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 Table 2; these are evaluated at the parameter values in Table 1.

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 and are combined into with –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 numbers and , the Jacobian matrix for this disease-free system isand 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:(13)

For our parameters, this means that stability occurs exactly when both and exceed cells. Therefore, from the parameterized equilibrium values in Table 2, 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 Table 1, 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 [19], [38]–[40]. When invading bacteria meet a naive immune system, the dynamics in Figure 2 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.

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 Figure 3. A larger absolute immune response, as evidenced by the antibody and B cell peaks, is stimulated by this secondary infection (Figure 3b); 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 (Figure 3a). 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.

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 (Figure 4). 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 [41]. Furthermore, elevated levels of IFN- have been shown in humans with shigellosis or following administration of candidate attenuated Shigella vaccines [20]–[27].

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 (Figure 4). From Table 2, we know that in the order of IgA and IgG established via vaccine are sustained at the joint disease-free equilibrium. Yet from Figure 4, it is clear that holding Shigella to small numbers requires much higher initial antibody levels; for instance, it takes IgA alone, 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 Figure 4 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. Figure 4b displays horizontal slices through the surface in Figure 4a. The center diagonal shows where equal amounts occur. From Figures 4a and 4b, 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 Figure 2, in which 20% of total IgA is present in the lumen at homeostasis. Thus, in Figures 4a and 4b, 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 Figures 4a and 4b but instead requiring that 100% of the total IgA migrates to the lumen. Figure 4c 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 becomes(14)

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 Figure 5. 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.

The model parameters have been chosen from the literature whenever possible (Table 1). 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 Table 1. We monitor the post-vaccine dynamics over 45 days. For any chosen parameter range, we measure and plot.

The biological values of many of these quantities are unknown (Table 1). 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 (, , and ), the rate antibodies neutralize Shigella ( and ), the B carrying capacities ( and ), the rate B differentiate into plasma cells upon antigenic stimulation ( and ), the number of plasma cells generated by proliferating antigen-activated B cells (), the rate that plasma cells are generated from antigen-activated B cells (, , , ), and the delay terms ( and the primary infection immune delay). Results of these simulations are shown in Figures 6, 7, 8.

Antibody half-lives have been measured clinically in humans [42]; 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 (, , and ), which we set to be equal to one another, from /d to /d. In Figure 6a, 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. Figure 6b 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 Figure 6c, wherein we plot peak Shigella numbers in the lumen (L), lamina propria (LP), epithelium (E) or engulfed in innate immune cells (I). In Figure 6d, 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 ( and ) from to /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 (Figures 6e–f). 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 (Figure 6g) and times (Figure 6h), 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 (Figure 6g); however, such rates lead to the nearly immediate elimination of the Shigella infection (Figure 6h) 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 ( and ) from to cells in Figures 6i–l. 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 (Figures 6i–j). Interestingly, the carrying capacity for Shigella-directed B cells does substantially impact the amount of Shigella present in the epithelium (Figure 6k). 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 (Figure 6k). 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 to cells. However, Figure 4 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 Figure 6k, 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 ) and at the number of plasma cells created per B cell from this differentiation and subsequent proliferation (). We vary the rates from to and see in Figures 7a–d 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 cells.

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 (Figures 7e–h). In these figures, the value of ranges from to . 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 (Figure 7h); 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 (Figure 7g). 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 (, , , and ) from nearly zero () to 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 (Figures 8a–b). The higher antibody presence has little effect on the non-epithelial Shigella peaks, but the bacterial peak in the epithelium is reduced (Figure 8c). 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 (Figure 8d). 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 to days, has little-to-no effect on any observed dynamics, as can be seen in Figures 8e–h. 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 or 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 to days, Figures 8i–l 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 infection.