The results of a single dose ) of stable vaccine antigen being introduced into the model system at time are given in Figure 2. The model simulation showed an immune response stimulated by vaccine antigen that lasted for several weeks: the timescales observed for the different cell types and antibodies were characteristic of those seen following immunisation of cattle [9].

Specifically the removal of vaccine antigen, , on a fast timescale was observed, together with the generation of a burst of short-term antibodies that peaked around the second week after immunisation. A longer-term antibody response was generated, which persisted over the host lifetime, remaining present to immediately respond to future antigen challenges. A sustained, elevated, population of memory B cells and long-lived plasma B cells was also produced, which constituted the generation of immunological memory.

We repeated the calculation but varied the initial dose of (stable) vaccine antigen: results are given in Figure 3. The magnitude of the immune response was clearly sensitive to the initial dose and the qualitative form of the temporal dynamics was maintained as we would expect. Although higher vaccine doses elicited stronger immune responses, there was an upper bound due to saturation of dendritic cells (i.e. the assumption that all local dendritic cells present are stimulated and addition of more vaccine does not lead to further immunological stimulation) but only at a much higher vaccine dose than observed experimentally.

We next considered the effect of vaccine stability on the immune response by varying the vaccine decay rate parameter : results are given in Figure 4. Vaccine stability had an impact on the magnitude of the immune response, with the more stable antigens stimulating stronger responses in all cell types. However, stability had negligible impact on the overall timing of the immune response.

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. [15]). Using the model it was possible to investigate the effect and observed cumulative benefit of repeat vaccination with vaccine formulations of differing structural stability. Figure 5 shows the simulation results when a second (identical) vaccine dose was introduced 29 days after the first, and observed for a further 42 days (71 days in total), reflecting the experimental protocol referred to in Figure 6. We performed this calculation both for stable and unstable vaccine antigens.

The system produced the response in IgM and IgG that would be expected empirically, namely, only a small difference in IgM () between the first and second dose but a much larger booster effect in IgG () for stable vaccine (blue data in Figure 6). This effect was much larger than can be explained by the sum total of doses alone. It was evident that the model is able to qualitatively capture immunological memory, which is a central feature of the adaptive immune response. As before, stable vaccines elicited a stronger immune response, especially following secondary vaccination. Antibody levels were maintained at a higher level, and for longer, with increasing vaccine stability.

Figure 6 shows experimental IgM and IgG levels in cattle immunised twice – an initial dose at and an equal booster dose at days – with two different vaccines. One vaccine was modelled on a conventional, commercial vaccine (data plotted in blue) that initiates the full immune response. The second was a modified vaccine (data plotted in green) that is designed not to fully initiate a T-cell dependent response. We note that in the latter case there was still some generation of IgG-secreting B-cells, and thus IgG (Figure 6, bottom right), although much reduced. The principle difference is a significant reduction in the IgG response to a booster dose of vaccine in the absence of a T-cell mediated response.

Although the data is best considered qualitatively, using the model we investigated the effect of repeat vaccination with such modified antigens: an inhibited T-cell independent response was represented by reducing and in equations (10)–(13); results are given in Figure 7. Since the T-cell mediated response is an intrinsic component of the long-term (adaptive) pathway that generates IgG, and assumed to require more sustained stimulation, while IgM is produced by the short term pathway only, we considered the data presented in Figure 6 as a guide to the response expected from stable and unstable (i.e. slowly and rapidly decaying) vaccine.

Although the model only aims to qualitatively replicate observed dynamics, we have plotted experimental results and simulation results together in Figure 8 for comparison. Bearing in mind that the success of the experimental protocol in removing T-cell dependence is unknown, we believe that the model is also able to produce good agreement with the observations for this scenario.

We have shown that the model for the within-host 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.

