Equilibrium analysis elucidates model dynamics between infections. Prior to a primary Shigella
infection, all variables lie at a trivial equilibrium where no Shigella
bacteria or immune components specific for Shigella
yet exist. (The trivial equilibrium disallows potential cross-reactivity with pre-existing antibodies created in response to a different bacterial infection. Inclusion of these must be done externally to the model via initial conditions.) A Shigella
infection is cleared unless the disease is so acutely severe that it kills the host; however, a B
population specific for the bacteria remains indefinitely and supports ongoing plasma cell and antibody creation. The model captures this behavior with its disease-free equilibria that have nonzero numbers for antibodies, plasma cells, and B
while simultaneously having no Shigella
bacteria in any spatial compartment. Three disease-free equilibria exist for the model: one with mucosal (IgA-type) but not systemic (IgG-type) immunity, one with systemic but not mucosal immunity, and a joint equilibrium with both. The values of the model variables at equilibrium are given in ; these are evaluated at the parameter values in .
Importantly, there is no completely nontrivial equilibrium of the model, which is consistent with the fact that a Shigella
infection is never persistent nor chronically latent. If, however, macrophage re-engulfment of Shigella
in the lamina propria is allowed–that is, if
are combined into
–then a persistently infected macrophage population develops that continually seeds the Shigella
infection and prevents bacterial clearance (not shown). This chronic state is not observed biologically; hence, macrophage re-engulfment is not allowed in the model and the Shigella
populations in the lamina propria are separate.
Stability of the trivial and disease-free equilibria can be determined by examining the eigenvalues of the Jacobian matrix for our system of differential equations at the equilibria. Since there is no truly nontrivial equilibrium and hence the Shigella
populations are zero at all equilibria, we examine the stability of the reduced system with
. For a generic equilibrium with B
, the Jacobian matrix for this disease-free system is
and the eigenvalues of the reduced system are
If the following two stability conditions are satisfied, the eigenvalues are all negative and hence the corresponding equilibrium is stable:
For our parameters, this means that stability occurs exactly when both
cells. Therefore, from the parameterized equilibrium values in , it is apparent that the joint (IgA and IgG) disease-free equilibrium is stable while the trivial equilibrium, IgA-only disease-free quilibrium, and IgG-only disease-free equilibrium are saddle points.
Thus, following a primary Shigella infection, the trivial equilibrium is not maintained and some level of permanent immunity is established. If only A-type or G-type immunity is generated initially, another infection could boost the system to the stable joint disease-free immune state. Whether these immune levels are sufficient to confer protective immunity to the host remains to be determined. Nevertheless, the model supports the hypothesis that a vaccine, which perturbs the immune system away from the trivial equilibrium, will establish a persistent nontrivial level of humoral immunity specific for Shigella.
We assess the behavior of the mathematical model via numerical simulations. Using a delay differential equation solver in MATLAB, we examine the bacterial and immune dynamics during a Shigella infection that occurs either prior to or months after the administration of a vaccine; the action of the vaccine mathematically is to shift the system from the trivial equilibrium to a disease-free equilibrium. Therefore, a primary Shigella infection initializes with trivial variable values while a post-vaccine Shigella infection starts at a disease-free equilibrium. (We investigate a vaccine that elicits both IgA and IgG, and accordingly the model initializes at the joint IgA-IgG disease-free equilibrium for a post-vaccine infection.) Parameter values are given in , and their impact is discussed in the Parameters section.
We establish an infection by assuming enough Shigella
is ingested to introduce a population of 1,000 bacteria into the gut lumen. Since a minimum of 100–1000 Shigella
bacteria clinically cause disease, this is sufficient to cause infection 
. When invading bacteria meet a naive immune system, the dynamics in result. Shigella
grows and migrates unfettered in the host during the incubation period before sufficient numbers of naive B cells have been stimulated to create ASC and B
cells that target the infection. We have built a 3.5-day incubation period into the model prior to immune initiation; after this time, a B cell and antibody response is generated that eliminates the bacterial infection. IgA and IgG levels peak roughly 10–21 days after infection and equilibrate after about one month. Plasma cells and B
cells also reach new homeostatic levels after about a month. Crucially, these levels are consistent with a disease-free equilibrium’s nontrivial immune levels rather than with the trivial equilibrium at which the system began. Immune activity clears the Shigella
infection to below one bacterium in the lumen and lamina propria in 20 days, but bacteria that enter epithelial cells escape humoral immune targeting and grow without restraint. It should be noted that Shigella
’s infection of the epithelium may be controlled by cytotoxic T cells and thus in reality this epithelial population would be more controlled; however, T cell activity against Shigella
remains largely undefined; thus we have not included them in the model. As a result, true Shigella
dynamics in response to both humoral and cell-mediated immune responses are outside the purview of this model. For our purposes, the essential fact shown by the primary infection model is that Shigella
can survive well in the epithelium and thus a vaccine that effectively protects against shigellosis must prevent most epithelial entry.
