be equal to the milk yield from all the S and R animals plus the milk yield from all the clinical I and subclinical I animals. If we really wanted to be fancy, we could even model the effect of having high SCC on the price received for the milk. If one of the control interventions in your Staph aureus model was providing intramammary treatment during lactation, you would need to remember to create a compartment for animals being actively treated since their milk would be withheld from the bulk tank to prevent antimicrobial residues.
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