Idea Transcript
Modelling and dynamics of intramammary infections caused by Corynebacterium species Amira RACHAH ∗ , Gunnar DALEN ∗
∗† ,
Olav REKSEN ∗ , H˚avard NØRSTEBØ∗† , John W. BARLOW ‡ ,
Norwegian University of Life Sciences, Department of Production Animal Clinical Sciences, 0033 Oslo, Norway † TINE As, ˚ P.O. box 58, N-1430 As, ˚ Norway ‡ Department of Animal Science, University of Vermont, Burlington VT 05405, USA
Abstract—We present a deterministic mathematical model that describes the transmission dynamics of intramammary infections (IMI) caused by Corynebacterium spp. (Corynebacterium species) in lactating dairy herds. Longitudinal, quantitative, dynamic models are likely to be valuable for predicting infections outbreak risk, quantify the effectiveness of response tactics and performing response planning. Previous investigations on IMI have considered the transmission of several bacterial pathogens, but not Corynebacterium spp., in modelling investigations. We have established a new Corynebacterium spp. Sn ISw compartmental model subject on appropriate unknown parameters that we estimated by using a deterministic approach, based on real data fitting procedure. The real data, from which the parameters of the model are estimated, were obtained in a field trial conducted in a New York dairy herd (US).
I.
I NTRODUCTION
Mastitis (inflammation of the mammary gland) is the most frequently occurring disease of dairy cows. It is a significant animal and economic issue in the dairy farming industry, and affects both milk production and milk quality. Mastitis caused by bacterial infection is routinely diagnosed through bacterial culture. Corynebacterium spp. are considered to be minor udder pathogens [7], [8], [17], [31], [21], [22]. The development of decision support tools for detection and management of intramammary infections (IMI) remains the subject of extensive research. IMI with Corynebacterium spp. are generally mild causing limited loss in milk production. However, significant elevation of somatic cell count (SCC) has been observed [8], [9], [10], [37], [6]. Pathogen-specific transmission patterns have been described for other major and minor mastitis pathogens [33], [5] but not for Corynebacterium spp. This has motivated us to construct a model of the transmission dynamics of Corynebacterium spp. IMI throughout the lactation period. Mathematical modelling of infectious diseases for animals, is a powerful tool for understanding infection dynamics by providing useful predictions about the potential transmission of infections and the effectiveness of control measures [23], [15], [29], [2]. It is only recently that a small number of studies have integrated the principle of compartmental models to describe the dynamics of mastitis transmission and the overall effects of interventions for the modelling of infectious diseases in animals [1], [25], [26], [38], [36], [3], [35], [13], [17], [18]. Mathematical models depend on appropriate parameter values that are often unknown and must be estimated from
the real data. In this paper, we develop a framework for efficient estimation of Corynebacterium spp. IMI model with transmission parameters obtained through real data fitting procedure using nonlinear least squares method for nonlinear ordinary differential equations (ODEs) [34]. The real data were collected from a 13-month longitudinal observational study in one commercial dairy herd in New York, US. The transmission dynamics were further evaluated by the basic reproduction number R0 , which is defined as the expected number of secondary cases that arises per infectious individual in a fully susceptible population [11], [12], [30]. II.
DATA COLLECTION
Data were obtained from a 13-month longitudinal observational study in one commercial dairy Holstein herd in New York, US. Details on the herd, microbial analyses and sampling framework has been published previously [33]. Mastitis control practices, including pre- and postmilking teat disinfection and the use of blanket dry-cow therapy, were standardized. Quarter milk samples were collected for bacteriological diagnosis monthly from lactating cows. In total 371 cows were sampled, and the median number of cows sampled each month was 210. In addition, samples were collected within 3 days after parturition and whenever animals were moved to or from the lactation pen. Altogether 14401 quarter samples were analyzed from this herd. Trained field technicians collected the scheduled monthly quarter milk samples. Additional quarter milk samples were collected by farm personnel that had received training for this. All samples were collected according to recommended guidelines [16]. Samples collected monthly were kept on ice after collection and during transport to the laboratory, where they were frozen. Additional samples collected by farm personnel were frozen immediately after collection. Samples were thawed in the laboratory and bacteriological culture was performed according to standard procedures [16]. Samples with culture results presenting more than 3 morphologically different colony types were treated as contaminated and excluded from further analyses. A quarter was considered to have an infection (IMI) with Corynebacterium spp. when adhering to at least one of the following criteria: 1) ≥ 1000 c.f.u./ml of the pathogen was cultured from a single sample, 2) ≥ 500 c.f.u./ml of the pathogen were cultured from two out of three consecutive milk samples, 3) ≥ 100 c.f.u./ml were cultured from three consecutive milk samples, or 4) ≥ 100 c.f.u./ml were cultured from a clinical sample, where c.f.u respresents colony-forming units.
c 978-1-5090-5454-1/17/$31.00 2017 IEEE
Positive bacteriological culture results that did not meet any of the above criteria were classified as transient colonizations.
