Controlling tick-borne diseases through domestic animal management: a theoretical approach

Rosalyn Porter; Rachel Norman; Lucy Gilbert

doi:10.1007/s12080-010-0080-2

Theoretical Ecology

August 2011, Volume 4, Issue 3, pp 321–339 | Cite as

Controlling tick-borne diseases through domestic animal management: a theoretical approach

Authors
Authors and affiliations

Rosalyn PorterEmail author
Rachel Norman
Lucy Gilbert

Original Paper

First Online: 20 May 2010

Received: 22 July 2009

Accepted: 04 May 2010

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9 Citations

Abstract

Vector-borne diseases are of global importance to human and animal health. Empirical trials of effective methods to control vectors and their pathogens can be difficult for practical, financial and ethical reasons. Here, therefore, we use a mathematical model to predict the effectiveness of a vector-borne disease control method. As a case study, we use the tick-louping ill virus system, where sheep are treated with acaricide in an attempt to control ticks and disease in red grouse , an economically important game bird. We ran the model under different scenarios of sheep flock sizes, alternative host (deer) densities, acaricide efficacies and tick burdens. The model predicted that, with very low deer densities, using sheep as tick mops can reduce the tick population and virus prevalence. However, treatment is ineffective above a certain threshold deer density, dependent on the comparative tick burden on sheep and deer. The model also predicted that high efficacy levels of acaricide must be maintained for effective tick control. This study suggests that benignly managing one host species to protect another host species from a vector and pathogen can be effective under certain conditions. It also highlights the importance of understanding the ecological complexity of a system, in order to target control methods only under certain circumstances for maximum effectiveness.

Keywords

Louping ill virus Mathematical model Sheep tick mop Tick borne Animal management

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Notes

Acknowledgements

R. Porter was funded by a CASE studentship from NERC and the Macaulay Land Use Research Institute. We thank the Game and Wildlife Conservation Trust for access to the sheep farm we used for counting tick burdens on sheep. We thank Edward Jones, Peter Goddard and the anonymous referees for comments on the manuscript.

Appendix: Sensitivity analysis

Sensitivity analysis was conducted following the methods of Watts et al. (2009). The analysis was done for both the high and low sheep tick burden models at different deer densities, with and without sheep. Each of these models was run 500 times selecting parameters randomly from a range ±1% of the parameter used throughout the paper. The outputs from these simulations were then correlated against the individual parameters and interactions between parameters within the groups concerning grouse dynamics, tick dynamics and viral dynamics, respectively. The results from this analysis are given in Table 2.

Table 2

Significant correlation (Pearson’s correlation coefficient, P < 0.05) between model parameters and the model predictions of grouse density (per square kilometre) and virus prevalence within the grouse population, respectively

