A mathematical model to study resistance and tolerance to infection at the animal and population levels: application to E. coli mastitis

Abstract

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.

Keywords: infection; mathematics; resistance; tolerance.

Figure 1

Typical changes in the within-host concentrations of pathogens, response cells, and hosts performance according to the scenario of response to infection.

Figure 2

Size of the endemics according to the scenario of response to infection.

Figure 3

Concentrations of pathogens and response cells for different levels of resistance to infection and to disease, and host performances for different levels of direct and indirect tolerances. Values of the parameters are in Table 1.

Figure 4

Area under the curves for the host performances and number of infected in the population for the first 100 time-units and expressed as the percentages of their respective maxima.