Dynamics of the mathematical model for a primary infection.
Secondary immune responses are elicited when a vaccine is given that elicits LPS-directed IgA and IgG humoral immune responses and shifts the immune system to the joint disease-free equilibrium. After the host system re-equilibrates, 1,000 wild-type Shigella bacteria are introduced to the gut. The dynamics of this secondary Shigella infection are depicted in . A larger absolute immune response, as evidenced by the antibody and B cell peaks, is stimulated by this secondary infection (); this is consistent with the fact that memory cells elicit stronger and more rapid reactions than naive cells. The vaccine decreases the duration of the bacterial infection by nearly half–from 20 days to 11.5 days until complete clearance (). It also lessens the severity of the infection, as seen by comparing the height of the “lamina propria” and “innate” Shigella peaks. (The luminal peak is fixed to 1,000 by initial conditions.) Also noteworthy are the slowed dynamics of the epithelial cell infection, which suggests the host may experience temporarily reduced symptoms resulting from epithelial destruction. Nevertheless, the vaccine does not fully prevent the epithelial infection, which means the host must instead hope that a primary cytotoxic T cell response can clear it.
Dynamics of the mathematical model for a secondary or post-vaccine Shigella infection.
We next ask if Shigella
’s infection of epithelial cells could be prevented purely by anti-LPS antibodies if they were available in larger supply (). Here, we boost antibody numbers above vaccine levels, for instance via a pre-infection serum injection of antibodies, while B cell numbers remain at levels established by a vaccine. We examine how many IgA and IgG molecules must be present to restrain Shigella
’s epithelial population to a given day-45 threshold. By day 45, a wild-type Shigella
infection will have cleared; thus, we assume that if we can control the bacterial population for long enough through antibody responses, other host’s immune responses such as CMI (e.g., cytotoxic T cells or production of IFN-
and other pro-inflammatory cytokines that activate macrophages and enhance their ability to kill intracellular Shigella
) will be sufficient to clear the infection. In fact, recent publications have begun to show that CMI responses are important. For example, the clearance of a primary Shigella
infection is impaired in the absence of T cells. Additional studies demonstrated that following reinfection, IL-17A and IL-22 producing T cells are primed by Shigella
and the IL-17A produced by these cells restricts bacterial growth 
. Furthermore, elevated levels of IFN-
have been shown in humans with shigellosis or following administration of candidate attenuated Shigella
The number of antibodies needed to control an epithelial Shigella infection is shown.
The maximum tolerable day-45 threshold should be fixed as a number of Shigella
that can infect the epithelium without the host becoming severely symptomatic; as this value is unknown, we vary the peak number of bacteria allowed and examine the corresponding antibody requirements (). From , we know that in the order of
IgA and IgG established via vaccine are sustained at the joint disease-free equilibrium. Yet from , it is clear that holding Shigella
to small numbers requires much higher initial antibody levels; for instance, it takes
IgG alone, or
IgA and IgG together in the GI tract to keep the Shigella
epithelial population at day 45 below 100 bacteria. This four-orders-of-magnitude increase in initial antibody levels could be difficult to elicit biologically and may be untenable.