	Grouse density per km²												Infection prevalence
	Low						High						Low						High
	4w	4wo	7w	7wo	15w	15wo	4w	4wo	7w	7wo	15w	15wo	4w	4wo	7w	7wo	15w	15wo	4w	4wo	7w	7wo	15w	15wo
a _g	0.22	0.22	0.10	0.10			0.19	0.23									0.26	0.20					0.20	0.22
b _g	−0.98	−0.98	−0.40	−0.10			−0.98	−0.98	−0.31	−0.24							−0.89	−0.76		0.11			−0.76	−0.88
s _g	−0.11	−0.17						−0.17
α							0.12
γ																							0.09
a _g × b _g	0.96	0.96	−0.39				−0.95	−0.95	−0.30	−0.24							−0.87	−0.74		0.11			−0.74	−0.85
a _g × s _g				0.09													0.18	0.14					0.19	0.15
a _g × α	0.14	0.13					0.22	0.16									0.13	0.13					0.09	0.17
a _g × γ	0.14	0.14					0.15	0.18									0.18	0.15					0.21	0.16
b _g × s _g	−0.98	−0.98	−0.40	−0.10			−0.98	−0.98	−0.30	−0.23							−0.89	−0.77		0.11			−0.75	−0.87
b _g × α	−0.97	−0.97	−0.41	−0.10			−0.97	−0.97	−0.30	−0.24							−0.89	−0.77		0.11			−0.76	−0.88
b _g × γ	−0.97	−0.97	−0.40	−0.10			−0.97	−0.97	−0.30	−0.24							−0.89	−0.76		0.10			−0.75	−0.87
α×s _g	−0.10	−0.14						−0.12
α×γ							0.10
γ×s _g	−0.10	−0.13						−0.10															0.11
a _t												−0.12		0.10
b _t	−0.10																0.10		−0.11
s _t				−0.10						−0.10						0.10						0.09
a _t × b _t	−0.10											−0.10							−0.12
a _t × s _t
b _t × s _t	−0.11																0.09						0.10
p	0.09										0.13											0.12
β ₁
β ₂															−0.11									0.10
β ₃			0.35	0.87	−0.81	−0.88			0.35	0.85	−0.89	−0.80	−0.16	−0.37	−0.33	−0.85	−0.16	−0.40	−0.19	−0.34	−0.34	−0.84	−0.48	−0.11
β ₅																					0.09
β ₆				0.10	0.12											−0.11
β ₇			0.33		0.40				0.33			0.34	−0.13		−0.30				−0.15		−0.32			0.11
β ₁ × β ₂									−0.09				0.10
β ₁ × β ₃			0.35	0.87	−0.81	−0.88			0.35	0.85	0.89	−0.80	−0.16	−0.37	−0.33	−0.85	−0.16	−0.40	−0.19	−0.34	−0.34	−0.84	−0.48	−0.11
β ₁ × β ₅																					0.09
β ₁ × β ₆				0.09	0.13											−0.11
β ₁ × β ₇			0.33		0.40				0.33			0.34	−0.13		−0.30		0.10		−0.15		−0.32			0.11
β ₂ × β ₃			0.35	0.87	−0.81	−0.88			0.35	0.85	−0.89	−0.80	−0.16	−0.37	−0.33	−0.85	−0.16	−0.40	−0.19	−0.34	−0.34	−0.84	−0.48	−0.11
β ₂ × β ₅									−0.11												0.10			0.10
β ₂ × β ₆				0.10	0.12											−0.11
β ₂ × β ₇			0.33		0.40				0.33			0.34	−0.13		−0.30		0.10		−0.15		−0.32			0.11
β ₃ × β ₅			0.35	0.87	−0.81	−0.88			0.35	0.85	−0.89	−0.80	−0.16	−0.37	−0.33	−0.85	−0.16	−0.40	−0.19	−0.34	−0.34	−0.84	−0.48	−0.11
β ₃ × β ₆			0.28	0.70	−0.59	−0.66			0.29	0.64	−0.67	−0.59	−0.13	−0.27	−0.25	−0.68	−0.16	−0.28	−0.14	−0.26	−0.28	−0.64	−0.40
β ₃ × β ₇			0.35	0.87	−0.81	−0.88			0.35	0.85	−0.89	−0.80	−0.16	−0.37	−0.33	−0.85	−0.16	−0.40	−0.19	−0.34	−0.34	−0.84	−0.48	−0.11
β ₅ × β ₆				0.10	0.12											−0.11
β ₅ × β ₇			0.32		0.40				0.33			0.34	−0.13		−0.30		0.10		−0.15		−0.32			0.11
β ₆ × β ₇			0.26		0.41				0.28			0.31	−0.11		−0.24				−0.11		−0.26

The bold values in this table show the values that account for more than 20% of the variance in the data

Non-significant correlations are not shown. The model was run for four, seven and 15 deer per square kilometre, with and without 50 sheep per square kilometre treated with acaricide of 100% efficacy. The parameters are grouped as grouse, tick and viral dynamic parameters and their corresponding interactions w with, wo without

The relative effect on the model outputs of changing each parameter individually by ±10% was also investigated. The percentage change on the model output is given in Table 3 whenever the percentage change is greater than 10%. The deer density for each model was chosen to allow the grouse to reach an intermediate density with the parameters used throughout the paper.

