A Carriage-Structured Model of Meningococcal Meningitis: Backward Bifurcation, Seasonal Recurrence, and the Reach of Conjugate-Vaccine Strategies in the African Meningitis Belt

1. Introduction

Bacterial meningitis remains a leading cause of death and long-term neurological disability in sub-Saharan Africa. The African meningitis belt, a band of twenty-six countries stretching from Senegal to Ethiopia, experiences a characteristic epidemiology found nowhere else: hyperendemic transmission of Neisseria meningitidis punctuated by explosive dry-season epidemics and, over longer horizons, by irregular multi-annual waves recurring every five to twelve years [6,11,12]. The introduction of the monovalent serogroup A conjugate vaccine MenAfriVac from 2010 virtually eliminated serogroup A disease [14], but epidemics caused by serogroups C, W, X and Y have continued, and a pentavalent conjugate vaccine (Men5CV/MenFive) has since been prequalified and recommended for introduction across the belt [13].

Two features of meningococcal biology make the disease unusual to model. First, transmission is carried almost entirely by asymptomatic nasopharyngeal carriage; invasive disease is a rare complication of carriage—on the order of one per cent of carriage episodes or fewer—and invasive cases do not themselves transmit [4,5]. A model in which the notified case is the unit of transmission therefore describes the wrong process. Second, the recurrence of epidemics on the multi-annual timescale is not mechanistically settled: candidate explanations include seasonal forcing of the transmission rate through Harmattan dust and low humidity, seasonal forcing of the invasion rate through mucosal damage or transient co-infection, and the slow accumulation and waning of population immunity [1,12]. Irving et al. [7] showed by simulation that seasonal forcing of transmission alone cannot reproduce the observed irregular multi-annual dynamics, and that waning immunity or an analogous slow process is required. That finding has not since been revisited with the tools of bifurcation theory.

Transmission-dynamic models have been used to evaluate conjugate-vaccine strategies in the belt [8,9], most recently to assess multivalent-vaccine rollout, with vaccine effectiveness against carriage identified as the critical remaining unknown [10]. Here we take a complementary route: rather than fitting a detailed simulation model, we build a parsimonious carriage-structured model whose structure is transparent enough to analyse, and we ask what its equilibrium and bifurcation structure imply for prevention. Our contributions are: (i) a demonstration that ${\mathcal{R}}_0$ is a carriage quantity, essentially decoupled from disease incidence (Proposition 2); (ii) a closed-form backward-bifurcation condition, obtained by the centre manifold method, showing that bistability is driven by a threshold on carriage-blocking immunity and that the invasive-disease compartment cannot enter it (Theorem 2); (iii) a characterisation of the seasonally forced model’s attractor structure that locates multi-annual recurrence in a band of immunity-waning rate; and (iv) a scenario analysis linking the carriage-efficacy threshold to whether sustained immunisation eliminates or merely postpones epidemics.

2. Model Formulation

We partition a homogeneously mixing population of constant size N, normalised to unity, into five compartments: susceptible S; vaccinated V; asymptomatic carriers C; individuals with invasive disease I; and recovered/immune R. Only carriers transmit. The force of infection is

$$\lambda \left(t\right)=\beta \left(t\right)C,$$

(1)

where $\beta \left(t\right)$ is the (possibly seasonally forced) transmission rate. New carriage acquisition draws from the susceptible pool, from the vaccinated pool at reduced rate $(1-\sigma )$, and from the immune pool at reduced rate $(1-\epsilon )$; the parameter $\epsilon \in [0,1]$ measures protection against re-acquisition of carriage among those with immunity, and $\epsilon <1$ encodes the leaky, serogroup-specific nature of meningococcal immunity. Carriage clears at rate $\alpha$ or, with small probability, proceeds to invasive disease at rate $\rho$. The compartmental structure is shown in Figure 1.

Individuals are susceptible (S), vaccinated (V), asymptomatic carriers (C), invasively diseased (I), or recovered and immune (R). Carriers are the sole transmitting class: susceptibles and vaccinees acquire carriage at rates $\lambda S$ and $(1-\sigma )\lambda V$, and the immune re-acquire it at $(1-\epsilon )\lambda R$, where the force of infection $\lambda =\beta \left(t\right)C$ (dashed) is driven by carriage prevalence, not by disease. Carriage either clears to immunity at rate $\alpha$ or, rarely, progresses to invasive disease at rate $\rho$; because $\rho \ll \alpha$, almost all carriage resolves without disease. Invasive disease is epidemiologically terminal, draining only to R at recovery rate $\gamma$ and contributing nothing to onward transmission. Immunity wanes back to susceptibility at rate $\omega$ and vaccine protection at rate $\theta$, while the mortality rate $\mu$ acts on every compartment. The governing system is