Figure 9 shows the sensitivity analysis for a LHS scaling factor of 4 (i.e. each parameter is uniformly distributed between to 4 times its estimated value). It is evident that the qualitative behaviour of the model was preserved for up to this multiple; comparable IgM () responses following initial and booster vaccination were seen and a significant enhancement in the magnitude of IgG () response was maintained. Other calculations (not shown) indicated consistent model behaviour up to multiples of at least 6. This indicates that the model of the T-cell independent and T-cell dependent antibody responses is robust against large uncertainty in the model parameters.

A key feature of the dynamics is the booster effect in IgG (), and the metric we used to evaluate whether this is preserved was the ratio of peak responses following the first and second dose of vaccine; explicitly:A systematic exploration of the peaks ratio observed for IgM and IgG is shown in Figure 10: ratios greater than 1 were usually observed over parameter multiples up to at least 6 times nominal. As parameter uncertainty increased (i.e. the scaling factor increased) the observed magnitude of theT -cell dependent antibody response decreased, on average (but with higher variability).

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 “short-term” and “long-term” 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 T-cells and specialist B-cells 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 () passes, via the lymphatic system, into the lymph nodes. Here the antigen markers on the capsid surface stimulate a cascade of responses that will, in time (following a delay ), generate targeted B-cells () which, in turn, release into the body specifically-targeted antibodies, namely immunoglobin IgM () [22]. This antibody binds with the free capsids to form a capsid-antibody complex () that, although inert, is involved in the long-term pathway (see below). After the antigenic stimulus is removed, the specific B-cell population, and consequently antibody production, will decline to negligible levels over a period of a few weeks. We assume that this pathway has no inherent memory so that any future insults, either by virus or vaccine, will initiate the process anew. The T-cell independent pathway does not produce long lived memory cells [23].

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 short-term pathway. Unactivated DCs () are present throughout the body, and can become activated either by direct uptake of the capsid particles through macropinocytosis () or by uptake of the capsid-antibody complex via the FC receptor mediated process (). These activated DCs migrate to the lymph-node to present the antigen to naive T-cells, which in time (following a delay ), mature to either effector T-cells () or memory T-cells (). Effector T-cells in turn stimulate development of a new line of differentiated B-cells capable of producing immunoglobin IgG antibodies (); these B-cells may be short-lived () or long-lived plasma () cells, depending on whether they colonize in the bone marrow. The latter are often referred to as “niche” cells since they settle in niches in the bone marrow. In addition, effector T-cells also stimulate memory B-cells () that interact with memory T-cells and vaccine capsid () to generate additional effector T-cells: this T-cell dependent antibody response allows for a much more rapid response to “booster” vaccine doses – or exposure to live virus – in future.

The full integrated system, comprising of the short-term and long-term pathways, is illustrated in Figure 1 in terms of the population variables and parameters defined in Tables 1 and 2 respectively, and is in broad agreement with that described by Goodnow et al. [24].

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 40-fold difference in the decay rate of live virus [13]. Given the enhanced immunogenicity engendered by more stable capsid particles we use the relative capsid decay rate as a metric of stability in the model.

The model equations are derived from the pathways shown in Figure 1 under the assumption of mass action laws and exponential decay where appropriate. The variables represent concentrations i.e. cell number per unit volume. Most parameters represent rates, either production (), conversion () or decay (). In the absence of data on exactly how delays in the system are distributed, we model the time for B-cell and T-cell generation as simple discrete temporal delays.

The temporal evolution of the vaccine antigen is determined by its decay rate, uptake by IgM (, uptake by unactivated DCs () and uptake by IgG (), which are explicitly given by:(1)The decay rate parameter describes the stability of the vaccine capsid (with a half-life given by ). Here and are the rate at which vaccine-IgM complex and vaccine-IgG complex form respectively, while is the rate of vaccine uptake by micropinocytosis.

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 Michaelis-Menten kinetics. Unactivated DCs () are taken up by antigen () or complex () so that:(2)Here and are the level of vaccine and complex respectively resulting in 50 saturation of unactivated DCs. is the rate of uptake of complex by the FC activated pathway, while is the rate of dendritic cell migration.

