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MBW:Modelling the dynamics of a complex life-cycle parasite

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ABSTRACT

Amphibian malformations are a growing problem at many wetlands across the United States. The main cause of these deformities is thought to be the complex life cycle parasite, Ribeiroia ondatrae. This parasite must pass through bird, snail, and amphibian hosts to complete its life cycle, leading to complex transmission dynamics. We constructed a model for the stage of transmission that occurs between the parasite and tadpoles. A model of transmission dynamics at this stage of the parasite’s life cycle can help identify variables that might be important in changing the rate at which amphibians become infected and malformed. We are particularly interested in understanding the roles of predators and changing climate on the transmission of this parasite to tadpoles. We simulated our model with varying levels of predators in the system, and completed a sensitivity analysis to help clarify which parameters had the strongest effect on the model solutions. See APPM4390:Stage-Based Population Model Analysis for another demonstration of sensitivity analysis.

Project Categorization

Mathematics Used

In this project a system of 4 partial differential equations are used to model the dynamics of the populations of parasite and tadpole/host. There are equations for the host organism for the change in the uninfected population and the infected populations. There are differential equation for the different stages of life of the parasite, miracidia and cercariae.

Sensitivity analysis on the parameter for transmission of cercariae to tadpoles and the consumption of cercariae by predators is performed using Simbiology.

Type of Model

The model used is a model of transmission dynamics at different stages from the parasite to the tadpoles. A Simbiology program was used to implement this model and analyze the sensitivity of the parameters in the population.

Biological System Studied

The biological systems studied in this model are the parasite Ribeiroia ondatrae and the 2nd intermediate host the tadpole. The parasite studied is only able to live with the help of a host organism.

INTRODUCTION

Brief History of Macroparasite Life Cycle Modeling

The term macroparasite includes two groups of organisms, helminths and arthropods, of health significance to humans, domestic animals, and wildlife. Macroparasites tend to produce persistent infections, with the host supporting populations of the parasites for long time periods due to continual reinfection. The group of helminths includes roundworms (nematoda), tapeworms (cestoda), and flatworms (trematoda). Macroparasites possess life cycles with either direct transmission, such as human hookworms Ancylostoma duodenale and Necator americanus, or indirect transmission through one or more intermediate hosts, such as Schistosoma mansoni. Mathematical modeling of these parasite systems stems primarily from a medical interest in controlling human diseases such as malaria. Anderson and May (1979) provides a strong foundation of mathematical modeling of macroparasite life cycle dynamics. Visit APPM4390:Population Biology of Infectious Diseases to learn more about modeling macroparasite dynamics.

Amphibian Malformations and Ribeiroia ondatrae

In the mid-1990s, scientific and public awareness of amphibian malformations, primarily including missing, extra or deformed limbs, was growing (Figure 1).
Frog.jpg
Currently, 71 amphibian species have been documented with malformations from across the United States (Lannoo 2008). While no single cause can be identified to explain all amphibian malformations, and multiple causes can interact synergistically (Blaustein and Johnson 2003), many amphibian malformations can be directly attributed to infection with Ribeiroia ondatrae, a complex life-cycle macroparasite (Johnson et al. 1999). The mechanism underlying the development of malformed limbs has not been completely described; however, it is clear that mechanical disruption of amphibian limbs at specific developmental stages can induce malformations. For more information on amphibian parasites and malformations, please visit the Johnson Lab website.

Life Cycle of Ribeiroia ondatrae

The life cycle of R. ondatrae is complex, including three hosts connected by larval, or immature, stages of the parasite (Figure 2). The definitive host is the host where the parasite develops into an adult and reproduces sexually. In the case of R. ondatrae, the definitive host is a frog-eating bird. In the bird host, the adult parasites shed eggs in the host feces that enter the aquatic environment. Once the eggs have reached the water they complete development, hatching into free-living infective stages called miracidia. The miracidia then seek out and infect the second intermediate host, an aquatic snail. Inside the snail the parasite reproduces asexually into a series of stages called, sporocysts and rediae. These larval stages then continue to reproduce into a second free-living infective stage called cercariae that are shed from the snail into the aquatic environment. The cercariae then actively seek out a tadpole as the second intermediate host. The cercariae primarily infect the developing limb buds of the amphibian, and induce malformations in the host. Once cercariae infect their tadpole hosts, they lose their tails and encyst to form metacercariae in the amphibian. The metacercariae remain encysted until the amphibian is consumed by a bird and then the cycle continues. It is hypothesized that the parasite induces the malformations in the amphibian host as a morphological modification to ensure transmission to the definitive host by increasing the likelihood of successful predation.

Rib.jpg

Abiotic and Biotic Influences

The numbers of amphibian species and populations affected by this parasite and the severity of resulting malformations may be related to anthropogenic environmental change, such as climate change and loss of biodiversity.

