mathematical models of neospora caninum infection in dairy cattle: transmission and options for...

Mathematical models of Neospora caninum infection indairy cattle: transmission and options for control

N.P. Frencha, *, D. Clancyb, H.C. Davison c, A.J. Trees c

aDepartment of Clinical Veterinary Science, University of Liverpool, Leahurst, Neston, South Wirral, UKbStatistics and Operational Research Division, Department of Mathematical Sciences, University of Liverpool, Liverpool, UK

cVeterinary Parasitology, Liverpool School of Tropical Medicine/Faculty of Veterinary Science, University of Liverpool, Pembroke

Place, Liverpool, UK

Received 15 April 1999; received in revised form 21 July 1999; accepted 21 July 1999

Abstract

The transmission and control of Neospora caninum infection in dairy cattle was examined using deterministic andstochastic models. Parameter estimates were derived from recent studies conducted in the UK and from thepublished literature. Three routes of transmission were considered: maternal vertical transmission with a highprobability (0.95), horizontal transmission from infected cattle within the herd, and horizontal transmission from an

independent external source. Putative infection via pooled colostrum was used as an example of within-herdhorizontal transmission, and the recent ®nding that the dog is a de®nitive host of N. caninum supported theinclusion of an external independent source of infection. The predicted amount of horizontal transmission required

to maintain infection at levels commonly observed in ®eld studies in the UK and elsewhere, was consistent with thatobserved in studies of post-natal seroconversion (0.85±9.0 per 100 cow-years). A stochastic version of the model wasused to simulate the spread of infection in herds of 100 cattle, with a mean infection prevalence similar to that

observed in UK studies (around 20%). The distributions of infected and uninfected cattle corresponded closely toNormal distributions, with S.D.s of 6.3 and 7.0, respectively. Control measures were considered by altering birth,death and horizontal transmission parameters. A policy of annual culling of infected cattle very rapidly reduced the

prevalence of infection, and was shown to be the most e�ective method of control in the short term. Not breedingreplacements from infected cattle was also e�ective in the short term, particularly in herds with a higher turnover ofcattle. However, the long-term e�ectiveness of these measures depended on the amount and source of horizontalinfection. If the level of within-herd transmission was above a critical threshold, then a combination of reducing

within-herd, and blocking external sources of transmission was required to permanently eliminate infection. # 1999Australian Society for Parasitology Inc. Published by Elsevier Science Ltd. All rights reserved.

Keywords: Cattle; Horizontal and vertical transmission; Mathematical models; Neospora caninum

1. Introduction

Neospora caninum infection in dairy cattle is

now recognised as a major cause of abortion and

International Journal for Parasitology 29 (1999) 1691±1704

0020-7519/99/$20.00 # 1999 Australian Society for Parasitology Inc. Published by Elsevier Science Ltd. All rights reserved.

PII: S0020-7519(99 )00131-9

* Corresponding author. Tel.: +44-151-794-6031; fax: +44-

151-794-6028.

E-mail address: [email protected] (N.P. French)

economic loss to farmers worldwide [1±4]. Thishas stimulated research in a number of importantareas, including: quantifying the impact of N.caninum infection on production [5±8], develop-ing tools for identifying infected cattle [9] andimproving our understanding of the life-cycle ofthe parasite [10]. In this study we use a theoreti-cal modelling approach, based on experimentaland ®eld data, to examine the transmission of N.caninum in herds of dairy cattle and explore theimpact of a range of control options.

A previous modelling study of N. caninuminfection in closed dairy herds has explored therole of horizontal and vertical transmission in themaintenance of infection and the e�ects of pre-ferential selection of family lines on the trans-mission process (NP French, HC Davison, DClancy, M Begon, AJ Trees. Modelling ofNeospora species infection in dairy cattle: the im-portance of horizontal and vertical transmissionand di�erential culling. Proceedings of theSociety for Veterinary Epidemiology andPreventive Medicine, Ennis, 1998;113±22). Theconclusions of this work were that: (1) verticaltransmission alone would not sustain the infec-tion in a herd of cattle; and (2) in the absence ofany other route of transmission, preferentialselection of infected family lines would result in apotentially large, but transient, increase in theprevalence of infection.