Under the assumption that the level of vaccine in the lymph nodes is proportional to the total level, the dynamics of the short-term antibody secreting B-cells and subsequent short-term IgM concentration are given by(3)(4) and are the natural decay/removal rate of short term antibody secreting cells (ASCs) and IgM respectively, while and are the rate of production of short-term ASC induced by vaccine and the rate of IgM production by short-term ASC. The production of these B-cells takes a time and is thus dependent on the vaccine concentration at a time in the past. IgM is produced in proportion to the number of B-cells and is lost through binding with vaccine antigen and through decay. In the absence of vaccine , both and decay to zero as expected.

The generation of the vaccine–antibody complex through the long-term 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:(5)(6)(7)Here , and are the decay rates of complex, macropinacytosis-activated DCs and FC-activated DCs respectively.

The presentation of vaccine or complex derived antigenic peptides by activated DCs to naive T-cells generates two classes of T-cells, with the differentiation of naive T-cells assumed to take a time . Furthermore, memory T-cells can be converted to effector T-cells by memory B-cells in the presence of vaccine, while memory B-cells themselves are produced in the presence of effector T cells. B cell development is supported by T helper cells.

The rate of decay of effector T-cells, memory T-cells and memory B-cells is given by , and respectively, so that:(8)(9)and(10)Here and are the rate of effector T-cell production by macropinocytosis activated DCs and FC activated DCs respectively, and are the rate of memory T-cell production by macropinocytosis activated DCs and FC activated DCs respectively, is the rate of memory B-cell production and is the rate at which memory T-cells are converted to effector T-cells.

Effector T-cells give rise to B-cells, a proportion of which settle in the bone marrow where their life span is significantly enhanced (long-lived plasma cells). IgG is produced by both types of T-cell dependent B-cells and can form complex with vaccine, giving:(11)(12)(13)It is assumed that negligible IgG is produced by the short-term pathway, so that represent only T-cell dependent immunoglobulin. The rates , , and represent short term T-cell dependent ASC production, long-term T-cell dependent ASC production, IgG production rate by short-term T-cell dependent ASC and IgG production rate by long-term T-cell dependent ASC respectively; , and are the rates at which short-term T-cell dependent ASC, long-term T-cell dependent ASC and IgG decay.

Vaccine antigen () is first introduced as an impulse at time into a system of DCs (), and we therefore impose the following initial conditions:and for , with all other variables equal to zero. The system is defined completely by the closed equations (1)–(13) and these initial conditions. When a “booster” vaccine dose is applied at time , we introduce an impulse of the formthis means that immediately after the timepoint the value of has increased by a quantity compared to immediately before time . All model variables are given in Tables 1.

The immunology of FMDV has been intensively studied in a variety of in-vivo and in-vitro contexts. Because of this, and perhaps unusually for an immunological system, we are able to make good estimates for many of the parameters in our detailed model.

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 Table 2, together with sources for each parameter estimate. Although estimates for a number parameters are not available – and there is no way at present to extract these numbers experimentally – it is still possible to make progress by numerical simulation and analysis of the model without them. We have undertaken sensitivity analysis to capture the uncertainty these introduce, and noted which parameters primarily result in the scaling of certain variables. In what follows we therefore restrict our attention to the relative rather than absolute levels of IgG ( and IgM () in response to different scenarios, and when comparing output to experimental data.

A systematic procedure for investigating the range of behaviours in complex dynamic models is Latin Hypercube Sampling (LHS) – see [25]. It is a statistical analysis technique that permits a comprehensive exploration of the model parameter space in a computationally efficient manner. Its efficiency stems from the fact that each value of each parameter is used only once. The model is run for different combinations of model parameters and the outputs can be described by basic descriptive statistics, though here we are primarily focussed on the robustness of the qualitative behaviours of the model. Essentially the issue is one of investigating the effect of uncertainty in determining model parameters on the model outputs. It is not our intention to review the LHS methodology here, rather we note that it has been successfully applied to deterministic mathematical epidemiology models of comparable size and complexity to the immunological model described above [26], [27] and in systems engineering applications as well [25].

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 [25]. The parameters used in the model are itemised in Table 2.