An immuno-epidemiological model for Johne’s disease in cattle
- Maia Martcheva^{1},
- Suzanne Lenhart^{2}Email author,
- Shigetoshi Eda^{3},
- Don Klinkenberg^{4, 7},
- Eiichi Momotani^{5} and
- Judy Stabel^{6}
Received: 15 September 2014
Accepted: 2 February 2015
Published: 19 June 2015
Abstract
To better understand the mechanisms involved in the dynamics of Johne’s disease in dairy cattle, this paper illustrates a novel way to link a within-host model for Mycobacterium avium ssp. paratuberculosis with an epidemiological model. The underlying variable in the within-host model is the time since infection. Two compartments, infected macrophages and T cells, of the within-host model feed into the epidemiological model through the direct transmission rate, disease-induced mortality rate, the vertical transmission rate, and the shedding of MAP into the environment. The epidemiological reproduction number depends on the within-host bacteria load in a complex way, exhibiting multiple peaks. A possible mechanism to account for the switch in shedding patterns of the bacteria in this disease is included in the within-host model, and its effect can be seen in the epidemiological reproduction model.
Introduction
Johne’s disease (JD) in dairy cattle is a chronic infectious disease in the intestines caused by the bacillus, Mycobacterium avium ssp. paratuberculosis (MAP). MAP in a contaminated environment infects cattle through oral route. Contaminated colostrum and milk from infected cows are important sources of infection for calves. Actual infection occurs when MAP bacilli are phagocytosed by M-cells covering the dome of the Peyer’s patches [1] and transported to macrophages. In the early stages of the infection, some of the MAP will not be destroyed by macrophages and will grow in those cells until cellular immunity will be generated. To develop specific cellular immunity, macrophages differentiate into epithelioid cells and then intracellular growth of the MAP will be suppressed. Epithelioid cells form specific structures, granulomas, which act to restrict MAP growth inside and destroy them gradually. Some of the MAP in the granuloma will survive and enter a period of dormancy until reactivation. Reactivation of MAP will start slowly in the subclinical stage in which intermittent shedding of MAP will start. Granulomatous enteritis develops during the subclinical stage and accelerates in the clinical stage. Histological studies of infected areas reveal the presence of many infected macrophages, but very little extracellular bacteria have been observed [2].
This disease exhibits a variety of shedding patterns of MAP into feces, usually with a brief initial shedding period and then followed by a long latent period. In later stages, some infected cattle progress to weight loss, diarrhea, and reduced milk production. The mechanisms of the pathogen and the reaction of the immune system that cause this long latent period are not well understood. Also the connection of underlying immunology responses and the variable shedding patterns is difficult to explain. See the following papers in this issue related to these mechanisms and their relationship with data on shedding patterns, the growth of granulomas, and the length of the latent period [3,4].
To better understand the mechanisms of this disease and to later choose control strategies, we will illustrate through modeling how the immune system of infectious cows may have an effect at the epidemiological herd level. In particular, we are interested in understanding how the host immune responses influence the epidemiological reproduction number of JD in a farm or a geographical region. We will study the complex immuno-epidemiology of JD through explicitly linking of epidemiological processes to the immune system dynamics.
Linking models at the two scales, immunological and epidemiological, has been done recently [5,6]. Different approaches have been used for such models, and some work has used decoupling assumptions to deal with the two scales separately [7-9]. In this paper, we are following the nested approach, introduced by Gilchrist, Coombs and Sasaki [10,11], in which the within-host model is independent of the between-host model but feeds into the between-host model. Our immuno-epidemiological model consists of two components: a time-since-infection dependent immunological model (within-host) and an epidemiological model (between hosts) whose transmission rates and virulence depend on the within-host variables. We will illustrate the basic ideas of linking a within-host model with a between-host model. For this illustration, we use a relatively simple model for the epidemiological dynamics. The representation of epidemiological reproduction number shows the dependence on specific within-host populations. Since there may be a type of switch in the within-host system that accounts for the variability of the shedding patterns, we give a way to illustrate a possible switching mechanism and show its effect on the epidemiological reproduction number. We use the stimulation rate of the immune response from infected macrophages as this switch.
