# Appendix two: modelling in veterinary epidemiology and economics

## 1. Introduction

One of the major problems in veterinary epidemiology and economics lies in the estimation of the relationships between the many different factors determining a disease process in a livestock population.

There are two approaches to estimating such relationships: the empirical and the theoretical approach. The empirical approach involves going out into the real world to observe and monitor, while the theoretical approach involves attempts to deduce how the system being investigated works and thus the effect that one factor has on another. The latter approach essentially involves building a model of the particular system being investigated. Models are a representation of a system, which allow the behaviour of the system to be simulated under controlled conditions. In engineering, models are often physical (e.g. an aeroplane wing in a wind tunnel) whereas in epidemiology and economics they are invariably mathematical. Thus instead of being represented by physical structures, the system is represented by mathematical relationships.

The difference between the two approaches is best illustrated by a simple example. Suppose that it was necessary to determine the percentage of male calves born in a cattle population. An empiricist would take a sample of calves and count the number of males. A theoretically inclined person might catch a cow and a bull, examine their reproductive system and deduce that since "X" and "Y" spermatozoa are produced in equal numbers and are of approximately equal viability and motility, the proportion of male calves would be approximately 50%.

In more complex situations, both approaches have weaknesses. If it were necessary to estimate the relationship between foot-and-mouth disease (FMD) vaccination and milk production in dairy cattle in an FMD-endemic area, one would be unlikely to obtain useful results by simply measuring milk production in a sample of vaccinated and unvaccinated herds. This is because milk production is influenced by many factors other than FMD, and these would tend to confuse the results. Worse still, some factors will most likely be related to both FMD vaccination and milk production. For example, farms with better management will tend to have a higher output of milk and to use FMD vaccine. Thus it would be wrong to attribute higher milk production in vaccinated herds to the vaccination alone. The theoretical approach might not be very helpful either for it is unlikely that we shall ever achieve a complete quantitative understanding of either the epidemiology of FMD or of the dairy production system. The solution is to model those parts of the system that are understood, and to estimate those relationships that are not by observation and experiment.

## 2. Types of model

There are many different types of model based on different techniques with varying degrees of complexity that are used in the fields of veterinary epidemiology and economics. To describe all types of model and the techniques used is beyond the scope of this manual. We will, therefore, concentrate on two models which are particularly useful in the economic assessment of disease. At this stage, two basic distinctions need to be made.

Models may be either dynamic or static. A dynamic model will show the behaviourof a system over time, whereas a static model will only describe the steady-state situation representing the equilibrium that the system should eventually reach. For our purposes, equilibrium can be described as that situation when output and growth have settled at their steady-state or constant values. A dynamic model might show the daily offtake of milk that would be produced by a herd over 20 years, whereas its static counterpart would show only the average daily milk production per head (or per livestock unit) that would be produced when the system had settled to equilibrium. Dynamic models generally involve much more computation than static models and, as such, they normally require a computer for their effective use.

Models may also be deterministic or stochastic. A deterministic model will describe the situation which would arise if all the variables had average values, while a stochastic model allows the variables to take values from a range of values according to some probability distribution. For example, we could make a deterministic model of the sex ratio of 100 calves by the formulae:

M = pN
F= (1-p)N

where:
N = total number of calves,
M = number of male calves,
F = number of female calves, and
p = the probability of a calf being male.

Thus, for 100 calves:

M = 0.5 x 100 = 50
F=(1 - 0.5) x 100=50

A stochastic model of the same system, i.e. based on binomial distribution, would tell us that there is a probability of the number of males being any number from 0 to 100, and that the mean number of males would be pN = 50 with a standard error of:

We may say with 95% confidence that in an sample of 100 calves, the number of males will be within two standard errors of the mean i.e. in the range 40 to 60. Thus, stochastic models can take into account the effect of chance, but often at a considerable computational cost. Whether the cost is justified depends on how important the effect of chance is seen to be by the user of the model. Generally, if the population to be modelled is large, a deterministic model may give sufficiently good results, as illustrated in Table 1.