Since this initial study was conducted, therehave been a number of developments that havegiven new insight into the epidemiology of the

disease. There is evidence of two potentialsources of horizontal transmission; via a caninede®nitive host [10] and cow-to-calf transmissionvia pooled colostrum or milk [11]. Furthermore,studies in the UK have given more precise esti-mates of transmission parameters (see [12], thisissue, for further details). This paper attempts touse our current understanding of this disease tomodify and improve existing models. Thisapproach is intended to complement and informfuture studies and be part of the long term goalof reducing the impact of N. caninum infectionon cattle production.

2. The basic deterministic model

2.1. Model structure

The model uses a system of non-linear di�eren-tial equations to describe the transmission of N.caninum infection between dairy cattle. Its beha-viour is explored using analytical and numericalmethods. Fig. 1 shows the structure of themodel. Our current understanding of the diseasesuggests that cattle are either persistently infected(Y) or uninfected (X), and there is no evidence ofa recovered or immune class. The birth rate par-ameter, r, is the rate at which female cattle areadded to the milking herd, and is r1 for infectedand r2 for uninfected cattle. Importantly, this isnot simply the birth rate of cattle in the herd, asthis would also include male calves and beef

Fig. 1. Schematic representation of a deterministic model of N. caninum.

N.P. French et al. / International Journal for Parasitology 29 (1999) 1691±17041692

crosses. For simplicity, no time delay wasincluded to allow for the period between birthand ®rst calving, which is usually between 2 and3 years, although numerical simulations includingsuch a delay had little e�ect on the behaviour ofthe model. The death rate parameter m is the rateat which cattle are culled or removed from theherd, and is m1 for infected and m2 for uninfectedcattle. Both r and m are given values that areconsistent with national herd demographics andherd age structures in the UK. In order to modelthe impact of parasite virulence on both birth(e.g. abortion) and death (mortality and culling)processes, the birth and death parameters forinfected cattle are ®xed and the culling rate isalways greater than the birth rate m1>r1. Herdsize is then maintained by allowing the netgrowth of the uninfected population to vary.These demographic processes are consistent withthose observed in longitudinal ®eld studies [7].The e�ect of preferentially selecting infectedfamily lines has been described elsewhere, and isnot included in this study (NP French, HCDavison, D Clancy, M Begon, AJ Trees.Modelling of Neospora species infection in dairycattle: the importance of horizontal and verticaltransmission and di�erential culling. Proceedingsof the Society for Veterinary Epidemiology andPreventive Medicine, Ennis, 1998;113±22).

2.2. Routes of transmission

We consider three routes of transmission ofwhich only one, vertical transmission, has beenproven to occur in naturally infected cattle.

2.2.1. Vertical transmissionVertical transmission is de®ned here as the

transfer of infection from dam to calf via thetransplacental route. The probability of verticaltransmission (f) used in all models is 0.95, basedon a recent study conducted in the UK [12].Other studies have estimated this parameter to be0.81 [8] and 1.00 [6].

2.2.2. Horizontal transmission dependent on herdinfection prevalence: infection from within the herd

Here the force of infection is a function of cur-rent infection prevalence and is modelled using a`true' mass action process with a parameter b.The putative spread of infection via pooled colos-trum or milk [11] would be an example of such atransmission process whereby the rate of newinfections would depend on the current pro-portion of infected animals in the herd.

2.2.3. Horizontal transmission independent of herdinfection prevalence: infection from outside theherd

This type of transmission process is modelledusing a constant per-capita force of infectionwith a parameter t. An independent cycle ofinfection in another host [10, 13] would be anexample of this type of transmission.

At present, we have no information on therelative contribution of di�erent sources of hori-zontal infection, so neither b nor t can be givenreliable values based on either ®eld or experimen-tal data. However, we do have estimates of therate of post-natal seroconversion [12, 6]. Thesecan be used to explore the criteria necessary formaintaining stable equilibrium prevalences andto examine the role and behaviour of potentialsources of horizontal transmission.

2.3. Model equations

The model (Fig. 1) can be described by the sys-tem of di�erential equations:

dY=dt � r1fYÿ m1Y� bXY=N� tX, �1�

dX=dt � �r2 ÿ m2�X� r1�1ÿ f�Yÿ bXY=Nÿ tX,

where the net growth of the population ofuninfected cattle (r2ÿm2) is allowed to vary:

r2 ÿ m2 � ��m1 ÿ r1�Y �=X: �2�

Equivalently, we can model the prevalence ofinfection y:

dy=dt � r1fyÿ m1y� by�1ÿ y� � t�1ÿ y�, �3�where x=1ÿy.