From we can also parse the relative effectiveness of IgA versus IgG. We assume equal rate parameters but different spatial dynamics for IgA and IgG; IgG removes bacteria in the lamina propria while IgA is made in the lamina propria but functions in the lumen. displays horizontal slices through the surface in . The center diagonal shows where equal amounts occur. From , we see that IgA and IgG are nearly equally effective alone, with IgG being slightly more potent. However, we must examine the details more carefully to discern true differences in antibody efficacy. These figures show the total amount of IgA in comparison with IgG. However, IgA is distributed across two spatial compartments: the lamina propria, where it is formed, and the lumen, where it functions antimicrobially. The amount of IgA in the lumen versus the lamina propria initially is consistent with the ratio from , in which 20% of total IgA is present in the lumen at homeostasis. Thus, in , 20% of the total IgA acts nearly comparably to 100% of IgG, which does not have spatial compartmentalization. This suggests that A-type antibodies may actually be more effective than G-type on a per-molecule basis. We check this by repeating the simulation done to create but instead requiring that 100% of the total IgA migrates to the lumen. shows that this has little overall effect, but careful examination of the intercepts shows that IgA is now slightly more potent than IgG. Thus, if only one type of antibody response can be elicited by a vaccine, either a mucosal (IgA) or a systemic (IgG) response will be about equally effective. An IgA-only response may require more total antibodies to sustain a sufficient luminal concentration, but each individual IgA molecule is indicated to be slightly more efficacious. If both IgA and IgG can be elicited instead of only one, there is an order-of-magnitude drop in the total amount of antibody needed for protection.
Another option to improve the efficacy of the vaccine’s control on the epithelial invasion is to modify the vaccine targets. We have shown that large amounts of IgA and IgG must be present for a vaccine targeting Shigella
outer membrane components such as LPS to be effective. What if we additionally include an antibody population capable of specifically targeting epithelial entry? To examine this question, we alter the model to allow IgG to nonmechanistically modulate the rate at which in the lamina propria Shigella
enters epithelial cells. Equation 5
The inclusion of epithelial targeting by antibodies has the desired effect of almost entirely preventing bacterial invasion of epithelial cells, as can be seen in . Although the epithelial population is fractionally higher than zero, this negligible bacterial population that succeeds in circumventing these tightened immune constraints will likely be eliminated through other host defenses. It should be noted that in this altered model we imperfectly assume that the same IgG population can target both LPS and epithelial entry; a future, mechanistic model will separate these populations. However, this simple, nonmechanistic approach demonstrates that targeting epithelial entry can be a successful strategy in theory and is worth pursuing in more detail. Future work must also take into account other important issues, such as potentially brief availability of epithelial entry proteins.
Bacterial and immune dynamics when the model includes antibody targeting of epithelial entry.
The model parameters have been chosen from the literature whenever possible (). No parameters have been fitted. While a detailed search of parameter space is outside the scope of this current study, we explore the role of individual model parameters on the model dynamics by conducting further simulations in which we vary a single parameter while leaving the rest at the values in . We monitor the post-vaccine dynamics over 45 days. For any chosen parameter range, we measure and plot.
- The magnitude and timing of peak total antibody numbers.
- The magnitude and timing of the peaks in Shigella numbers in each spatial compartment.
- The time of extinction of Shigella in non-epithelial compartments, which we define as having less than one Shigella bacterium total in the lumen, lamina propria, or engulfed populations.
- The time until antibody decays to 10% of its peak value.
The biological values of many of these quantities are unknown (). The goal of this study is to determine the degree of dependence of the predicted outcomes on the underlying parameters. We primarily focus on varying parameters about whose values we are most uncertain in the context of shigellosis. These are the antibody decay rates (
), the rate antibodies neutralize Shigella
), the B
carrying capacities (
), the rate B
differentiate into plasma cells upon antigenic stimulation (
), the number of plasma cells generated by proliferating antigen-activated B cells (
), the rate that plasma cells are generated from antigen-activated B
), and the delay terms (
and the primary infection immune delay). Results of these simulations are shown in , , .
Effects of various parameters on post-vaccine dynamics are evaluated.
Effects of various parameters on post-vaccine dynamics are evaluated.
Effects of various parameters on primary infection dynamics are evaluated.
