^{*}

Edited by: Andrea B. Doeschl-Wilson, The University of Edinburgh, UK

Reviewed by: Vinca Russell, University of Edinburgh, UK; Beatriz Villanueva, INIA, Spain; Liesbeth Van Der Waaij, Wageningen University, Netherlands

*Correspondence: Johann C. Detilleux, Quantitative Genetics Group, Faculty of Veterinary Medicine, University of Liège, Liège, Belgium. e-mail:

This article was submitted to Frontiers in Livestock Genomics, a specialty of Frontiers in Genetics.

This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and subject to any copyright notices concerning any third-party graphics etc.

A mathematical model is proposed that describes the colonization of host tissues by a contagious pathogen and the early nonspecific immune response, the impact of the infection on the performances of the host, and the spread of the infection in the population. The model obeys specific biological characteristics: Susceptible hosts are infected after contact with an infected one. The number of pathogenic units that invade a susceptible host is dependent on the infectious dose provided by the infected host and on the ability of the susceptible host to resist the invasion. After entry in host, pathogenic changes over time are expressed as the difference between the intrinsic logistic growth rate and the Holling type II kill rate provided by the immune response cells. Hosts have different ability to restrict reproduction of the pathogen units. The number of response cells actively recruited to the site of infection depends on the number of the pathogenic units. Response cells are removed after having killed a fixed number of pathogenic units. The effects of the number of pathogenic units on the performances of the host depend upon its levels of tolerance to the deleterious effects of both pathogenic and response cells. Pre-infection costs are associated to tolerance and resistance levels. Estimates of most biological parameters of the model are based on published experimental studies while resistance/tolerance parameters are varied across their allowable ranges. The model reproduces qualitatively realistic outcomes in response to infection: healthy response, recurrent infection, persistent infectious and non-infectious inflammation, and severe immunodeficiency. Evolution across time at the animal and population levels is presented. Effects on animal performances are discussed with respect to changes in resistance/tolerance parameters and selection strategies are suggested.

Many conservation and selection programs (e.g., FAO,

Resistance traits are broadly defined as host traits that reduce the extent of pathogen infection. They include traits that reduce pathogen transmission at contact and pathogen growth rate once infection has occurred (Kover and Schaal,

Tolerance, on the other hand, is defined as the host's ability to reduce the effect of infection on its fitness. Although fitness measurements include different life-history traits, only performance (e.g., growth, milk or wool) is considered as it is very important in farm animals. Tolerance may be targeted to reduce damage directly inflicted by the pathogen (direct tolerance) or caused by the immune response (indirect tolerance). Little is known about underlying mechanisms of tolerance (see one example in Medzhitov,

Costs are associated with both resistance and tolerance because energy is required to maintain immune-competence and to mount an efficient immune response, as shown in various empirical studies (Boots and Bowers,

Unfortunately, levels and costs associated to resistance and tolerance are usually difficult to obtain in field studies under pathogen attack. Given these technical difficulties, their relative importance is here investigated using mathematical simulation studies. Hence, the main objective of the paper is to investigate and the effects of resistance and tolerance on the spread of an infectious disease and on the performances of the animals within a closed population, for a range of realistic scenarios. At the animal level, a comprehensive model is constructed that incorporates important biological characteristics associated with the early immune response to infection.

The model has two main components, each with two parts. The first system of equations describes the changes in cell concentrations associated with the infection. The second expresses the effects of the infection on host performances. Both are made stochastic rather than deterministic to capture the variability inherent in biological processes.

The system of equations elaborates on a previous discrete susceptible–infected–susceptible model (Detilleux,

For one individual, the within-host model follows the dynamics of response cells and pathogen populations:
_{t} is the concentration of pathogens and _{t} is the concentration of response cells at time _{t + Δt} new pathogens while pathogens present within the host multiply (_{t + Δt}) and are killed by response cells (_{t + Δt}). In the absence of infection, _{t + Δt} response cells reach the tissues while an extra-concentration (_{t + Δt}) is recruited and removed (_{t + Δt}) in case of infection. All concentrations are homogeneous Poisson processes: the number of events in time interval (

The symbol _{t + Δt} represents the concentration of pathogens effectively transmitted and inoculated to one host after contact with a number I of infective hosts, each infected with ^{i}_{t} pathogens. It is governed by the equation:
^{i} = fraction of ^{i}_{t} the infected host excrete during an effective contact. Stated otherwise, β ^{i}_{t} is the infective dose released by an infected host and ν represents the host anti-infection resistance.