N.P. French et al. / International Journal for Parasitology 29 (1999) 1691±1704 1693

The basic reproductive ratio, Ro, for Eq. (1) is:

Ro � �r1f� b� tN �=m1, �4�which is the sum of the vertical and horizontaltransmission components divided by the averagelifespan of an infected individual.

Figs. 2 and 3 show the behaviour of Eq. (1)for a range of horizontal transmission rates thatgive equilibrium prevalences consistent withthose observed in the ®eld [14]. The parametervalues were f=0.95, m1=0.3, r1=0.2, and theinitial values are X=99 and Y=1. In Fig. 2t=0.025 and in Fig. 3 b=0.025. Given the in-itial values and parameters, the prevalenceincreases monotonically to a stable equilibriumabove zero. Figs. 2 and 3 show that the system ismore sensitive to changes in the rate of infectionvia a constant per capita exposure (t) than awithin-herd, prevalence-dependent, process (b).

2.4. Intermittent high-dose exposure

Here we consider a source of horizontal trans-mission that occurs intermittently rather thancontinuously. An example of this could be feedcontaminated with oocysts from dog faeces [13].Fig. 4 simulates the e�ect of a high level of con-tamination that lasts for 1 month every 5 years.The high level of horizontal transmission resultsin a sharp increase in the infection prevalence,which is then maintained by vertical transmissiongiving a `saw-tooth' pattern of infection preva-lence.

2.5. The criteria for maintaining an infectionprevalence above zero

The prevalence of infection at equilibrium isidenti®ed by solving the system when the rates of

Fig. 2. The dynamic behaviour of a deterministic model of N. caninum infection in dairy cattle with varying levels of within-herd

transmission (b). The other parameter values were f=0.95, m1=0.3, r1=0.2 and t=0.025.


change are set to zero (dY/dt=dX/dt=0). If theonly route of horizontal transmission was via awithin-herd transmission process, then the preva-lence of infection at equilibrium y* would beeither endemically stable at:

y* � �r1fÿ m1 � b�=b �5�(provided that 0< m1ÿr1f< b)

or y* � 0:

If m1ÿr1f>b, then the equilibrium at y*=0 isstable.

If there is a combination of within-herd andconstant per capita horizontal transmission then,provided that r1f<=m1, the prevalence ofinfection at equilibrium would be the positive

root of the quadratic equation:

ÿb� y�2 � �b� r1fÿ m1 ÿ t�y� t � 0: �6�De®ning

k � r1fÿ m1 � bÿ t

and

c � �p k2 � 4tb�,then

y* � �k� c�=2b or �kÿ c�=2b: �7�

If y(0) is the value of y at time=0, andv=(k+ c)/2b is the positive equilibrium point,then the solution of Eq. (3) is:

Fig. 3. The dynamic behaviour of a deterministic model of N. caninum infection in dairy cattle with varying levels of external trans-

mission (t). The other parameter values were f=0.95, m1=0.3, r1=0.2 and b=0.025.


y�t� � v� fc� y�0� ÿ v� exp�ÿct�g=fc� b�1

ÿ exp�ÿct�� y�0� ÿ v�g: �8�

Whatever the source of horizontal infection, itfollows from Eq. (3) that the force of horizontaltransmission or incidence rate of post-natal infec-tion (l) needed to maintain an equilibrium preva-lence of y* would be:

l � � y*=�1ÿ y*��m1 ÿ r1f�: �9�If this were achieved via a within-herd process

alone, the transmission coe�cient would be:

b � �m1 ÿ r1f�=�1ÿ y*�: �10�This solution allows us to quantify the inci-

dence rate of post-natal infection required tomaintain a range of equilibrium prevalences

(Fig. 5). It can be seen that the force of infection

required to maintain a given equilibrium preva-

lence is higher if the parasite is more virulent (i.e.

has a greater negative impact on reducing fertility

and/or increasing mortality). Furthermore, the

amount of horizontal transmission required to

maintain equilibrium prevalences in the observed

range of 7±45% [13, 14] is between 0.85 and 9.0

per 100 cow-years (when m1=0.3 and r1=0.2).

