Conceived and designed the experiments: DS BC CG JW. Performed the experiments: RJT VS KL CG DS. Analyzed the data: KL RJT DS CJR JW. Contributed reagents/materials/analysis tools: VS JW RJT CJR DS. Wrote the paper: DS CJR RJT JW.
The authors have declared that no competing interests exist.
Foot and mouth disease virus causes a livestock disease of significant global socioeconomic importance. Advances in its control and eradication depend critically on improvements in vaccine efficacy, which can be best achieved by better understanding the complex withinhost immunodynamic response to inoculation. We present a detailed and empirically parametrised dynamical mathematical model of the hypothesised immune response in cattle, and explore its behaviour with reference to a variety of experimental observations relating to foot and mouth immunology. The model system is able to qualitatively account for the observed responses during invivo experiments, and we use it to gain insight into the incompletely understood effect of single and repeat inoculations of differing dosage using vaccine formulations of different structural stability.
Foot and mouth disease virus (FMDV) causes a highly infectious disease of clovenhoofed animals that has significant global socioeconomic impact. Foot and mouth disease (FMD) severely affects the welfare and productivity of highvalue farm animals that are important to food security, including cattle, sheep and pigs (
Vaccination is one of the principle methods available for disease control and eradication, and mathematical modelling has been identified as playing a vital role in helping formulate effective strategies
FMD occurs as seven main serotypes (O, A, C, SAT1, SAT2, SAT3, Asia1) with numerous antigen subtypes within each strain. Vaccines tend to be most effective against the specific strain they are designed to elicit protection. However, there is an urgent need to develop better FMD vaccines which protect against a wider range of strains and, more importantly, confer longerlasting host protection than existing formulations. Commercially available FMD vaccines are based on inactivated virus grown in largescale cell culture. In many commercial livestock herds repeat vaccination is necessary to sustain host protection and, although costly, this approach is used in many parts of the world where FMD is endemic or sporadic. Consequently, improved vaccines would contribute significantly to reducing the economic burden imposed by FMD and improving food security.
Generating more effective FMD vaccines depends critically on developing a better understanding of the basic host immunological responses both to infection by wildtype virus strains and to vaccination with antigenic formulations. Much detailed experimental work on immune mechanisms has been undertaken
However, key issues relating to vaccine immunogenicity remain unresolved. Juleff
Specifically, we develop and parametrise a detailed dynamical model of the proposed withinhost adaptive immunological response mechanism to inoculation with vaccine formulations of differing structural stability. The model is able to qualitatively account for empirically observed dynamics of the various constituent cell types in the coordinated immune response to the presence of antigen, as well as the generation of immune memory, thereby giving confidence that the proposed mechanism is an appropriate one. Moreover, it shows how repeat host vaccination and compensation of structural stability for dose can be used to maintain elevated levels of host protection.
The mathematical model complements existing experimental approaches to FMD immunology, and is intended to be used as a framework within which to formalise thinking about hypothesised immune mechanisms, and in the development of future experiments. Given the current level of knowledge with regard to bovine immunology, and the difficulties of deriving quantitative data on key factors from experiments, we believe that at present a coarsescale model is the most appropriate for investigating this system.The model as it is presented and parameterised gives confidence that currently proposed immune mechanisms are sound, and furthermore it can be used as a point of departure to explore possible outcomes before additional experimental work is undertaken. In this way we aim to advance understanding of which potential future improvements in vaccine technology will be most efficacious.
We describe a hypothesised immune response to the presence of vaccine antigen (see
For variable definitions see
Variable  Interpretation 

vaccine concentration 

short term antibody secreting cell concentration 

short term Tcell dependent antibody secreting cell concentration 

long term Tcell dependent antibody secreting (longlived plasma) cell concentration 

memory B cell concentration 

effector T cell concentration 

memory T cell concentration 

antigenantibody complex concentration 

nonactivated dendritic cell concentration 

activated DC level via macropinocytosis 

activated DC level via the FC activation process 

short term memory antibody IgM concentration 

long term memory antibody IgG concentration 
Parameter  Interpretation  Value 
Source/justification 

Dendritic cell migration rate  3.0 wk 
95% repopulation within 1 week after local removal 

IgM production by shortterm ASC rate  2.0 wk 
observed initial growth rate (see 

IgG production rate by shortterm Tcell dependent ASC  ” 


shortterm ASC production rate  6.9 wk 


short term Tcell dependent ASC production rate  ” 


longterm Tcell dependent ASC production rate  ” 


memory Tcell production by FC activated DCs rate  0.17 wk 


memory Tcell production by macropinocytosis activated DCs rate  ” 


effector Tcell production by FC activated DCs rate  ” 


effector Tcell production by macropinocytosis activated DCs rate  ” 


memory Bcell production rate  0.17 wk 


IgG production rate by longterm Tcell dependent ASC  1.36 wk 


memory to effector Tcell conversion rate  1 wk 


vaccineIgM complex formation rate  2 wk 
Observed in less than 1 hour 

vaccineIgG complex formation rate  ” 


Vaccine uptake rate by micropinocytosis  1.4 wk 
5% of final take up achieved within 6 hours (unpublished data) 