In the next section, we introduce our within-host model, and then we discuss its estimated parameter values and stability analysis. We include a mechanism to switch from low bacterial load to high bacterial load. The third section gives the between-host model linked to the within-host model. We show how the epidemiological reproduction number can be represented in terms of the equilibrium points of the within-host model.
Materials and methods
A within-host immunological model of JD
We give some brief background of immunology of MAP leading to the within-host model.
Essential immunology of MAP
MAP enters the body of a cow from environmental sources, including fecal material, at birth or early in life through milk. After entry, MAP travels to the intestines of the affected animal and infects the macrophages located in the Peyer’s patches [1]. In the early stages of the disease the bacterium prevents the macrophages from destroying it and the infected macrophages are kept dormant. The infection persists in the affected animal in the subclinical stage. Histological studies of infected areas reveal the presence of many infected macrophages but the amount of free bacteria has not been clearly documented. It appears that bacteria, that are released when an infected macrophage is destroyed, are immediately engulfed by new macrophages. To account for this observation, we include in the model uninfected macrophages and infected macrophages but no free bacteria. Immune responses to MAP are reviewed more thoroughly in [2] in this special issue. Early in the infection, a cellular immune response is activated through T cells and other cells. The cellular immune response is accompanied by proinflammatory cytokines, such as interferon-γ (IFN-γ) [12]. The cellular immune response is effective in controlling the infection, so that during the subclinical stage the shedding is often minimal or intermittent [13-16]. In the later stages of infection the cellular immune response wanes and a humoral response is activated in the form of B-cells and antibodies [17]. This response appears to be less effective in the controlling the infection and often increased shedding is observed at this stage of the infection. For this reason we include in the model the cellular immune response only.
The immunological model
Parameters and dependent variables and their interpretation
Variable or parameter | Meaning |
---|---|
M _{ u } (τ) | Number of uninfected macrophages at time τ |
M _{ i } (τ) | Number of infected macrophages at time τ |
C (τ) | Number of immune cells at time τ |
r | Total recruitment rate for uninfected cells |
α | Infection rate of uninfected macrophages by infected macrophages |
d _{1} | Natural death rate for uninfected macrophages |
δ | Death rate of infected macrophages |
d _{2} | Clearance rate for immune cells |
k | Stimulation rate of immune response from infected cells |
ε _{1} | Reciprocal of half saturation constant for killing of infected macrophages |
ε _{2} | Reciprocal of half saturation constant for immune response stimulation rate |
ε _{3} | Reciprocal of half saturation constant for killing of infected macrophages |
c | Reciprocal of half saturation constant for macrophages infection rate |
p | Killing rate of infected macrophages by immune response. |
Fitting the immune model to calf data and parameter estimation
Estimating parameters in within-host models can be achieved through fitting to time series data and estimates of lifespans. We use calf data and procedures reported in [21,22]. Neonatal Holstein dairy calves were obtained from status level 4 herds with no reportable incidence of JD in Minnesota at 1–2 days of age. Calves were housed in Biosafety Level-2 containment barns for the duration of the study and experimentally inoculated by feeding milk replacer containing 2.6 × 10^{12} live MAP obtained by scraping the ileal mucosa from a clinically infected cow as previously described in [21]. Calves were dosed on days 0, 7, and 14 of the study. All procedures performed on the calves were approved by the Institutional Animal Care and Use Committee (National Animal Disease Center (NADC), Ames, Iowa). The clinical isolate of MAP was obtained from the ileum of a clinical cow at necropsy that had shed high numbers of MAP in the feces.