These are consistent with the 1.9 and 8.5/

100 cow-years observed in seroconversion

studies [12, 15]. If this were achieved by a within-

herd, prevalence-dependent process, the equival-

ent b transmission coe�cients would be 0.12 and

0.29.

Fig. 5 illustrates a number of further important

points. Firstly, if all horizontal infection is via a

Fig. 4. The dynamic behaviour of a deterministic model of N. caninum infection in dairy cattle with intermittent high-dose ex-

posure. The parameter values were f=0.95, m1=0.3, r1=0.2


mass action process, there is a threshold b coe�-cient below which a stable equilibrium prevalenceabove zero is not possible (i.e. the disease wouldnot become endemic). The threshold values canbe found where the lines intercept the Y axis andthe prevalence of infection is zero. However, ifhorizontal transmission includes a constant percapita process, then the equilibrium prevalencewill always be above zero. The relationshipbetween the incidence of horizontal infection andequilibrium prevalence follows a logistic curve,which suggests that the amount of horizontalinfection needed to achieve very high equilibriumprevalences increases very rapidly. If this wereachieved by a within-herd, prevalence-dependentprocess, there is a very rapid increase in the btransmission coe�cient. This may explain why

infection prevalences greater than 70% are rarelyobserved in the ®eld.

By simulating the prevalence of infection incohorts of animals at equilibrium, we can alsoshow that the age±prevalence pro®le is consistentwith the relatively ¯at age±seroprevalence pro®lesobserved in ®eld studies [14]. In a herd with anequilibrium prevalence of 22%, m1=0.3 andm2= m1/1.6 [7] the prevalence of infectionincreased with age from 19.7% in the youngestcalves to only 24.3% in 10 year old cattle.

3. The stochastic model

So far, we have looked at a deterministicmodel for the spread of N. caninum. In very large

Fig. 5. The relationship between equilibrium prevalence of infection with N. caninum, horizontal transmission and parasite virulence

(varied by increasing the mortality/culling rate parameter for infected cattle, m1). Parameter values were f=0.95 and r1=0.2.


populations, deterministic models can provide a

good description of the spread of disease.

However, for a herd of a few hundred animals,

the population is su�ciently small that stochastic

e�ects are important. Therefore, we now consider

a stochastic version of the model described in

Section 2. The rates shown in Fig. 1 are now

interpreted as average transition rates in a ran-

dom process. If the population at time t consists

of X uninfected and Y infected cattle, then in the

small time interval from t to t+Dt, the probabil-

ities of the various possible transitions are as

shown in Table 1.

For the deterministic model, we were able to

ensure that herd size remained constant by allow-

ing r2ÿm2 to vary according to Eq. (2). In our

stochastic model we will assume that all of the

parameters r1, m1, r2, m2, b, t, f, N remain con-

stant over time. The population size will be

allowed to vary between Nmin and Nmax.

Provided that Nmin<=X+ Y<=Nmax, the

population evolves according to the transition

rates given in Table 1. Whenever X+ Y= Nmin

and a death occurs, a new animal is instan-

taneously bought into the herd. This will be an

infected animal with probability P, uninfected

with probability 1ÿP, where P is the probability

that a purchased cow is infected prior to its arri-

val in the herd. Whenever X+ Y= Nmax and a

birth occurs, an animal is instantaneously culled.

This animal is chosen from the herd in such a

way that, with probability X/(X+ rY), a suscep-

tible animal is culled, while with probability rY/

(X+ rY), an infected animal is chosen (r being a

®xed parameter of the model).

Since herd size varies, the within-herd infectionrate bXY/N could perhaps be replaced by bXY/(X+ Y) but, since X+ Y is kept approximatelyconstant, this would make little di�erence, so forsimplicity we use bXY/N with N constant.

Using the same parameter values as in Section2, we take f=0.95, m1=0.3, r1=0.2, t=0.025,b=0.025. From Thurmond and Hietala [7], aplausible value for r is r=1.6. That is to say, theculling rate for infected cattle is 1.6 times thatfor uninfected animals. It therefore also seemssensible to take m2= m1/r=0.1875. We takeP=0.06 from estimates of the prevalence ofinfection in the national herd [16]. In our choiceof r2, we will be guided by Eq. (2). Using thesame equilibrium prevalence as in Section 2(v=0.22), we suppose that at equilibrium,r2ÿm2=(m1ÿr1)v/(1ÿv). We therefore taker2=0.1875+(0.1�0.22/0.78)10.2157, which isconsistent with the observation that uninfectedcattle have a lower risk of abortion comparedwith infected cattle [6]. Finally, we takeNmin=95 and Nmax=105 to ensure a herd sizeof around 100, and N=100.