Table 1. Ninety-five percent confidence limits for the percentage of males in calf populations of different sizes.

 Population size 95% confidence limits for % of males 10 18.4-81.6 100 40.0-60.0 1 000 46.8 - 53.2 10 000 49.0-51.0

A common method for introducing a stochastic, or chance, element into models is the Monte-Carlo technique. Although we shall not be using it in our examples, the technique is worth explaining because it is very simple to apply when dichotomous variables are involved, which is often the case in disease modelling.

If a model is examining individual cows, it is necessary to decide whether their calves will be male or female, as it would be absurd to introduce a calf that was half male and half female. The programme would generate a random number with a value between 0 and 1. If the random number is less than the probability of a calf being male (0.5), the calf would be male, otherwise it would be female. This technique is applicable in many other situations: e.g. Does the cow conceive to the service today? Does an animal become ill with a disease today? If diseased, does the animal die today?

## 3. Examples of models used in veterinary epidemiology and economics

We will now describe two models which are particularly useful in the economic assessment of disease control ac tivities. The first of these will be a simple dynamic model of a cattle herd and the second will be a static model which relates a series of herd productivity parameters to the quantity of offtake produced per unit of feed resource, under certain conditions.

### 3.1 The basic parameters required for herd modelling

The main biological parameters required for herd modelling incorporate data on mortalities, fertility and output.

Mortalities. Data on mortalities are normally incorporated in the form of death rates. Often age-specific death rates are used, which are death rates occurring in specific age categories (e.g. 0 to 1, 1 to 2, 2 to 3, 3 to 4, 4 to 5 years etc) in a specific time period, usually a year.

Alternatively, different age categories may be used, such as the mortality rate in calves between birth and weaning (calf mortality rate), the mortality rate in young stock between weaning and maturity, and the mortality rate in adults (often given as a constant for the different adult age categories).

Sometimes it is necessary to be even more precise and to use age/sex specific death rates. This is because in certain production systems mortalities may be higher in one sex of the same age category.

Survival rates are also often used. These are obviously I minus the death rate, if the rate is expressed as a decimal, or 100 minus the death rate if the rate is expressed as a percentage.

Fertility parameters. Data about fertility are normally incorporated in the form of parturition rates (i.e. calving, farrowing, kidding rates etc as appropriate). These are normally expressed as the number of live births occurring in a specified population of females in a specified time period, usually a year. Age-specific parturition rates are sometimes used. In the case of species where multiple births are common, it may also be necessary to specify the number of offspring per parturition.

Sometimes reproductive performance is specified in terms of a parturition interval instead of a parturition rate. This is normally expressed in terms of the average time interval between parturitions. In cases of species for which single births are the rule, the annual parturition interval and the annual parturition rate can be derived from one another by the following formula:

Annual parturition rate = 1/Parturition interval if the parturition interval is given in years; or

Annual parturition rate = 365/Parturition interval if the parturition interval is given in days.

Output. The next category of parameters used in herd models are those determining the physical quantities of output. These are frequently specified in the form of offtake (sales, slaughter and culling) and yields (milk, wool, eggs etc).

Offtake covers the removal of animals from a herd or flock for all reasons other than mortality and emergency slaughter due to illness. A distinction is often made between culling and the sale of surplus animals, with culling usually referring to sale of old or unproductive animals for slaughter. Offtake is determined by the livestock producer and may vary according to external circumstances. Both offtake and culling can also be expressed in the form of rates and are usually calculated using age and sex categories.

Yields are usually given either in relation to some other parameter (e.g. lactation yield per parturition) or in terms of annual amounts for specific age/sex categories. The average annual milk yield of a dairy animal can be derived from the formula (12/I) Y, where Y is the average lactation yield for the particular category of animal and I is the calving interval in months.

The values of offtake and yields are determined by applying prices to the output data. For offtake, prices are normally given in relation to age/sex categories, while for yields they are given in terms of per unit of the appropriate commodity produced.