Complex uptake rate by FC activated pathway  19 wk 
50% of final take up achieved within 6 hours (unpublished data) 
ltirow2* 
unstable vaccine decay rate  39 wk 
invitro halflife of 3 hrs 
stable vaccine decay rate  19 wk 
invitro halflife of 6 hrs 


antigenantibody complex decay rate  17 wk 
loss: 50% in 5 hrs, 90% in 30 hrs 

macropinacytosisactivated DC decay rate  13 wk 
lifespan of 2–3 days (unpublished data) 

FCactivated DC decay rate  ” 


effector Tcell decay rate  4.7 wk 
cleared within 1 week (unpublished data) 

memory Tcell decay rate  0.51 wk 
7% loss per day ( 

memory Bcell decay  0.43 wk 
6% loss per day ( 

shortterm ASC decay rate  0.69 wk 
half life of 1 week (unpublished data) 

shortterm Tcell dependent ASC decay rate  ” 


longterm Tcell dependent ASC decay rate  0.01 wk 
halflife of order one year 

IgG decay rate  1.36 wk 


IgM decay rate  0.17 wk 


halfsaturation of vaccine uptake by unactivated DCs  20 c  Saturation at 30–40 times standard dose (unpublished data) 

halfsaturation of complex uptake by unactivated DCs  0.1 c  

baseline nonactivated dendritic cell concentration  1 c  

timedelay in Bcell response to vaccine  0.14 wks  observed within 1 day (unpublished data) 