Table 3

The percentage change (not shown if <10%) of the model predictions of total grouse density and infection prevalence in the grouse population after changing each parameter individually by ±10%

	Grouse density per km²		Infection prevalence
	Low	High	Low	High
a _g +	13	11	21	22
a _g −	−12		−21	−22
b _g +	−11		−12	−11
b _g −	16	12	11	11
s _g +
s _g −
α +
α −			12	12
γ +
γ −
a _t +	−51	−45
a _t −	297	553	−37	−71
b _t +	248	113	−31	−13
b _t −	−51	−35
s _t +		12
s _t −		−11
p +
p −
β ₁ +
β ₁ −
β ₂ +	−12	−12
β ₂ −	17	15
β ₃ +
β ₃ −
β ₅ +	−45	−39
β ₅ −	247	424	−32	−56
β ₆ +
β ₆ −
β ₇ +		39
β ₇ −		−22

The results are given for both the high and low sheep tick burden models run with 11 and seven deer per square kilometre, respectively

Correlation effects In general, the grouse dynamic parameters (birth and death rates) and corresponding interactions show high correlation with the predicted grouse density only for low deer density. This is the same for both models and can be explained by the lack of ticks at this deer density. At low deer densities, the tick population cannot be maintained at a high enough density to allow LIV to persist so the disease dynamics are not important. As a consequence, the grouse population dynamics are governed by the natural birth and death rates.

The grouse dynamics parameters show high correlation with the virus prevalence for high deer densities. At high deer densities, the tick population is large enough to allow disease persistence, and the grouse population is regulated by the disease. If the natural death rate is increased, this will reduce the grouse population already at a low density, which reduces the proportion of infected grouse. The tick dynamic parameters do not show a high correlation with any of the model outputs.

Of the viral dynamics, β ₃, the rate at which grouse ingest ticks, has the highest correlation both with the grouse density and virus prevalence at higher deer densities. At the intermediate deer density, the ingestion of ticks shows a positive effect on the grouse population. This seems counterintuitive as ingestion is another route of infection, but here, the tick population is small enough for the grouse to be able to consume a sufficient quantity that the overall effect is to reduce the tick population and thus the virus prevalence. However, at higher deer densities, the tick population is large and the ingestion of ticks has a limited effect on the tick population and the overall effect on the grouse population is negative; now, ingestion is essentially just another route of infection.

β ₇, the rate at which immature ticks attach to sheep, also shows a positive correlation with the grouse population for higher deer densities. At low deer densities, the tick population is already small. At higher deer densities, the increased attachment and subsequent death of ticks that attach to sheep would be expected to have a positive effect on the grouse population as the tick population is reduced and with it the virus prevalence.

Individual parameters The parameters that have the largest disproportionate effect on the model outputs are the same for both the high and low sheep tick burden models. Where there are differences, the effect is only just beyond what would be expected and may be simply due to the effect of the particular deer density. “How do alternative hosts, such as deer, impact on the effectiveness of treated sheep?” section discusses the impact of different deer densities on the model predictions.

The tick parameters show a highly disproportionate effect on the model predictions for the grouse density. In particular, decreasing a _t the tick birth rate, increasing b _t the death rate and decreasing β ₅ the tick biting rate on deer which will all reduce the tick population have a huge positive effect on the predicted grouse population. Although a positive effect would be expected as a reduction in the tick population will reduce the virus prevalence, the magnitude of the effect is an order of magnitude higher than expected. This can be explained by the sheer size of the tick population and a small relative change can be a large change in terms of actual numbers. We are at a point where a small change in the tick population has a large effect on the grouse population and so the effect is disproportionate. Consequently, the model is sensitive to the estimates of these parameters.

Although a change to those parameters increasing the tick population does have a negative effect on the grouse population predictions, the magnitude is not as extreme.

In the high sheep tick burden model, a change to β ₇ the rate at which immature ticks attach to sheep has a disproportionate effect on the grouse population predictions. The explanation for this may be due to the effect this parameter in the high sheep tick burden model has on the tick population and small relative changes now make sufficiently large numerical changes to show a large effect on the grouse population.

A smaller, but still disproportionate, effect can be seen by changing a _g the grouse birth rate, b _g the natural grouse death rate and β ₂ the rate at which infected ticks bite grouse. Interestingly, although β ₃ the rate at which ticks are ingested by grouse showed high correlation for both models, it does not have a disproportionate effect on model outputs.

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Authors and Affiliations

Rosalyn Porter
- 1
- 2
Email author
Rachel Norman
- 1
Lucy Gilbert
- 2

1.Department of Computing Science and MathematicsUniversity of StirlingStirlingUK
2.Macaulay Land Use Research InstituteAberdeenUK

Controlling tick-borne diseases through domestic animal management: a theoretical approach

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