Once the above parameters have been determined, the composition of the herd or flock must be defined in order to form a basis for the projection. This normally involves defining the number of animals in each age/sex category or setting targets for certain categories. In the latter case, these may be expressed in terms of numbers (e.g. 100 breeding cows) or as ratios (e.g. cow:bull ratio).

### 3.2 Dynamic herd models

The links between the parameters that are necessary to de rive a dynamic herd model are illustrated on the following example.

The model described in Table 2 is a dynamic and deterministic model showing the number of females by age group and the number of male calves produced each year. The parameters required to build this are the mortality and calving rates for each age group. The survival rate for each group is calculated as 1 - age-specific mortality rate. As in most herd models, the parameters used to calculate survival rate, number of births etc expected during a year, are applied to the numbers of animals in the appropriate age/sex categories at the start of the year in question. This generates an end-of-year figure which is shown in the output table for the following year.

We may now calculate the number of immature females in the 1-2 year age group for year 2. This will be the number in the 0-1 age group for year 1, multiplied by the survival rate for the 0-1 age group i.e. 0.92 x 30 = 27.6. The decimals are normally rounded off to the nearest whole number since we cannot have 27.6 animals. The same procedure is applied to all other age groups, except that the number moving into the 10 + age group for year 2 will be the sum of the number in the 9-10 age group for year 1 multiplied by the survival rate plus the number in the 10 + age group for year 1 multiplied by the survival rate.

Next we need to calculate the number of calves born in year 1. This will be the sum of the cow numbers in each group from the 3-4 age group onwards multiplied by the calving rate for each group. Half of the calves will be male, and should be entered in a box at the bottom of the table for year 1, and half will be female which will be entered in a box for year 2.

The process can be repeated to calculate herd structures and male calf production for as many years as required. It can also be used to model the herd and flock structures of other livestock species, and the model can be given a stochastic element by applying the Monte-Carlo technique. The last four columns of the table have been left blank for the reader to try the process.

The calculations involved in the model are simple and can be easily programmed into a programmable calculator or computer. The model can be extended to include culling rates, the fattening of male calves, milk production and many other factors. If, for example, we wished to include an annual culling rate of 10% (0.1) in the age groups over 4 years of age, the number of animals in the 5-6 year age group in year 2 would be the number of animals in the 4-5 year age group in year 1 x [ 1 - (mortality rate + culling rate)] = 18x[1-(0.05+0.10)] = 18x0.85 = 15.3 or 15.

### 3.3 Incorporating the effect of disease into herd models

Dynamic herd models are useful in that they allow some of the dynamic effects of disease losses, such as reduction in fertility, to be evaluated on a "with" and "without" basis. They are also useful for simulating the effects of measures designed to improve animal productivity.

For example, the effect of a disease outbreak in a herd may be modelled by applying an increased death rate, an increased culling rate, lowered milk yield, decreased parturition rate etc to different age/sex categories. In order to use the model, we need to determine the effect of disease on various productivity parameters; once this is done, we can model its impact on output. This is much easier than trying to observe the effect on output directly. Information on the effects of disease on productivity parameters can be obtained from surveys or experiments. Normally, the effect of disease is manifested as the difference in the value of a parameter for an infected animal; e.g. a growing animal affected by FMD might suffer a 3-month delay in reaching maturity. The basic parameter values are usually estimated in the "with disease" situation. We therefore need to calculate the mean parameter value when the incidence of disease is 0, so that the model can be run for a "with" and a "without disease" situation, and the output values compared. The general formula is:

A0 = A ± Er

where:

A = the mean parameter value with the disease,
A0 = the mean parameter value without the disease,
E = the disease effect on the parameter, and
r = the incidence of the disease.

The sign used in the equation depends on whether the disease effect is likely to increase or decrease the mean parameter value.