The concentration of pathogens resulting from reproduction (_{t + Δt}) is controlled by their multiplication rate, here assumed to be logistic:

In the equation, the per-capita growth rate (γ) is a function of the ability of pathogens to multiply until they reached their maximum concentration (_{B}). This behavior has indeed been observed in well-mixed

Concurrently to infection, response cells are activated to kill _{t + Δt} pathogens:

In the second part of equation [1], _{t + Δt} is the normal concentration of response cells in the tissue environment:

When pathogens are present, an extra-concentration of cells is recruited:
_{m} is the half-saturation constant. Then, if _{t} is low, _{t + Δt} ~ Poisson [μ C_{t}/K_{m}], and reaches _{t + Δt} ~ Poisson [μ C_{t}] when _{t} is high.

The symbol _{t + Δt} represents the extra-removal of response cells after infection:

The rate is called the numerical response rate (change in predator concentration as a function of change in prey concentration) and corresponds to the above Holling type II functional rate.

A response cell kills on average θ pathogenic cells before removal.

Only the effects of pathogen and cells concentrations on hosts performances are considered. All other effects, such as resource intake, management or age are assumed fixed. Then,
_{t} is the performance of the host in the presence of _{t} pathogens and _{t} response cells. The parameters _{B} and _{C} are the maximum performance lost per pathogen (virulence) and response cell, respectively. The parameters λ_{b} and λ_{c} are scaling parameters representing the relative ability of the host to tolerate damages caused by pathogens and immune cells. If λ_{b} = λ_{c} = 1, the host is completely tolerant and produces at the initial level (_{b} = λ_{c} = 0, the host is not tolerant at all. Although unrealistic (Can an animal be totally tolerant or un-tolerant?), the scaling parameters set the limits for _{t} with a maximum at _{1} and a minimum at _{1}−B_{t} L_{b}−C_{t} _{C}.

Resistance and tolerance are associated with a redistribution of resources away from performance:
^{Max} is the maximal level of performance reached when levels of resistance and tolerance are null (ρ = λ_{b} = λ_{c} = 0). The parameter _{ρ} is the relative costs of resistance while _{b} and _{c} are the relative costs of tolerance to pathogens (direct) and response cells (indirect).

Values for parameters describing a healthy and early inflammatory response to infection are from studies on ^{7}cells/μL, and 10^{6} bacteria/μL, respectively. The value 10^{6} bacteria/μL is the highest concentration observed in neutropenic cows (Rainard and Riollet, ^{7} cells/μL is the highest somatic cells concentration observed in a field survey of mastitis in Belgium (Detilleux et al., _{1}) are normally distributed with mean of 100 cells/μL and standard deviation of five cells/μL (Djabri et al., _{t} is small, and up to five bacteria are killed when _{t} is high.

_{B} |
Maximum concentration of pathogens | 10^{6}/μL |

_{C} |
Maximum concentration of response cells | 10^{7}/μL |

^{Max} |
Maximum performance | 100 units |

γ | Pathogen logistic growth rate | 1 pathogen/μL/h |

τ | Time for a response cell to capture and kill pathogens | 1 h/cell |

θ | Pathogen concentration killed per response cell | 10 pathogens/cell |

Contact rate between hosts | 0.1/h | |

_{M} |
Pathogen concentration such that response cells reach the infection site in 1 time unit | |

Healthy response (scenario A) | 10 cells/μL | |

Recurrent infection (scenario B) | 10000 cells/μL | |

α | Pathogen clearance rate | |

Healthy response (scenario A) | 0.005 pathogen/cell/h | |

Persistent infectious response (scenario C) | 0 pathogen/cell/h | |

ω | Recruitment/elimination rate of response cells during health | |

Healthy response (scenario A) | 0.5 cells/h | |

Persistent non-infectious response (scenario D) | 0.01 cells/h | |

μ | Extra-recruitment rate of response cells during infection | |

Healthy response (scenario A) | 2 cells/μL/h | |

Immuno-depression (scenario E) | 0 cells/μL/h | |

β | Infectiousness | U[0; 0.01] |

_{C} |
Loss associated with each response cell | U[0; 25/_{C}] |

_{B} |
Loss associated with each pathogen | U[0; 25/_{B}] |

_{ρ}, _{b}, _{c} |
Resistance, direct and indirect tolerance costs | U[0; 0.1] |

ν | Resistance to infection | |

Low | U[0; 0.001] | |

Average | U[0; 0.01] | |

High | U[0.009; 0.01] | |

ρ | Resistance to disease | |

λ_{b}, λ_{c} |
Direct and indirect tolerances | |

Low | U[0; 0.1] | |

Average | U[0; 1] | |

High | U[0.9; 1] |

Outcome of the inflammatory response is not always health. To determine whether the model could reflect such reality, scenarios for the inflammatory response, other than the healthy response (scenario A), were tested by modifying the values of the parameters (Kumar et al.,