Antibody half-lives have been measured clinically in humans 
; yet, survival times for antibodies present in the lamina propria may be lower due to time spent localizing to the lamina propria, transcytosis, washout, and other factors. To better understand how antibody survivorship times affect immune and bacterial dynamics, we vary the antibody decay rates (
), which we set to be equal to one another, from
/d. In , we plot the magnitude of the total post-vaccine antibody peak, which sums lamina propria IgG, lamina propria IgA, and luminal IgA. As the natural antibody death rate increases, the peak number of antibodies decreases because the time window over which antibodies present at the peak were created is broader. gives the time at which the antibody peak occurs as well as the time at which only 10% of the peak remains following the infection. Since we limit the time frame to 45 days, the total antibody population never decays to 10% of the peak for small antibody death rates. Comparing these figures, we see that smaller antibody peaks require less time to be reached. Despite the changing antibody peaks, the bacterial peaks do not vary much with the antibody death rates, as seen in , wherein we plot peak Shigella
numbers in the lumen (L), lamina propria (LP), epithelium (E) or engulfed in innate immune cells (I). In , we plot Shigella
peak times, which also are independent of this parameter. However, we also plot the time at which the total Shigella
population in the lumen, lamina propria, and engulfed in innate cells drops below one bacterium, corresponding to Shigella
extinction in non-epithelial compartments. This extinction time does increase with antibody death rates, likely because the amount of antibody capable of removing bacteria decreases.
We next vary the rate that antibodies neutralize Shigella
in the lumen and lamina propria (
/antibody/day. Our model value of
/Ab/day was chosen for being the largest neutralization rate for which a non-epithelial Shigella
infection takes at least a day to peak in simulations; it was not chosen as an optimal value that fits more biologically stringent Shigella
peak behaviors. By varying the value, we see that higher neutralization rates correspond to faster, and thus lower magnitude, antibody peaks (). Only for smaller antibody peaks do antibody levels decay to 10% of the peak within 45 days. Little change in peak antibody numbers occurs for a neutralization rate lower than
/Ab/day, which suggests that this is a lower bound on antibody effectiveness. This is confirmed by examining the Shigella
peak magnitudes () and times (), which are identical for all neutralization rates below about
/Ab/day. However, higher neutralization rates lower the peak bacterial load nonlinearly and drive non-epithelial Shigella
infections to extinction within 0–3 weeks. The epithelial peak also never reaches above
bacteria at day 45 for naturalization rates larger than
/Ab/day (); however, such rates lead to the nearly immediate elimination of the Shigella
infection () and thus may not be biologically feasible. Whether increasing the neutralization ability of individual antibodies is possible and effective in clinical parameter ranges should be explored in more detail with future modeling.
Since the number of antigen-specific B
cells needed to confer immunity to Shigella
is unknown and likely can vary with antigen targets, we vary the carrying capacities (i.e., the maximum sustainable cell numbers in the absence of infection) for IgA- and IgG-B
cells in . Increases in the carrying capacities induce roughly the same order-of-magnitude increase in the peak number of total antibodies but does not much affect the timing of the peak or the time at which all but 10% of the peak antibody remains (). Interestingly, the carrying capacity for Shigella
cells does substantially impact the amount of Shigella
present in the epithelium (). If
or more Shigella
-specific IgA- and IgG-B
cells can be sustained, the peak number of Shigella
in a post-vaccine infection at day 45 remains below
bacteria (). This is due in part to the resulting order-of-magnitude increase in both initial (disease-free equilibria) and peak antibody numbers when the carrying capacities are changed from
cells. However, suggests the presence of
total antibodies prior to infection would still not be sufficient to confer protection; this figure assumes a pre-infection boost of antibodies above vaccine levels (via a serum injection of antibodies, for instance) without a corresponding increase in B
cells or other immunity. This makes clear that an antibody boost alone is not sufficient for immune protection. However, an antibody increase resulting from an underlying boost in B
cells can confer protection if high enough B
cells numbers are reached, perhaps because higher antibody levels can then be sustained for longer times. Importantly, from , we see that altering the B
carrying capacity can improve the effectiveness of a vaccine enough that a purely anti-LPS Shigella
vaccine could be sufficient to confer immunity. This suggests that anti-LPS B
cells could serve as correlates of immunity and should be a key focus of future work in parallel with explorations of epithelial entry protein targeting.