Example: If the mean age of maturity of animals is 3.8 years in a cattle population with a 15% incidence of trypanosomiasis infection and the effect of the infection is an estimated 0.5-year delay in reaching maturity, then the mean age at maturity without trypanosomiasis would be:

3.8 - 0.5 x 0.15 = 3.725 years

Most cattle herd models use calving rate as a fertility parameter, but disease effects are frequently expressed as an extension of calving interval. In such circumstances it is necessary to change the calving rate into a mean calving interval before calculating the disease effect. This is done by the following formula:

Calving interval = 1/Calving rate

Example: It is estimated that, in a herd infected with brucellosis, 2% of pregnancies end in abortion. The aborting cows are estimated to suffer a 1-year extension to the calving interval. The calving rate with the disease is 80%. The calving rate without the disease is calcualted as follows:

Calving interval = 1/0.8 = 1.25
Mean calving interval A0 = A - Er = 1.25 - 1 x 0.02 = 1.23
Calving rate without disease = 1/1.23 = 0.813 or 81.3%

Disease effects on mortality and culling rates are simply additive. Thus, if a disease is estimated to cause 5% mortality in infected animals, and has an annual incidence of 20%, the average annual mortality caused in the whole population will be 0.05 x 0.2 = 0.01 or 1%. If the herd mortality rate from all causes is 5% per year, then without the disease, the mortality rate would be 5% - 1% = 4%.

### 3.4 Static herd productivity model

The main problem with a dynamic model is that it considers output on a per animal basis. This is a nuisance when we wish to determine the effect of disease on current productivity, because the model changes herd sizes and structures as the various parameters within the model are altered by the effects of the disease. It is then difficult to compare "with" and "without" disease results at the same point in time, because the population structures are different.

This problem can be overcome by the use of a static model, which assumes a herd at equilibrium with a growth rate of 0, so that all animals not needed to replace breeding stock are sold out of the herd as soon as they reach maturity. The model assumes that at equilibrium the system will use all of the available feed resources, and in this case the herd structure and production are implied by a set of parameters. The relationships within the model can be shown to be true for any species, and so we can deduce the effect of a change in any of the production parameters on the value of offtake with absolute certainty, given that certain conditions hold true. The relationships are illustrated in Figure 1.

Figure 1. Illustration of the relationship between productivity parameters and offtake using a static herd productivity model.

 Productivity parameters Mortality rates Culling rates Milk yield/lactation Parturition rate Prolificacy Weaning rates Ages at maturity Feed requirements ¯ MODEL ¯ Quantities of offtake per unit of feed supply per year Culled breeding females Culled breeding males Barren replacement females Mature replacement females Mature fattening males Litres of milk

Precise account is taken of all the interactions within the system. The value of this is best illustrated by an example. Suppose that a disease kills 10% of all growing animals, but has no other effects, and that it is not possible to purchase replacements at the time of mortality. Then one might calculate the economic loss as being the value of the offtake of mature animals lost. This would be an overestimate, however, because more breeding animals could be kept on the same feed resource when the mortality was occurring. To calculate how many animals could be kept on the feed resource would be difficult, the model would take this effect into account.

There are four categories of animals in the model: male and female breeding stock and male and female surplus or fattening animals which are sold out of the herd when they reach maturity. For each of these categories, replacement stock between the ages of weaning and maturity can be found in the herd. The parameters that are used in the model are listed in Figure 1. Having ascertained these parameters, the following steps are generally needed to construct a static model.

1. Work out the number of replacement breeding stock needed annually as follows:

- Fix the number of breeding cows.

- Apply a bull/cow ratio to derive the number of breeding bulls.

- Apply the appropriate death and culling rates to the breeding cow and breeding bull categories and thereby derive the numbers of adult breeding stock required annually. In the case of the adult replacement females, the numbers must be increased by a correction factor to take into account the percentage of these animals assumed to be barren.

2. Work out the maximum number of replacement breeding stock that could be produced as follows:

- Apply the appropriate parturition rates and numbers of births per parturition to the appropriate breeding female categories in order to derive the number of male and female calves born per annum.

- Apply the appropriate weaning rate to derive the numbers of calves weaned per annum.

- Apply the appropriate death rates to each category of weaned replacement breeding stock to derive the number of animals of each sex surviving to maturity.