Climate Change

Changes in temperature associated with climate change might alter the transmission dynamics o the parasite in a few ways. Trematodes are extraordinarily sensitive to changes in temperature; when the temperature is raised by 10°C they can experience growth rates that are an average of 8 times faster (Poulin 2006). The hosts of this parasite tend to show just a doubling in growth rate over the same temperature difference. This means that at higher temperatures, parasites will develop faster and have higher activity levels relative to their hosts.

Biodiversity Loss

Loss of biodiversity can be caused by human activities such as conversion of natural habitats to agriculture or urban areas, degradation of remaining habitat through chemical contamination, or introduction of exotic species (Novacek 2001, Wilson 1988). One of the components of biodiversity that may be important in controlling parasite transmission is predators of free-living infective stages. Predators may have a strong influence on transmission dynamics if the predation pressure reduces parasite density.

MATHEMATICAL MODEL

Host Equations:

Eqnsparams.jpg

ANALYSIS

We analyzed our model using the Simbiology toolkit in Matlab (Fig. 3) Simbiology.jpg

To test our model we began by adding values for the parameters and initial conditions according to three sets of conditions; laboratory conditions, field conditions, and field conditions at higher temperatures. Transmission parameters and initial conditions are likely to vary dramatically between these scenarios. Specifically, transmission of the parasite to tadpoles is considerably higher under laboratory conditions when tadpoles are confined in small containers with the parasites. Natural death rates of tadpoles are higher in field conditions than in the lab due to tadpole predation in the field but not in the lab. Transmission rates and death of parasites are higher at elevated temperatures because although the cercariae are more active while they are alive, they use up their energy resources and die faster at higher temperatures.

Param Lab Field High Temperature

αI .001 .001 .001

β .600 .010 .025

dc .028 .028 .070

dT .010 .300 .300

dI .010 .300 .300

e .100 .5 N/A

Initial conditions for variables were set at C0 = 1,000; T0=100; P0=100 for all three scenarios. We also ran a sensitivity analysis on the parameters that might be expected to change as a result of the effects of predation on cercariae as well as climate change.

RESULTS

Comparison of transmission dynamics in laboratory and field settings:

We found that when parameters were set to reflect the typical dynamics within laboratory experiments, transmission of the parasite occurred at a much faster rate (Figure 4). Furthermore, more tadpoles ended up infected when compared to field settings, and more metacercariae were produced overall. Labfield.jpg Comparison of effect of higher temperature on transmission dynamics:

We found that when parameters were set to reflect transmission dynamics at higher temperatures, transmission occurred at a faster rate, and more tadpoles were infected with the parasite (Figure 5). To reflect higher temperatures we multiplied the field parameter for parasite death rate and transmission rate by 2.5 to reflect the average metabolic response to a 10°C change in temperature. This is a conservative estimate of the changes expected since many studies have shown higher metabolic responses of parasites to temperature (Poulin 2006). We assumed that higher parasite activity levels would yield higher transmission rates, but also proportionally faster death rates. This is because temperature, activity levels, and death rates of cercariae are directly correlated (unpub data). It also seems that the total number of metacercariae encysted in tadpoles responds the most strongly to changes in temperature- almost doubling at higher temperatures. This is important biologically because pathology in tadpoles can often be dose-dependent. Temperatures.jpg

We also performed a sensitivity analysis on the parameter for transmission of cercariae to tadpoles (β) and on the parameter for the death rate of cercariae (dc). We found that the cumulative change in the numbers of cercariae and tadpoles over time was much more sensitive to changes in the value for the transmission parameter than they were to changes in the death rate of cercariae (Figure 6). This suggests that an accurate value for the transmission parameter is more important to model dynamics than an accurate value for the death rate of cercariae. If transmission does indeed increase with higher temperatures, it is likely that this effect will supersede any declines in infection due to higher cercarial death rates, since transmission dynamics are more strongly governed by transmission rate than by death rate of parasites. Temperaturesensitiv.jpg

Comparison of effect of predation on parasites on transmission dynamics:

We found that when we increased the number of predators on cercariae from 100 to 500, with the predation rate of cercariae equal to 0.5, the same as other field simulations, the total number of infected tadpoles declined, as did the total numbers of cercariae and metacercariae (Figure 7). The rate of infection also seemed to slow slightly. This difference suggests that the number of predators is important to the dynamics of the model. Predator.jpg


We also performed a sensitivity analysis on the parameter for the rate at which cercariae are consumed by predators (e). Overall the model dynamics do not seem particularly sensitive to the value given to the cercarial predation rate since the sensitivity was less than 1 for all variables considered (Figure 8). Not surprisingly, cercarial dynamics were the most sensitive to the value of this parameter. Predatorsensitiv.jpg

DISCUSSION

Future Questions

Several laboratory experiments are planned to explore the influence of temperature on the transmission dynamics of this parasite. The next step that we hope to take is to add a parameter for temperature directly into the model as a determinant of death rates of cercariae and transmission. We hope to use this model to clarify which variables are most sensitive to changes in temperature, and which temperature-sensitive parameters are most important to model dynamics as a whole. We have planned laboratory experiments for this summer that will explore the effects of temperature on transmission of the parasite to tadpoles which should help in determining more accurate parameters for the model. We would also like to perform a similar analysis on a transmission model for the dynamics between this parasite and its snail host.