ψIβSn I
θ sn µN
III.
M ODEL FORMULATION
Sn
In this study, we describe the transmission dynamics of Corynebacterium spp. IMI by a Sn ISw model. The model describes a population of lactating udder quarters divided into three compartments: •
Sn : naive susceptible quarters which represents quarters in a cow where none of the lactating quarters is harboring Corynebacterium spp.,
•
Sw : susceptible quarters in a cow, already harboring preexisting IMI from Corynebacterium spp. neighbouring quarters,
•
I: quarters affected with IMI caused by Corynebacterium spp.
The model is defined by a set of three nonlinear ordinary differential equations describing the change in proportion of quarters within each compartment over time: dSn (t) =θsn N µ + αI(t)P + αSw (t) dt − (ψw + ψI ) βSn (t)I(t), −µSn (t)
(1)
dSw (t) =θsw N µ + ψw βSn (t)I(t) + αI(t)(1 − P ) dt − λβSw (t)I(t) − αSw (t) − µSw (t), (2) dI(t) =θI N µ + ψI βSn (t)I(t) + λβSw (t)I(t) dt − αI(t)(1 − P ) − αI(t)P − µI(t),
(3)
where Sn represents the proportion of susceptible quarters in cows with no Corynebacterium spp. IMI; Sw represents the proportion of susceptible quarters in cows with Corynebacterium spp. isolated from one or more of the neighboring quarters in the same cow; I represents the proportion of infected quarters; β is the transmission parameter; α is the daily rate of cured quarters, N is the total population of quarters (N = Sn + Sw + I); P is a function for the estimation of proportion of cows with ψw factor of quarters infected, Sw approximated by ; and λ is the increased risk of within ψw I cows transmission for an Sw quarter. The number of new Corynebacterium spp. IMI from Sn was estimated by ψI βSn I; the number of new Corynebacterium spp. IMI from Sw was estimated by λβSw I; the number of cured quarters from I to Sn was estimated as αIP ; and the number of cured quarters from I to Sw was estimated as αI(1 − P ). The number of susceptible quarters from Sn to Sw equal to ψw βSn I; and the number of susceptible quarters from Sn to Sw is αSw . The daily rate of entering and exit of quarters to and from each compartment is µ; the proportion of µ entering Sn is θsn ; the proportion of µ entering Sw is θsw ; and the proportion of µ entering I is θI . The dynamics of state transitions are illustrated in Figure 1.
αIP
µS n
θ I µN αS w
ψwβSn I
I
µI θ sw µN
αI (1 − P) Sw
µS w
λβSw I
Fig. 1. Diagram of the mathematical model of transmission of intramammary infections with Corynebacterium spp. IMI. The boxes represent the state variables Sn = naive susceptible , Sw = susceptible quarter in a cow with a simultaneous infectionin one or more other quarters and I = quarter affected with an IMI by Corynebacterium spp. The arrows represent the flow rate between the compartments: Sn to I = βSn I, Sn to Sw = ψw βSn I, I to Sn = αIP , I to Sw = αI(1 − P ), Sw to Sn = αSw , Sw to I = λβSw I. Enteries of quarters to the different compartments are determined by θsn , θsw and θI multiplied by µ × N , where µ = daily rate of entry and ext of quarters to and from the lactation pen, and N = total population of quarters at any given time. P = function for the estimation of proportion of cows with ψw factor of quarters infected, approximated by ψSwI and λ is the increased risk w of within cows transmission for an Sw quarter.
IV.