- Subtract from this the number of male and female breeding replacements required to derive the proportion of weaned animals of each sex which must be retained as breeding stock.

3. Work out offtake and herd composition as follows:

- If appropriate, apply a relevant correction factor (to take into account variations in death rates between surplus and replacement stock) to the surplus numbers of weaned animals of each sex, in order to derive the number of animals that will be sold out of the herd as surplus when they reach maturity.

- Apply the appropriate culling rates to adult breeding stock and the proportion of barren heifers to the number of mature female replacements to derive the total offtake of animals in each of these categories.

- Calculate the total offtake.

- From the above calculations, the number of animals in each of the different categories in the herd can be calculated. The animal numbers can be summed together and the herd composition in percentage terms can be derived.

- Calculate the total annual milk yield by applying the appropriate variations of the formula (I 2/1)Y to the numbers of breeding cows in the relevant categories.

The steps outlined above do not correspond precisely to the actual steps used in the model demonstrated in Figures 2 and 3, which uses more complex mathematics to arrive at the results more quickly (such as defining the herd structure in terms of the ratios of other classes of stock to females of reproductive age), but the principles are similar.

The model can be taken a stage further. Feed requirements in terms of livestock units can be specified for each of the four categories of mature stock, and the average requirement for the replacement stock can be calculated by assuming a linear growth from no feed requirement to the feed requirement at maturity, making the appropriate allowance for mortality. The mean feed requirement for growing animals tends to be less than half the feed requirement at maturity, since there are more animals in the younger age groups. The whole of the model can then be standardised on one livestock unit, which is not defined in the model. Thus different quantities of grassland, or combinations of concentrates and forage making up the requirements of one livestock unit can be applied to a herd using the production parameters given. The results of the model are then given in terms of the average combination of livestock on one livestock unit of feed resource and the value of output specified in terms of that one unit.

Used in this way, the model can be applied to any species of livestock. It can compare the efficiency say, cattle and goats in their utilisation of a feed resource. Moreover, the herd or flock being modelled need not be located solely in one geographical area, so systems where animals are bred in one area and fattened in another can be simulated.

The model has no stochastic element, which means that it gives expected production and tells us nothing about the potential variability in individual herds. It is most useful, therefore, in predicting the behaviour of national herds, where the changes in mean parameter values can be expected to be slow.

We will now illustrate the use of this type of model in detail. The example makes use of a microcomputerised static model whose output is illustrated in Figures 2 and 3. The model programme can be obtained on request from the authors.

Example: Suppose that foot-and-mouth disease is endemic in an extensive cattle production system in, say, Kenya. What would be the estimated annual loss due to the disease? The following parameters for the system were estimated:

 Cattle population 2.8 million Mortality rate in animals over 6 months 4% per annum Cull rate in cows 5% per annum Cull rate in bulls 7% per annum Milk offtake 450 litres/lactation Calving rate 65% per annum Calf weaning rate 85% per annum Age of heifers at first calving 4 years Age of bulls at maturity 2.5 years Age of steers at sale 4.5 years Offtake values (KSh) * Culled cows 1800 Culled bulls 2000 Surplus heifers 2500 Mature steers 2200 Milk 2.00/litre

* For the purpose of this exercise the exchange rate is KSh 10 = US\$ 1.

Livestock units

 Breeding female 1.0 Mature fattening female 1.0 Bull 1.25 Mature fattening steer I.25

The estimated annual incidence of foot-and-mouth disease was 30%, and the effects of the disease were estimated as follows:

· 1% of the animals affected died.

· 2% of the affected cows and bulls were culled.

· Cows produced milk for 6 months after calving. If a lactating cow was affected, 20% of the lactation yield was lost.

· 10% of pregnant cows affected with the disease aborted and had calving intervals extended by I year.

· Non-pregnant cows suffered a l-month extension to the calving interval.

· Calves were weaned at 6 months; 8% of the suckling calves affected died.