To investigate the behaviour of the stochasticmodel, Monte Carlo simulation was used. Forthe above parameter values, 5000 simulationswere carried out, each with initial conditionsX=100, Y=0. The mean and S.D. of each of Xand Y, together with the correlation between Xand Y, were recorded at 1 year intervals over aperiod of 100 years. Fig. 6 shows the develop-ment over time of the mean number of infectedcattle in the herd, together with the mean totalherd size. An idea of the amount of randomvariability is given by the curves plotted two

Table 1

The probabilities of transition from one state to another in a stochastic model of N. caninum infection in dairy cattle

Event Probability

A susceptible gives birth to a susceptible (X, Y) 4 (X+1, Y) r2X DtAn infected gives birth to a susceptible (X, Y) 4 (X+1, Y) r1(1ÿf)Y DtAn infected gives birth to an infected (X, Y) 4 (X, Y+1) r1fY DtA susceptible dies or is culled (X, Y) 4 (Xÿ1, Y) m2X DtAn infected dies or is culled (X, Y) 4 (X, Yÿ1) m1Y DtAn infection occurs from outside the herd (X, Y) 4 (Xÿ1, Y+1) tX DtAn infection occurs from within the herd (X, Y) 4 (Xÿ1, Y+1) b(XY/N) Dt


standard deviations either side of the mean num-ber of infecteds. It is clear from Fig. 6 that notonly does the mean number of infected cattlesettle down to a stable value, but so does theS.D. To investigate further the probability distri-bution of number of infected cattle in equili-brium, a further 5000 simulations were carriedout with the same parameter values, in this caserecording the (X, Y) values at t=100 years. Theobserved distributions of X and Y were eachfound to correspond closely to Normal distri-butions. Observed probability distributions for Xand Y are shown in Fig. 7, together with theor-etical Normal distribution curves, computedusing the observed mean and S.D. values fromsimulation (mean X=82.24, S.D. X=6.99,mean Y=17.95, S.D. Y=6.31). The correlationbetween X and Y at t=100 years was aroundÿ0.9, a consequence of the fact that herd size iskept approximately constantÐif herd size werekept exactly constant, then the correlation

between X and Y would be exactly ÿ1, perfectnegative correlation.

4. Control strategies

We return to the deterministic model toexplore the impact of a range of control optionson the prevalence of infection. Although this willnot provide information on the full range of out-comes for herds of di�erent sizes, it will give agood average estimation of the e�ect of controlmeasures over time.

It is important to emphasize that we are hereconsidering options for control of infection withN. caninum, rather than of abortion due to N.caninum. The following strategies were con-sidered:

1. Culling infected cattle. This was simulated byincreasing the mortality parameter to m1=2.0.

Fig. 6. The dynamic behaviour of a stochastic model of N. caninum in dairy cattle showing the mean infection prevalence (22 S.D.)

and mean herd size. Parameter values were f=0.95, m1=0.3, r1=0.2, m2=0.1875, r210.2157, t=0.025, b=0.025, N=100

and P=0.06.


This e�ectively ensures that infected cattle sur-vive on average for just 6 months, and couldbe achieved by annual testing of the herd andculling seropositive cattle.

2. Not breeding replacements (or other cattlethat would be a source of infection to theherd) from infected cattle. This was simulatedby reducing the birth rate parameter tor1=0.

3. Preventing within-herd, prevalence-dependenttransmission (e.g. not feeding colostrum ormilk from infected cows to calves) by reducingthe parameter b=0.

4. Preventing constant environmental exposure(e.g. blocking transmission from the de®nitivehost by avoiding contamination of feed withdog faeces) by reducing parameter t=0.