PARAMETERS ESTIMATION
In this section, we present the deterministic approach that we used in the estimation of parameters of the Sn ISw model from real data. In traditional biological modelling studies, veterinarians used to apply statistical techniques (based on some assumption) for the estimation of parameters of bacterial pathogens transmission models. Estimation of parameters is an important step for describing transmission dynamics, since effective realtime decision making and management of epidemic outbreaks based on modelling depends on the ability to estimate epidemic parameters as the epidemic unfolds [4], [27], [28], [32]. Our mathematical approach is based on real data fitting procedure using nonlinear least squares method for nonlinear ordinary differential equations (ODEs). According to the definition of the least squares method, the best-fit curve of a given type is the curve that has a minimal sum of deviation squared from the real data. The mathematical description of our approach is as follow: Find estimates of the unknown parameters β, α, µ, θsn , θsw , θI , ψw , ψI , λ of the Corynebacterium spp. IMI model, for a given real data described by the points (tj , Ireal,j ) , j = 1, 2, . . . , n where tj is the time and Ireal,j is the value of proportion infected at the time tj , which will be fitted by the I-solution of the ODEs that describe the Sn ISw model, given by Isiml,j such that the sum of the squares of the deviations between Ireal,j and Isiml,j is minimized. These deviations d for each data point are given by d = Ireal,j − Isiml,j , j = 1, 2, . . . , n where Isiml,j is the proportion infected at the time tj obtained from the resolution of the nonlinear ordinary equations of the Corynebacterium spp. model. According to the nonlinear least squares method,
minimize
n X
(Ireal,j − Isiml,j )
2
(4)
j=1
subject to
equations of dynamics (1) − (2) − (3)
where the unknown parameters, in this optimization problem, are β, α, µ, θsn , θsw , θI , ψw , ψI , λ.
0.6 Proportion infected simulated Real proportion infected of US data
0.5
Proportion infected
the best fitting has the property that d21 + d22 + . . . d2n = P Pcurve n n 2 2 j=1 dj = j=1 (Ireal,j − Isiml,j ) = minimum. The fitting procedure is associated with the numerical resolution of the nonlinear ordinary differential equations of the Corynebacterium spp. model. Then we can formulate our mathematical approach for the estimation of parameters, as follows:
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0.1
0
V.
50
R ESULTS
Firstly, we present the distribution of quarter samples. Secondly, we present the result of the estimation of parameters of the Sn ISw -model, by using the US dairy herd data, and the numerical resolution of Corynebacterium spp. IMI model by using theses parameters. 3251 milk samples were culture positive for Corynebacterium spp. IMI. 1350 positive cultures were associated with an IMI. Distribution of quarter samples related to our categorization is shown in table I. ≥ 5000cf u/ml ≥ 1000cf u/ml ≥ 500cf u/ml ≥ 100cf u/ml No Corynebacterium spp. cultured Total number of quarter samples
TABLE I.
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350
Lactation days
In this section, we present the numerical resolution of the estimation of the parameters of Corynebacterium spp. model by using the US data.
Number of quarter samples 258 382 480 2131 11150 14401
100
Proportion 1.79 % 2.65 % 3.33 % 14.80 % 77.43 % 100 %
D ISTRIBUTION OF QUARTER SAMPLE CULTURE RESULTS .
The parameters are estimated by resolving the optimization problem (4), that is, by minimizing the sum of the squares of the deviations made between the proportion infected as given by the data and the corresponding ODEs Sn ISw -model. The optimization problem is solved by using the nonlinear programming solver fminsearchbnd of Matlab. The optimization problem was solved in combination with the numerical resolution of the ODEs of the Sn ISw -model. The nonlinear ordinary equations of the model are solved numerically by using ode45 solver, which is based on the Runge-Kutta (4, 5) method [14]. Figure 2 shows the numerical resolution of the estimation parameters approach which is based on data fitting procedures using nonlinear least squares method for nonlinear ordinary differential equations. The figure shows the result of the fit of the Corynebacterium spp. IMI Sn ISw -model to the real data. By using the estimated parameters, the proportion infected simulated passes through the majority of the points of the proportion infected of the real data trajectories (see figure 2). Table II consists of the results of the estimated parameters of the Corynebacterium spp. IMI Sn ISw -model to the real data.