· Growing animals suffered an average delay of 6 weeks in reaching maturity.

The parameter values "with" and "without" FMD can be estimated as follows:

Mortality rate
Mortality rate due to FMD = 0.01 x 0.3 =. 0.003
Mortality rate without FMD = 0.04 - 0.003 = 0.037

Weaning rate
Calf mortality rate = 1 -Weaning rate = 1 - 0.85 = 0.15
Incidence of FMD in calves during the 6-month preweaning period = 0.3 x 0.5 = 0.15
Calf mortality rate due to FMD = 0.08 x 0.15 = 0.012
Calf mortality rate without FMD = 0.15 - 0.012 = 0.138
Weaning rate without FMD = 1 - 0.138 = 0.862

Cull rate
Cull rate due to FMD = 0.02 x 0.3 = 0.006
Cull rate without FMD in cows = 0.05 - 0.006 = 0.044
Cull rate without FMD in bulls = 0.07 - 0.006 = 0.064

Calving (parturition) rate
Pregnancy rate = Gestation period/Calving interval
= Gestation (in years) x Calving rate
= 0.75x0.65= 0.4875

Non-pregnancy rate = I - 0.4875 = 0.5125
Effect of FMD on mean calving interval in affected animals = (1/12 x 0.5125) + (1 x 0.4875 x 0.1) = 0.0914583 years
Mean calving interval without FMD = 1.5384615 - 0.0914583 x 0.3 = 1.511024 years
Mean calving rate without FMD = I/1.511024 = 0.6618

Milk offtake
The cows were in milk for 6 months, so the FMD incidence rate during the lactation period will be half the annual incidence rate i.e. 0.3/2 = 0.15.
Mean amount of milk lost per lactation = 450 x 0.2 x 0.15 = 13.5 litres
Mean amount of milk without FMD = 450 + 13.5 = 463.5 litres

Age at maturity
Incidence of FMD for the growing period in:
Heifers = 4x0.3 = 1.2
Bulls = 2.5x0.3 = 0.75
Steers = 4.5x0.3 = 1.35

Age at maturity without FMD:
Heifers = 4 - 6/52 x 1.2 = 3.862 years
Bulls = 2.5 - 6/52 x 0.75 = 2.413 years
Steers = 4.5 - 6/52 x 1.35 = 4.344 years

These parameters can be fed into the model as indicated in Figures 2 and 3, which show the productivity of cattle "with" FMD and "without" FMD.

Estimating the economic effect of FMD

In order to estimate the effects of FMD we need to compare offtakes in the "with" and "without" situation. To be able to do this we need to determine the total carrying capacity of the area in livestock units, since the model calculates the value of offtake per livestock unit.

The total carrying capacity of the area can be estimated as follows. For each class of stock, the feed requirement will be the product of the number of animals of that type and the feed requirement per animal. Thus from Figure 2:

Breeding females = 0.313806* x 2 800 00 x 1 = 878 657
Breeding males = 0.0125522* x 2 800 000 x 1.25 = 43 933
Replacement females = 0.122716* x 2 800 000 x 0.448763* = 154 197
Replacement males = 0.00363414* x 2 800 000 x 0.584054* = 5943
Fattening females = 0.19721000* x 2 800 000 x 0.448763* = 247 802
Fattening males = 0.350082* x 2 800 000 x 0.531343* = 520 838
Total carrying capacity = 1 851 370 LU

The value of production in each situation can then be estimated by multiplying the value of the total offtake per livestock unit carrying capacity (as determined by the model) by the total carrying capacity in livestock units.

The value of the annual production lost because of foot-and-mouth disease is therefore (769.212870 x 1 851 370) - (730.428550 x 1 851 370) = 71 804 126 KSh.

* Determined by the model; see Figure 2.

Figure 2. Static simulation of the productivity a cattle population located in an area with endemic foot-and-mouth disease (30% annual incidence).