For future research investigating questions related to biodiversity loss, laboratory experiments are planned that will identify potential parasite predators and measure consumption rates at various predator densities. This data will be used to verify the predictions of the model and identify the threshold density of predators or feeding rates that will influence transmission.

One of our ultimate research goals is to model the complete life cycle of R. ondatrae. The next step will be to describe the dynamics of the parasite with the first intermediate hosts, aquatic snails. This will allow us to examine similar questions about the influence of climate change and biodiversity loss on other life cycle stages of the parasite. Finally, dynamics of the definitive, bird hosts can be modeled with field and laboratory investigations focused on identifying the rate at which parasites enter the bird host through successful predation of infected amphibians as well as the rate at which eggs are released into the aquatic environment.

Further Analysis

       For further study of the dynamics of this parasite, Bifurcation Analysis, Stability Analysis as well as adding a Delay could also be used. Using these techniques, not only would scientists have a better understanding of the dynamics, but also how to prevent more loss of amphibians.

CONCLUSIONS

Amphibian Conservation

Our model will allow us to investigate how anthropogenic change influences the dynamics of parasite populations and transmission to amphibian hosts. Through the interaction of parasites and amphibians we can use this information to determine what may be causing increased malformations in certain species of amphibians in particular environments. Then we can provide conservation biologists with information on which anthropogenic effects have the greatest influence on the parasite life cycle, and are therefore key to reducing amphibian malformations

Importance of Modeling Macroparasites

Our model of macroparasite life cycle dynamics could be an important foundation for the study of other amphibian parasites, such as Echinostoma trivolvis, another indirectly transmitted trematode parasite that can potentially reduce growth and survival of amphibian hosts. Macroparasites are also responsible for other diseases of wildlife, domestic animals and humans. By creating a mathematical model to study transmission and parasite life cycle dynamics it may be possible for researchers to determine the best management strategies for disease control. Other studies on the dynamics of parasites include Host-Parasitoid Models.

A Current Paper

Preedy, K.F., P. G. Schofield, M. A. J. Chaplain, S.F. Hubbard. 2007. Disease induced dynamics in host parasitoid systems: chaos and coexistence. J. R. Soc. Interface 4:463-471.

Katherine Preedy has done further research on disease-induced dynamics in host-parasitoid systems. In this current paper on the dynamics of a host-parasitoid system, the models of Anderson and May and the impact of disease on host populations is discussed. Preedy cites Anderson and May's paper and discusses that this is a simple mathematical model to analyze the impact of parasites on host populations. Similarly to paper written by Anderson and May, a system of ordinary differential equations that represent the host and parasite populations is used to represent the dynamics of each population. Preedy studies two parasitic populations on one host organism. The assumptions of this paper are that the host population has logistic growth with an intrinsic rate of growth and a carrying capacity. She also assumes that infection is random and there is no recovery from the infection. Preedy begins with a model that does not consider how spatial effects might influence the populations, but later introduces a model that accommodates for this. The spacio-temporal model introduces a zero flux boundary condition and random motility coefficients. In solving the system of ODE's the equations are nondimensionalized.

The results of the paper suggests that if one species of parasite is removed there are no coexistent fixed points and the disease continues to persist due to the presence of the other parasites. Preedy's results also suggest that if there is no infection than one of the species will become extinct. This shows that the parasites are interdependent on each other and the infection of a parasite can actually benefit the less efficient competitor in the system. The existence of all species is necessary for the whole system to exist.

REFERENCES

Anderson, R.M. and R.M. May. 1979. Population biology of infectious diseases: Part II. Nature 280:455-461.

Blaustein, A.R., P.T.J. Johnson. 2003. The complexity of deformed amphibians. Frontiers in Ecology and the Environment 1:87-94.

Johnson, P.T.J., K.B. Lunde, E.G. Ritchie, and A.E. Launer. 1999. The effect of trematode infection on amphibian limb development and survivorship. Science 284:802-804.

Poulin, R. 2006. Global warming and temperature-mediated increases in cercarial emergence in trematode parasites. Parasitology 132:143-151.

Preedy, K.F., P. G. Schofield, M. A. J. Chaplain, S.F. Hubbard. 2007. Disease induced dynamics in host parasitoid systems: chaos and coexistence. J. R. Soc. Interface 4:463-471.

Lannoo, M. 2008 Malformed frogs: the collapse of aquatic ecosystems. University of California Press. Berkeley, California. pp. 203-215.

Novacek, M.J. (ed.). 2001. The Biodiversity Crisis, Losing What Counts. The New Press. New York.

Wilson, E.O. (ed.). 1988. Biodiversity. National Academic Press. Washington, D.C.

Further Investigation