Fig. 2. Fit of the Sn ISw model (described by the nonlinear ordinary differential equations (1), (2) and (3) to the real data using nonlinear least squares method for ODEs method, which minimized the deviation between the model output and the real data. Parameter α β µ θsn θsw θI ψw ψI λ
Value 0.037 0.065 0.01 0.451 0.19 0.041 3.6 1 1.202
TABLE II. E STIMATED PARAMETER VALUES OBTAINED BY APPLICATION OF THE MATHEMATICAL APPROACH BASED ON REAL DATA FITTING PROCEDURES USING A NONLINEAR LEAST SQUARES METHOD FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS .
To interpret the results, we describe the evolution of each group over time and the links between them. Figure 3 shows that the naive susceptible group begins to plummet due to how infectious the bacteria is. Simultaneously, the infected group’s proportion begins to rise, as seen in figure 2 and the susceptible within group begins to rise, as seen in figure 4. The curves showed on the figures 2, 3 and 4 are obtained through numerical resolution of the ODEs (1), (2) and (3) by using the parameters estimated through optimization. The key parameter in determining the behavior of the mathematical model is the basic reproduction number, R0 defined by R0 =
β . µ+α
The numerical value of R0 computed from the parameters estimated is equal to 1.36. The fact that the basic reproduction number R0 is greater than 1, explains the peak showed on the curve of the proportion infected in the figure 2. The biological interpretation of this peak and this value of R0 explain that Corynebacterium spp. IMI are contagious. The infection increases until the mid of the lactation and afterwards starts to decrease without any intervention.
By mathematically modelling the changes in prevalence by lactation stage, we demonstrate that the level of Corynebacterium spp. IMI vary throughout both the lactation period and the dry period. This way of estimating parameters by combining simulation and optimization is not yet common in mastitis research. Our approach expresses real change over time and should be applied in future studies. It will then provide a basis for finding the best time point for interventions such as teat disinfection and dry cow therapy, and include these in future optimal control programs.
1 0.9 0.8
Proportion of Sn
0.7 0.6 0.5 0.4 0.3 0.2
VII.
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Lactation days
Fig. 3. Proportion of naive susceptible quarters simulated by using the Sn ISw -model, where the parameters of the model are estimated by using; transmission parameter β = 0.065, α = 0.037, µ = 0.01, θsn = 0.451, θsw = 0.19, θI = 0.041, ψw = 3.6, ψI = 1, λ = 1.202. The initial values are Sn,0 = 0.7521, Sn,0 = 0.21 and I0 = 0.0378 obtained from the real data.
1 0.9 0.8
Proportion of Sw
0.7 0.6 0.5
C ONCLUSION
Novel mathematical description of the transmission dynamics of Corynebacterium spp. IMI is presented in this work. The model allows for understanding of the transmission of Corynebacterium spp. IMI, and to help to devise strategies to minimize the impact of the infection. The parameters of the obtained model were estimated by using a deterministic approach based on real data fitting procedures using nonlinear least squares method for nonlinear ordinary differential equations (ODEs). The numerical resolution of the ODEs of the model with the parameters fitted from the real data revealed the behavior of the transmission dynamics of the Corynebacterium spp. throughout the lactation period. The simulated curve of proportion infected that fitted the real proportion infected demonstrated that Corynebacterium spp. are contagious. Corynebacterium spp. IMI increases after calving until the peak in mid lactation. The IMI are however to some extent resolved by the cows towards drying off without any intervention.
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R EFERENCES
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[3] Fig. 4. Proportion of susceptible within quarters, simulated by using the Sn ISw -model, where the parameters of the model are estimated by using; transmission parameter β = 0.065, α = 0.037, µ = 0.01, θsn = 0.451, θsw = 0.19, θI = 0.041, ψw = 3.6, ψI = 1, λ = 1.202. The initial values are Sn,0 = 0.7521, Sn,0 = 0.21 and I0 = 0.0378 obtained from the real data.
[4] [5]
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VI.
D ISCUSSION
The Sn ISw model simulation fits the actual on farm situation throughout the lactation period of the cows. By modelling the prevalence of Corynebacterium spp. IMI, we correctly describe the actual changes in Corynebacterium spp. IMI throughout the lactation. The results indicate that Corynebacterium spp. are contagious. This is shown by the increase in Corynebacterium spp. IMI after calving until the peak in mid lactation and the corresponding basic reproduction number of 1.36. The IMIs are however to some extent resolved by the cows towards drying off without any intervention. Furthermore, the cure rate during the dry period is high. Since blanket dry cow therapy was used in the study herd, we cannot quantify the degree of self-cure.
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