5. Preventing all sources of horizontal trans-mission by reducing both b and t to 0.

Fig. 8 shows the impact of these control strat-egies on a herd with a 22% prevalence of infec-tion at equilibrium, and parameter values off=0.95, m1=0.3, r1=0.2, b=0.025, t=0.025.A policy of culling infected cattle rapidly reducedthe prevalence of infection and, in the shortterm, was the most e�ective control strategy. Notbreeding replacements from infected cattle alsorapidly reduced the infection prevalence, but wasless e�ective than culling. Even if there was nowithin-herd transmission, a selective breedingpolicy was less e�ective than culling. The leaste�ective strategy was blocking mass-action hori-zontal transmission. However, in the long term,neither a culling strategy nor a selective breedingpolicy eliminated infection. The only method ofeliminating infection from a herd, and maintain-ing zero prevalence, was to block any indepen-dent external source of transmission. Even this

Fig. 7. Observed probability distributions for susceptible and infected cattle from 5000 simulations of a stochastic model of N. cani-

num infection in dairy cattle. Parameter values were f=0.95, m1=0.3, r1=0.2, m2=0.1875, r210.2157, t=0.025, b=0.025,

N=100 and P=0.06.


method is e�ective only provided that the level ofwithin-herd transmission is below a criticalthreshold.

Fig. 9 repeated the control measures illustratedin Fig. 8, but the birth and death parameterswere increased to simulate a herd with a higherturnover of cattle. The di�erence between thebirth and death rate in infected cattle was keptthe same, but both were increased to m1=0.4and r1=0.3. By maintaining herd size, this e�ec-tively increased the net growth of the uninfectedpopulation, both before and after the controlmeasures have been implemented. It can be seenthat the e�ect of blocking horizontal routes oftransmission is largely unchanged but, under thisscenario, not breeding from infected cattlebecomes more e�ective and culling becomesslightly less e�ective.

5. Discussion

This study uses a theoretical modelling

approach to explore the epidemiology and con-

trol of N. caninum infection in dairy cattle. It has

been conducted in parallel with empirical

approaches at a time when our understanding of

the basic biology of the parasite and host±para-

site interactions has been advancing rapidly. This

combined multidisciplinary approach has high-

lighted gaps in our knowledge of the epidemiol-

ogy of this disease and directed research e�ort.

Even though we know little about the precise

mechanisms of horizontal transmission, we can

still highlight key features of the disease and

explore options for reducing the level of infection

in dairy herds.

Fig. 8. The impact of control measures on endemic N. caninum infection in dairy cattle using a deterministic model. The starting

parameters were f=0.95, m1=0.3, r1=0.2, t=0.025 and b=0.025. Increasing m1 to 2.0 simulates the e�ect of annual culling;

decreasing r1 to 0 simulates the e�ect of a selective breeding policy; decreasing t to 0 simulates blocking external sources of trans-

mission; and decreasing b to zero eliminates within-herd horizontal transmission.


It is evident that, even allowing for preferentialselection of family lines, imperfect, uniparental(i.e. maternal) vertical transmission alone will notsustain the infection in a herd. Even with a veryhigh vertical transmission probability, some formof horizontal transmission is required for the dis-ease to be endemic in a herd. This is consistentwith other theoretical studies of the dynamics ofdiseases with vertical and horizontal routes oftransmission [17, 18]. The force of horizontaltransmission required increases as the negativeimpact of the parasite on the reproductive ®tnessand survival of the host increases and, in agree-ment with [13, 14, 19], very high equilibrium pre-valences (>70%) would be increasingly di�cultto achieve. The source of horizontal transmissionis important in determining the stability of infec-tion within the herd. If infection enters the herdfrom outside, then very low levels of horizontaltransmission and low infection prevalences couldbe achieved and sustained. Conversely, if thesource of infection is within the herd, then there

is a critical transmission threshold, below whichthe infection would eventually decay down to astable prevalence of zero.

When modelling infectious processes in smallpopulations of animals, such as dairy herds, it isimportant to consider the contribution of ran-dom or stochastic e�ects. It can be seen fromthis study that, given identical input parameters,there is considerable variation around the meaninfection prevalence for a herd of 100 cattle. Thisvariability is Normally distributed and alonecould explain the distribution of infection preva-lences observed in ®eld studies. However, di�er-ences between herds [14] may also be attributedto variation in culling and breeding policy, cow±cow transmission rates (e.g. calving hygiene, feed-ing of pooled colostrum) and the presence of ade®nitive host [13].