Without information in the literature, values for the rates in equations for performance were drawn from uniform distributions. A convenient value of 100 units was given to ^{Max}. Individual levels in resistance and tolerance were drawn from distributions with different extreme values to have low (U[0, 0.1]), average (U[0, 1]), or high (U[0.9, 1]) levels. The maximum part of _{1} available to resistance and tolerance was set at ^{Max}/2. Individual tolerance and resistance costs were drawn from U[0, 0.1]. Highest direct (_{b}) and indirect (_{C}) loss associated with each pathogen were set at 25 × 10^{−6} units of performance lost per pathogen present, and 25 × 10^{−7} units of performance lost per response cell. The values for _{B} and _{C} were chosen to insure that _{t} remains positive when costs and cell and pathogen concentrations are highest.

All computations were done on SAS 9.1. Simulation steps were executed until _{t}) and concentrations of host cells (_{t}) and pathogens (_{t}) were expressed as the percentages of their maxima (^{Max}, _{C}, and _{B}, respectively), averaged over all animals and all replications, and plotted across time. Similarly, the number of infected hosts (_{t}) was expressed as the percentage of the total number of hosts in the population (50) and averaged over all replications. To sum up, area under the curves of _{t} (AUC_{P}) and _{t} (AUC_{I}) were computed for

This section starts with results about the ability of the model to simulate different scenarios of response to infection, at the animal and population levels. It follows by the effects of different levels of resistance and tolerance on a healthy response.

Typical within-host curves are shown in Figure

Figure

In Figure _{t}) during an episode of infection are shown for different levels of resistance to infection, concentrations of response cells (_{t}) are shown for different levels of resistance to disease, and host performances (_{t}) are shown for different levels of direct and indirect tolerance, all for hosts with a healthy response to infection. Peak of pathogen concentrations are high when both levels of resistance are low (Figures _{t} going from 98% at _{Max}) if the host is not tolerant (line “Low-Low”) to 10.9% if it is highly tolerant (line “High-High”). It is 7.2% and 8.3% if the host is tolerant to direct (line “High-Low”) or indirect (line “Low-High”) damages, respectively.

The area under the curves of performances (AUC_{P}) and number of infected hosts (AUC_{I}) are shown in Figure _{P} is the highest (most favourable) when hosts mount a healthy response to infection, are highly resistant to disease and infection, and not tolerant to both direct and indirect damages associated with the infection. It is the lowest when hosts are persistently infected, not resistant to disease and infection, and highly tolerant to both direct and indirect damages associated with the infection. The AUC_{I} is the highest (most favourable) when hosts are immunodepressed or persistently infected with low levels of resistance to infection and disease. It is the lowest when hosts mount a healthy or persistent response to infection and are highly resistant to infection and disease. Indirect and direct levels of tolerance have no effect on AUC_{I}.

A mathematical model is proposed to quantify the effects of resistance and tolerance on the spread of an infectious disease (here,

If they are necessary, models should also adequately reflect reality. Although simple, the model proposed here allows simulating scenarios that have all been observed in animals. For example, changes in pathogens and cell concentrations depicted in Figure

Within the range of selected values (Table

Values for costs and effects of resistance/tolerance on host performances were chosen arbitrarily because no information was found in the literature. An exception is the experiment of Råberg et al. (^{−1} × 10^{6} parasites. But even though they are based upon arbitrary values, effects shown in Figure

Another drawback of the model is that hosts in the population all present one particular scenario of response to infection (Figure

No link was considered between scenario of response to infection and costs of resistance/tolerance, considering that costs were constitutive and allocated in a pathogen-free environment (Rohr et al.,

In conclusion, the model is useful in shedding some light on the complex interactions between resistance/tolerance and performance but needs realistic values to better grasp the processes. To improve it, we are planning a small explorative study, funded by the European research group EADGENE_S, to measure animal levels of resistance/tolerance to bovine mastitis in herds located in Wallonia. Resistance will be measured by the number of bacteria colony forming units in milk of cows located in herds in which cows' opportunity to get infected is measureable (Detilleux et al.,

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

This study was supported by the European Research Group EADGENE_S.