timedelay in Tcell response to activated DCs  0.57 wks  observed within 4 days (unpublished data) 
wk: week; c
The system of delaydifferential equations (1)–(13), describing the dynamics of the variables listed in
The results of a single dose
Specifically the removal of vaccine antigen,
We repeated the calculation but varied the initial dose of (stable) vaccine antigen: results are given in
We next considered the effect of vaccine stability on the immune response by varying the vaccine decay rate parameter
The benefits of stability are not fully realised until a booster dose is applied (see
These model results suggested that antigen dose can compensate for stability within the estimated parameter ranges: specifically, increasing the vaccine dose elicits an immune response comparable with that of a more stable vaccine.
In the field, initial FMD vaccination often elicits transient host protection and repeat vaccination is necessary to maintain protective immunity: this is done at least twice a year where practical (e.g.
Plots give the median value (central bar), 25th–75th percentile (box) and extreme values (whiskers) unless considered outliers, in which case they are plotted separately (cross) for four (bottom: Tcell independent) or five (top: Tcell dependent) replicates (individual cattle). Data from
The system produced the response in IgM and IgG that would be expected empirically, namely, only a small difference in IgM (
Although the data is best considered qualitatively, using the model we investigated the effect of repeat vaccination with such modified antigens: an inhibited Tcell independent response was represented by reducing
All figures present variables on a log scale as percentages of their peak value.
Although the model only aims to qualitatively replicate observed dynamics, we have plotted experimental results and simulation results together in
Here the mean and range of each of the datasets from
We have shown that the model for the withinhost immune responses to immunisation with FMDV vaccine antigens of differing stability can lead to realistic qualitative behaviours using plausible sets of estimated model parameters. This indicates that the mathematical model is capable of successfully reflecting the inherent dynamics of the immunological response. However, our model was complex, involving a large number of state variables (13) and model parameters (34). The notional values for the model parameters were estimates but, inevitably, there was some range of uncertainty within which their precise values actually lie. When structural complexity of a mathematical model is allied with model parameter uncertainty it is useful and desirable to be able to systematically assess the likely variation in the results of the model in order to establish whether the sorts of behaviours seen in the output are robust over a broad range of parameter choices. For models to be useful the successful application of such an uncertainty assessment gives confidence that the observed output is not simply a fortuitous combination of model parameters.
Here the median (solid line) is plotted together with the range of possible results, in 5 percentile steps (shaded) from 410 replicates (axes upper bound set at maximum of 95^{th} percentile range) on a log scale.
A key feature of the dynamics is the booster effect in IgG (
Plots give the median (red bar), 25–75th percentile (box plot), nonoutlier range (whiskers) and outliers (red cross) for each multiple set. In addition the ratio from the estimated parameter set is marked (magenta) together with the value
We have developed a mathematical model of the bovine immune response to FMDV vaccination, incorporating detailed representations of the Tcell independent and Tcell dependent antibody responses. Such models have helped to further our understanding of the mechanisms involved in adaptive immune responses to host challenge with live pathogens and vaccine antigens
The model is able to replicate
We investigated whether it was possible to mitigate deficiencies in vaccine stability by inoculating with a higher dose of vaccine. This is a question of some significant practical importance. The model results suggest that it is possible to compensate for poor stability with increased dose, but empirically it is found that whilst this is true there is a saturation effect
Results suggest that vaccine stability may not have a pronounced impact on the timing of the T cell response, but will affect its magnitude – and hence duration. Future work will look in more detail to how these predictions compare with experimental evidence that vaccine produced from different virus serotypes can differentially stimulate T cell responses. In addition, we were able to use the model to account for experimental results involving conventional and modified vaccine formulations. Results presented here motivate experimental study of additional serotypes with assumed differential stability and Tcell response inducing properties.
The system achieves good qualitative agreement with empirical observations of the system response to booster vaccine doses, and suggests that stable vaccine benefits more from multiple doses. We undertook a LHS sensitivity analysis which demonstrated the robustness of the immunological model under significant parameter uncertainty. However, it will be necessary to provide better estimates of some of the parameters in the model: in particular, we are considering how the deficit in data regarding the
The model provides a consistent representation of immune responses to vaccination and will be used to inform future experimental investigations aimed at enhancing commercial vaccine efficacy. Our model can also be utilised to interrogate the immune response to Tdependent (TD) and Tindependent (TI) antigens. Antigens that require Tcell help to orchestrate a high affinity classswitched serological response are termed TD antigens. TI antigens are able to initiate a serological response in the absence of Tcell help but show little germinal centre formation or Bcell memory. There are two types, type I (polyclonal Bcell stimulant) or type II (nonpolyclonal stimulant). It has been demonstrated that TI antigens can be altered, via conjugation of a protein carrier, to produce a TD immunological response resulting in induction of a more sustained immunological memory response
Mathematical modelling of withinhost immunological responses to infection with replicating viruses has been valuable in clarifying and describing the most significant immunological interactions and mechanisms
We aim to describe the classical adaptive immune response in as simple a way as possible while including all essential components. The system consists of two principle pathways, referred to as the “shortterm” and “longterm” pathways, which are described in detail below. The process is driven by the presence of antigen (in this case vaccine capsid), which initiates the production of Tcells and specialist Bcells that are capable of producing the appropriate antibodies. These antibodies combine with free capsid (and, more importantly, with the relevant wildtype virus if it is present in future) to form a complex that is then removed from the system.
Vaccine capsids are designed not to interfere or inhibit any of the immune response processes, unlike most pathogens and many other external agents that the immune system is required to deal with. FMDV vaccines appear successful in this regard, and we therefore consider the system as closed.
Following inoculation vaccine capsid (
Vaccine capsid is also delivered to the lymph nodes by specialist antigen presenting dendritic cells (DCs), which stimulate a different set of responses to that of the shortterm pathway. Unactivated DCs (
The full integrated system, comprising of the shortterm and longterm pathways, is illustrated in
The structural stability of the complex protein envelope surrounding the RNA of the virus (the capsid envelope), upon which most FMD vaccine formulations are based, appears to be influential in determining the degree of protection afforded by the vaccine. Comparison of the thermal stability of FMD A and SAT2 strains incubated at 49°C indicated an approximately 40fold difference in the decay rate of live virus
The model equations are derived from the pathways shown in
The temporal evolution of the vaccine antigen is determined by its decay rate, uptake by IgM (
Although only a tiny proportion of DCs will be activated through vaccination, it is expected that local saturation will occur temporarily. We therefore assume that there exists a preferred inactivated DC concentration to which levels slowly return following uptake, and that the activation rate of DCs obeys MichaelisMenten kinetics. Unactivated DCs (
Under the assumption that the level of vaccine in the lymph nodes is proportional to the total level, the dynamics of the shortterm antibody secreting Bcells and subsequent shortterm IgM concentration are given by
The generation of the vaccine–antibody complex through the longterm pathway is a result of vaccine binding with IgM and IgG, with loss as a result of uptake by unactivated DCs and natural decay. We do not distinguish between vaccine–IgM and vaccine–IgG complex, since once they are formed the properties of these complexes are very similar. DCs are activated by vaccine uptake via macropinocytosis or via FC activation, but with saturation, and lost through decay, giving:
The presentation of vaccine or complex derived antigenic peptides by activated DCs to naive Tcells generates two classes of Tcells, with the differentiation of naive Tcells assumed to take a time
The rate of decay of effector Tcells, memory Tcells and memory Bcells is given by
Effector Tcells give rise to Bcells, a proportion of which settle in the bone marrow where their life span is significantly enhanced (longlived plasma cells). IgG is produced by both types of Tcell dependent Bcells and can form complex with vaccine, giving:
Vaccine antigen (
The immunology of FMDV has been intensively studied in a variety of
Parameter values for the immune model are taken from the literature or estimated from recent experimental results at the Institute for Animal Health – Pirbright Laboratory; where possible, we have sought to do so from experiments relating to FMDV infection in cattle or inoculation with antigens. The nominal parameters for our model are summarised in
A systematic procedure for investigating the range of behaviours in complex dynamic models is Latin Hypercube Sampling (LHS) – see
The application of LHS to our immunological model was straightforward and we assumed that the model parameters have uniformly distributed probability distribution functions (pdfs) spanning a range from a given minimum to a given maximum; at this stage we have no other information that permits more complex pdfs to be proposed so we used the simplest representation. Following the standard LHS procedure we generated an appropriate LHS table, selected the resulting parameter combinations and performed ten times the number of simulations required to fully sample the parameter space
The authors would like to thank all those participants of the 9^{th} UK Mathematics in Medicine Study Group (Imperial College London; September 2009) who contributed at the inception of this work, especially Mainul Haque.