To explore how antigen-stimulated plasma cell creation affects bacterial and immune dynamics, we look more closely at the rates that B
-cells differentiate into new plasma cells in response to antigen (
) and at the number of plasma cells created per B
cell from this differentiation and subsequent proliferation (
). We vary the rates from
and see in that both the antibody and Shigella
peaks are insensitive to large differentiation rates. When antigen-activated B
cells differentiate less frequently into plasma cells, the peak magnitudes for both Shigella
and total antibody are higher, although the peak times vary only minutely. This increased antibody production despite lower ASC creation rates demonstrates that either (1) consistent with what we have seen previously, the minor increase in the time to the antibody peak is sufficient to create more antibodies, despite a lower plasma cell creation rate or (2) antigenic stimulation of B cell generation due to higher Shigella
levels increases sufficiently to compensate for lower B
differentiation rates. The extinction time for non-epithelial Shigella
drops only slightly when plasma cells are generated more quickly from B
When we examine how the number of plasma cells made per differentiating B
cell in the presence of antigen (
) affects antibody peak dynamics, we see that antibody levels rise and Shigella
numbers decay substantially when more plasma cells are made per B
cell (). In these figures, the value of
. Here, higher antibody levels can be achieved with smaller peak times because ASC are more plentiful. Unlike with plasma cell rates, the extinction time for non-epithelial Shigella
drops substantially when more plasma cells are generated per B
(); this suggests that it is the number, not the rate, of ASC production that matters for Shigella
clearance. Furthermore, if more than
plasma cells are created by each antigen-stimulated B
cell differentiation and proliferation, the day-45 bacterial load in the epithelium can be contain to less than
bacteria (). This assumes a B
carrying capacity of
cells and indicates that if the B
homeostatic number cannot be boosted, immune protection might be achievable if the plasma-cell-generating potential of each B
can be sufficiently increased.
To explore how the creation of new B cells from naive cells impacts infection dynamics, we first vary the creation rates of plasma cells from naive B cells (
) from nearly zero (
plasma cells/bacteria/day. After a vaccine, both Shigella
and antibody dynamics are completely insensitive to changes of creation rates from naive B cells since pre-existing immunity contributions dominate (not shown). Thus, we also evaluate primary infection behavior. The faster that plasma cells are created from naive B cells, the higher the antibody peak and the more swiftly it is reached during a primary infection (). The higher antibody presence has little effect on the non-epithelial Shigella
peaks, but the bacterial peak in the epithelium is reduced (). Furthermore, small plasma cell creation rates can hinder Shigella
clearance during a primary infection, but rates above
plasma cells/bacteria/day show little variation in Shigella
extinction times (). Our arbitrarily chosen value of
plasma cells/bacteria/day thus results in roughly the same dynamics as far higher plasma cell creation rates.
Lastly, we investigate how the time delays used in our model influence the results. We use two time delays: (1) the time delay (
) for new plasma cell and B
creation from naive B cells which serves as the delay component of the differential equations and (2) a numerically enforced initial delay before naive cell activation can occur during an infection. To evaluate the impact of these delays, we consider primary infection dynamics rather than post-vaccine dynamics, as the former is where variation will be most evident. Yet, the value of the delay component built into the model (
), which we range from
days, has little-to-no effect on any observed dynamics, as can be seen in . Theorizing that this might be due to the small value for the
s, which multiply the delayed Shigella
numbers, we set
plasma cells/bacteria/day and reevaluated the delay effect. We again found no variation in the dynamics relative to
. Hence, the use of delay differential equations at this stage was not essential to the observed results. However, we continue to use delay equations because it incorporates the biologically observed delay from naive B cell activation to effector or memory cell functionality without limiting our computational ability. Furthermore, it establishes a realistic modeling infrastracture that could be useful in future work.
The second delay, with which we allow the Shigella
infection to establish for
days before naive B cell activation, does impact the results. We implement this numerically by starting an infection at either a trivial or disease-free equilibrium but running the reduced system in which we eliminate the terms for the creation of B cells from naive cells (i.e., any terms with
are set to zero) for
days. During this time, pre-existing immunity is unimpeded in its function or ability to generate new cells or antibodies. After
days, the naive cell terms are added back in and the full system runs from where the reduced system left off. When we vary this incubation time window from
days, result. Time delays less than one day change the dynamics little relative to one another, but longer delays increase Shigella
peak numbers, which results in higher antibody peaks due to increased antigenic stimulation. Clearance of Shigella
varies only slightly even when the Shigella
peak magnitude doubles. In fact, the Shigella
extinction time without the incubation period is identical to the 20-day extinction time with a 3.5-day incubation period. Thus, quick naive B cell activation is not vital to clearance of a Shigella