The predicted levels of horizontal transmissionrequired to maintain the prevalences of infectionobserved in ®eld studies are consistent with thoseobserved in studies of post-natal

Fig. 9. The impact of control measures on endemic N. caninum infection in dairy herd using a deterministic model with a higher

turnover of cattle. The starting parameters were f=0.95, m1=0.40, r1=0.30, t=0.025 and b=0.025. Increasing m1 to 2.0 simu-

lates the e�ect of annual testing and culling; decreasing r1 to 0 simulates the e�ect of a selective breeding policy; decreasing t to 0

simulates blocking external sources of transmission; and decreasing b to 0 eliminates within-herd horizontal transmission.


seroconversion [6, 12, 15]. This implies that thedisease can be maintained at endemic equilibriumand, although N. caninum was only recentlydescribed in cattle, it is unlikely to be a long-term transient epidemic that would die out with-out intervention. However, we still need to testthe assumption that cattle that seroconvert haveacquired post-natal infections, have an increasedrisk of abortion and will vertically infect theiro�spring. We also need to determine the sourcesof horizontal infection. There is increasing evi-dence that the dog is a de®nitive host that couldbe an external source of infection to theherd [10, 13]. We have considered this by includ-ing constant and intermittent per-capita sourcesof infection in the model, with the assumptionthat the cycle of infection in the dog is main-tained independently of the herd of cattle.However, the cycle of infection in the dog couldalso be maintained by infection in the dairy herdthrough the consumption of infected placentae orother bovine material. If this was the case, thenthis route of transmission would be a delayedmass-action process dependent on herd infectionprevalence. Clearly, a greater understanding ofthe epidemiology of the disease in farm dogs andother potential wildlife hosts is required.

The ®rst paper describing colostrum as a po-tential source of infection considered it as a formof pseudo-vertical transmission [11]. We havealso considered colostrum and milk from infectedcattle as a source of within-herd horizontal trans-mission due to the common practice of feedingpooled colostrum and milk to neonatal calves,although, for simplicity, we have not modelledthis as an age-dependent process. Other forms ofwithin-herd horizontal transmission could alsooccur; for example, cow to cow transmission viainfected placentae or amniotic ¯uid [20], butfurther studies are needed to test such hypoth-eses.

The most e�ective control policy would needto take into account both the costs and bene®tsof each strategy. However, when considering theimpact of each measure on disease prevalencealone, the culling of infected cattle would appearto be the most e�ective in the short term(although it becomes less e�ective in herds with a

higher turnover of stock). The reason for this is

that culling removes both within-herd horizontal

and vertical sources of infection. The policy of

keeping infected cattle in the herd but not breed-

ing replacements from them would also be e�ec-

tive in the short term (and conversely becomes

more e�ective in herds with a higher turnover of

stock). However, this policy would not be as

e�ective as culling, because it would only directly

block vertical transmission and infected cattle

would remain in the herd for longer periods of

time. In the presence of horizontal transmission

(with the exception of within-herd transmission

below the critical threshold), neither culling nor

selective breeding would eliminate infection from

the herd. Culling would leave a residual level of

infection from external sources such as an inde-

pendent de®nitive host, and a policy of not

breeding from infected cattle would allow an ad-

ditional within-herd source of horizontal infec-

tion. In the presence of horizontal infection,

elimination of infection from a herd would there-

fore require a combination of control options.

The models developed here have provided a

mathematical framework for our understanding

of neosporosis in dairy cattle. They have incor-

porated some of the key features of the disease in

dairy cattle: (1) the very high probability of verti-

cal transmission, (2) the impact on mortality and

reproduction, and (3) the compensatory e�ect on

survival and reproduction of uninfected cattle.

We have also explored the role of horizontal

transmission and highlighted the need to identify

and quantify these sources of infection. The

models have been deliberately kept simple, with

few parameters and an analytical solution. Even

though many simplifying assumptions have been

made, the models behave in a plausible way and

the outputs are consistent with observational

®eld studies. Once we have a greater quantitative

understanding of the epidemiology, it may be

appropriate to build more complex models that

capture the age structure of the herd and describe

the cycles of infection in the putative de®nitive

host.


Acknowledgements

Many thanks are due to Professor M. Begonfor his help in model development and his com-ments on the manuscript. Dr H. Davison wasfunded by the Ministry of Agriculture, Fisheriesand Food, UK.

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