Infection of calves was determined by measurement of shedding viable MAP in the feces and recovery of MAP from tissues as described in [21]. After 12 weeks of incubation at 39 °C, viable organisms were determined by counting the number of colonies on the agar slants. IFN-γ and interleukin-10 (IL-10) were measured in cell-free supernatants to assess immune response to infection. Briefly, periperhal blood mononuclear cells (PBMCs) were isolated from the buffy coat fractions of whole blood and cultured at 2.0 × 10^{6}/mL in complete medium at 39 °C in 5% CO _{2} in a humidified atmosphere. In vitro treatments consisted of no stimulation (medium only), concanavalin A (10 μg/mL), and a whole-cell sonicate of MAP (10 μg/mL) [22].
Estimated parameter values
Parameter | Calf 9 | Calf 10 | Calf 11 | Units |
---|---|---|---|---|
r | 114 | 600 | 1425 | cells/month |
α | 0.47 | 0.4 | 0.6 | Month^{−1} |
d _{2} | 0.5 | 0.4 | 0.3 | month^{−1} |
k | 8.0 | 3.0 | 13.8 | (cells × month)^{−1} |
p | 3.8 | 1.4 | 16.1 | (cells × month)^{−1} |
ε _{1} | 0.105 | 0.111 | 0.65 | cells^{−1} |
ε _{2} | 0.105 | 0.111 | 0.65 | cells^{−1} |
ε _{3} | 0.0 | 0.0 | 0.0 | cells^{−1} |
c | 0.0001659 | 0.0001 | 0.0001 | cells^{−1} |
C (0) | 26.3 | 15.3 | 0.1 | cells |
c _{1} | 0.005 | 0.01 | 0.01 | unitless |
c _{2} | 0.048 | 0.057 | 0.05 | unitless |
Multiple equilibria and significance in terms of shedding
The immuno-epidemiological model of Johne’s disease
We imbed the immunological model (1) from the previous section into an epidemiological model of JD. The epidemiological model and the immunological model of JD are linked through the time-post-infection variable as well as the dependence of some epidemiological parameters on the within-host pathogen load.
A brief description of JD epidemiology
JD is present in many countries with livestock industry and, in the United States, the causative agent of the disease, MAP, was found in about 70% of dairy farms [24,25]. After a long incubation time [26], dairy cattle infected with MAP start shedding the pathogen into feces, colostrum and milk [27-29]. MAP bacilli shed into feces can survive longer than a year in the environment [30] and the fecal-oral route was shown to be the major transmission pathway in dairy farms [31]. However, there is also evidence for transmission through contaminated milk, colostrum and placenta (vertical transmission) [29,31,32]. Young animals are more susceptible to MAP infection than adults [33], and therefore, good management practices to prevent MAP ingestion at young age (<1 year old) are suggested to be effective to control JD in dairy farms [34].
The epidemiological component of the JD model
Parameters and dependent variables and their interpretation
Variable or parameter | Meaning |
---|---|
S _{ C } (t) | Number of healthy calves at time t |
S _{ A } (t) | Number of healthy adults at time t |
i (τ, t) | Density of infected cattle with age-post-infection τ at time t |
B (t) | Number of bacteria in the environment at time t |
b _{ S } | Birth rate for susceptible cattle |
q (τ) | Proportion of births of infected calves from infected cows |
β _{ C } | Susceptibility of calves to infection due to milk |
μ _{ C } | Natural death rate of healthy calves |
μ _{ A } | Natural death rate of healthy adults |
μ | Natural death rate of infected cattle |
a | Rate of progression of calves to adulthood |
β _{ A } | Susceptibility of adults to infection |
γ _{ C } | Proportion of newborn calves getting infected from the environment |
γ _{ A } | Rate of infection of healthy adults from the environment |
b _{ I } (τ) | Birth rate of infected adults τ units post infection |
κ (τ) | Infectivity of infectious cattle τ units post infection |
Where K _{1} and K _{2} are given constants. This function is chosen because it has a threshold effect: it is low for low values of B and at some threshold value of B increases fast to one.