Annual death rates

 Breeding female 4 Replacement female 4 Breeding male 4 Replacement male 4 Fattening male 4

Annual culling rates

 Breeding female 5 Breeding male 7

Survival-to-weaning rates

 Males 85 Females 85

Mean ages at maturity

 Replacement female at first parturition 4 Replacement male used for breeding 2.5 Surplus female at first parturition 4 Fattening males at time of sale 4.5

Fertility data

 No. of breeding females per breeding male 25 Parturition rate (%) 65 No. of offspring per parturition 1 Percentage replacement females barren 0 Offtake of milk/lactation (litres) 450

Mean feed requirement (L U)

 Mature animals Crowing animals Breeding female 1 Replacement female 0.448763 Breeding male 1.25 Replacement male 0.584054 Surplus female 1 Surplus female 0.448763 Fattening male 1.2 Fattening male 0.531343

Herd structure

 Class of stock Number/LU carrying capacity % of herd Breeding female 0.474598 31.380600 Replacement female 0.185595 12.271600 Breeding male 0.018984 1.255220 Replacement male 0.005496 0.363414 Surplus female 0.225243 19.721000 Fattening male 0.529462 35.008200

Offtake

 Class of offtake Offtake Value/unit Offtake value (Unit/LU/year) (KSh) (KSh/LU/year) Culled breeding females 0.023730 1800.00 42.713800 Culled breeding males 0.001329 2000.00 2.657750 Barren replacement females 0.000000 0.00 0 000000 Mature surplus females 0.068643 2500.00 171.607000 Mature fattening males 0.107812 2200.00 235.801000 Litres of milk 138.820000 2.00 277.640000 Total 730.428550

Figure 3. Static simulation of the productivity of a cattle population in the same area but free from foot-and-mouth disease.

Annual death rates

 Breeding female 3.7 Replacement female. 7 Breeding male 3.7 Replacement male 3.7 Fattening male 3.7

Annual culling rates

 Breeding female 4.4 Breeding male 6.4

Survival-to-weaning rates

 Males 86.2 Females 86.2

Mean ages at maturity

 Replacement female at first parturition 3.862 Replacement male used for breeding 2.413 Surplus female at first parturition 3.862 Fattening males at time of sale 4.344

Fertility data

 No. of breeding females per breeding male 25 Parturition rate (%) 66.18 No. of offspring per parturition 1 Percentage replacement females barren 0 Offtake of milk/lactation (litres) 463.5

 Mature animals Growing animals Breeding female 1 Replacement female 0.454006 Breeding male 1.25 Replacement male 0.588341 Surplus female 1 Surplus female 0.454006 Fattening male 1.2 Fattening male 0.538336

Herd structure

 Class of stock Number/LU carrying capacity % of herd Breeding female 0.470112 31.244800 Replacement female 0.158307 10.521400 Breeding male 0.018804 1.249790 Replacement male 0.004798 0.318875 Surplus female 0.221005 21.508800 Fattening male 0.528965 35.156300

Offtake

 Class of offtake Offtake Value/unit Offtake value (Unit/LU/year) (KSh) (KSh/LU/year) Culled breeding females 0.020685 1800.00 37.232900 Culled breeding males 0.001203 2000.00 2.406970 Barren replacement females 0.000000 0.00 0.000000 Mature surplus females 0.077844 2500.00 194.611000 Mature fattening males 0.112070 2200.00 246.554000 Litres of milk 144.204000 2.00 288.408000 Total 769.212870

Table 2. Dynamic model of a dairy herd.

 Age group (years) Calving rate Survival rate Number of females by age group by year Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year 7 0- 1 0.0 0.92 30 29 35 1- 2 0.0 0.96 28 28 27 2- 3 0.0 0.96 32 27 27 3- 4 0.25 0.95 29 31 26 4- 5 0.75 0.95 18 28 -29 5- 6 0.6 0.95 24 17 27 6- 7 0.6 0.95 15 23 16 7- 8 0.6 0.95 10 14 22 8- 9 0.5 0.92 8 10 13 9- 10 0.4 0.75 6 7 9 10 + 0.4 0.5 3 6 8 Male calves 29 36