Linking the immunological and the epidemiological models
where κ _{ T } is the half-saturation constant.
The linking above describes how the epidemiological parameters change with the within-host bacterial load. The resulting immuno-epidemiological model assumes that the within-host progression of the disease is averaged among all individuals and gives the population-level spread which results from the averaged immune dynamics.
Reproduction number of the immuno-epidemiological model
Whether JD persists on epidemiological level depends on the epidemiological reproduction number R_{0} which gives the number of secondary infections one infected individual will produce in an entirely susceptible cattle population. If R_{0} > 1 then JD will persist in the population; if however, R_{0} < 1, JD may be eliminated from the population, even in the case when the infection persists in some individual infected animals.
is the probability of survival in the infectious class. Each term of the reproduction number corresponds to a different mode of transmission of JD. The first two terms give the number of secondary infections of calves and adults who get infected through the direct transmission route. The third term gives the number of secondary infections, generated through environmental transmission. The integral in this term gives the amount of bacteria shedded by one infectious individual, 1/d gives the survival time of bacteria in the environment and f ^{′} _{ E } (0) gives the rate of infection when the bacteria in the environment are rare. The last term gives the number of xsecondary infections generated through vertical transmission.
Estimation of epidemiological parameters
To illustrate the impact of within-host prevalence of MAP on the epidemiological reproduction number, we need estimates for some of the epidemiological parameters involved in the model. Reference [30] gives a detailed account of the survival properties of MAP in various environments. This allows us to estimate d which we take in the range 1/15–1/3 months^{−1}. We also derive the shedding rate from the fits of the immune model to CFU data. The constant that links the MAP amount within a host M _{ i } to the CFU is given by c _{1}, and we set c _{1}=0.01.
Parameters and dependent variables and their interpretation
Parameter | Meaning | Value | Range |
---|---|---|---|
b _{ S } | Birth rate for susceptible cattle | 100 | 0.1–1000 |
b _{ S } | Per capita birth rate for susceptible cattle | 100 | 0.01–100 |
b | Constant in b _{ I } (τ) | 0.1 | 0.0001–1 |
q | Proportion of infected calves born to infected cows | 0.25 | 0–0.25 |
Q | Half saturation constant | 100 | 0–100000 |
β _{ C } | Susceptibility of calves to infection due to milk | 0.0183333 | 0.001–1 |
μ _{ C } | Natural death rate of healthy calves | 1/(10 × 12) | 1/(25 × 12)–1/(5 × 12) |
μ _{ A } | Natural death rate of healthy adults | 1/(5 × 12) | 1/(25 × 12)–1/(2 × 12) |
μ | Natural death rate of infected cattle | 1/(3 × 12) | 1/(5 × 12)–1/(2 × 12) |
a | Rate of progression of calves to adulthood | 1/12 | 1/(2 × 12)–1/(0.5 × 12) |
β _{ A } | Susceptibility of adults to infection | 0.00733333 | 0.001–1 |
γ _{ C } | Proportion of newborns infected from environment | 0.0001 | 0–1 |
γ _{ A } | Rate of infection of healthy adults from environment | 0.0000183333 | 0.0–3 |
κ _{ T } | Half-saturation constant for κ (τ) | 1000 | 10–10000 |
m | Proportionality constant for ν (τ) | 0.3 | 0.0001–1 |
v | Proportionality constant for ν (τ) | 0.01 | 0.0001–1 |
η _{ E } | Shedding rate of infected individuals | 0.01 | 0.001–0.1 |
d | Clearance rate of MAP from the environment | 1/5 | 1/15–1/3 |
K _{1} | Constant in f _{ E } | 1 | 1–10000 |
K _{2} | Constant in f _{ E } | 1 | 1–10000 |
Results
Implications of the within-host dynamics to the epidemiology of JD
Discussion
Mathematical modeling is an important tool in learning about infectious diseases. We believe that linking immunological and epidemiological models will give further important contributions to understanding of multi-scale biological processes and to leading the way in disease control.
Overview of insights of immunological model
The immunological model in the system describes the within-host dynamics of MAP. We fitted the model to CFU data of three calves. The fitting resulted in the estimation of a number of key quantities in the within-host dynamics of MAP. In particular we obtain values of the infection rate of uninfected macrophages, the stimulation rate of the immune response and others. These results are given in Table 2. The immunological model also reveals that saturating incidence in infected macrophages leads to multiple steady states of the bacterial load when ℜ_{0} > 1. In particular for some parameter regimes close to the estimated values, there are three nonzero equilibria, two of which are typically attracting. That allows for the MAP bacterial load to stabilize at two non-zero values, mimicking stable nonshedding and stable shedding. Switching between shedding and non-shedding can occur as a result of a short-term disturbance that a cow may experience. This suggests that prolonged no shedding, followed by prolonged shedding are two stable regimes where the switching occurs as a result of a stressor. Simulations show, however, that for some parameter regimes close to the estimated values, the lower equilibrium can be unstable and the switching between no shedding and shedding may occur spontaneously without any external disturbance. In this case the duration of the non phase depended on the initial infection and other factors.
Overview of insights of immuno- epidemiological model
The epidemiological reproduction number R_{0}, computed in terms of the within-host bacterial load, allows us to potentially infer when JD will persist or die out in a herd from the within-host bacterial load or shedding. At the very least, we can observe how the within-host bacterial load impacts the reproduction number, and hence, the prevalence of JD in a herd. JD is spread through three modes of transmission: environmental, vertical and direct. In general for directly transmitted diseases it is known that the epidemiological R_{0} exhibits a humped shaped dependence with respect to the pathogen load. Surprisingly, we find that the stronger the virulence caused by the immune response, the larger the pathogen load needed for the maximum to occur. Furthermore, we find that R_{0} also exhibits such a humped shape if the transmission is only vertical. In contrast with direct transmission, the peak in this case is more pronounced and occurs for lower pathogen load. The decline of R_{0} in both cases is attributed to the virulence. As JD is spread via all three modes of transmission, JD’s epidemiological R_{0} exhibits complex dependence on the within-host bacterial load with two peaks – the first one caused by vertical transmission and the second one caused by direct transmission. The non-monotone dependence of R_{0} on the within-host bacterial load does not allow us to infer in general that if the shedding of all cows in a herd is non-detectable, R_{0} and prevalence of JD will be low. In fact, no shedding (low pathogen load) in all cows may very well lead to high prevalence of JD in the herd. Because of that, control measures that reduce shedding or extend the time of the cows in a herd being non-shedders may not reduce the prevalence of JD.
Significance of results
We introduce a new way to model long periods of shedding and no shedding as well as the switching between the two. This leads to understanding that these are due to stable stationary patterns of the within-host bacterial load. Furthermore, most likely the switching occurs due to temporary disturbance of the infected cow. Furthermore, the new model extends the modeling techniques to immuno-epidemiological models with multiple within-host equilibria and their impact on the epidemiology of disease. Finally, we uncover that in the case of multiple modes of transmission, as with JDs, the epidemiological reproduction number can depend on the pathogen load in a complex fashion, experiencing multiple peaks. Ultimately we conclude that no shedding (low pathogen load) does not necessarily imply low epidemiological reproduction number or low prevalence in the herd.
Declarations
Acknowledgments
The authors acknowledge the support of the Within-host modeling of MAP infections Working Group at the National Institute for Mathematical and Biological Synthesis, sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award DBI-1300426, with additional support from The University of Tennessee, Knoxville. Lenhart is partially supported by the University of Tennessee Center for Business and Economic Research.
Authors’ Affiliations
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