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}S}{\mathrm{d}t}}& =\mu +\omega R+\theta V-\lambda S-(\mu +\phi )S,\hfill \end{array}$$

(2a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}& =\phi S-(1-\sigma )\lambda V-(\mu +\theta )V,\hfill \end{array}$$

(2b)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}C}{\mathrm{d}t}}& =\lambda \left[S+(1-\sigma )V+(1-\epsilon )R\right]-(\rho +\alpha +\mu )C,\hfill \end{array}$$

(2c)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}I}{\mathrm{d}t}}& =\rho C-\left[\gamma +\mu +\delta \left(I\right)\right]I,\hfill \end{array}$$

(2d)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}R}{\mathrm{d}t}}& =\alpha C+\gamma I-(\omega +\mu )R-(1-\epsilon )\lambda R,\hfill \end{array}$$

(2e)

with all parameters positive and defined in Table 1. Here $\mu$ is the birth/death rate, $\phi$ the routine vaccination rate, $\theta$ the vaccine waning rate, $\omega$ the natural-immunity waning rate, and $\gamma$ the recovery rate from invasive disease.

Case fatality is allowed to depend on health-system load through a capacity-saturating function

$$\delta \left(I\right)={\delta }_T{\displaystyle \frac{K}{K+I}}+{\delta }_U{\displaystyle \frac{I}{K+I}},$$

(3)

which interpolates between a treated case-fatality rate ${\delta }_T$ when case load I is small relative to capacity K and an untreated rate ${\delta }_U>{\delta }_T$ when facilities are overwhelmed. We stress at the outset—and prove in Section 5—that $\delta \left(I\right)$ affects burden but not the transmission dynamics or the bifurcation structure, because I is an epidemiological dead end.

Remark 1. 

The term $(1-\epsilon )\lambda R$ appearing in both the C and R equations is the modelling decision that distinguishes this system from a standard vaccination model. It states that individuals with protective immunity may still acquire and transmit carriage. This is biologically realistic for the meningococcus and, as we show, is the source of the model’s bistability.

3. Basic Properties

Let $\Omega =\{(S,V,C,I,R)\in {\mathbb{R}}_{+}^5:S+V+C+I+R\le 1\}$.

Proposition 1 

(Well-posedness). The region Ω is positively invariant for (2), and solutions with initial data in Ω remain non-negative and bounded for all $t\ge 0$.

Proof. 

On each face $x_i=0$ the corresponding derivative is non-negative (e.g. ${\displaystyle \frac{\mathrm{d}S}{\mathrm{d}t}}{|}_{S=0}=\mu +\omega R+\theta V\ge 0$), so the positive orthant is invariant. Summing (2) with $M=S+V+C+I+R$ gives ${\displaystyle \frac{\mathrm{d}M}{\mathrm{d}t}}=\mu -\mu M-\delta \left(I\right)I\le \mu (1-M)$, whence ${lim\; sup}_{t\to \infty }M\left(t\right)\le 1$. Thus $\Omega$ is positively invariant and all solutions are bounded. □

4. The Disease-Free Equilibrium and the Reproduction Number

Setting $C=I=0$ in (2) yields the disease-free equilibrium (DFE)

$$E_0=(S^0,V^0,0,0,0),S^0={\displaystyle \frac{\mu +\theta }{\mu +\theta +\phi }},V^0={\displaystyle \frac{\phi }{\mu +\theta +\phi }}.$$

(4)

We compute ${\mathcal{R}}_0$ by the next-generation method [15]. The infected subsystem is $(C,I)$. Because carriage produces disease but disease produces neither carriage nor further disease, the new-infection and transition matrices are

$$F=\left(\begin{array}{cc}{\beta }_0\Phi & 0\\ 0& 0\end{array}\right),\mathcal{V}=\left(\begin{array}{cc}{K}_C& 0\\ -\rho & d\end{array}\right),$$

(5)

where ${\beta }_0$ is the mean transmission rate, $K_C=\rho +\alpha +\mu$ is the total carriage exit rate, $d=\gamma +\mu +{\delta }_T$, and

$$\Phi \equiv S^0+(1-\sigma )V^0={\displaystyle \frac{(\mu +\theta )+(1-\sigma )\phi }{\mu +\theta +\phi }}$$

(6)

is the effective susceptibility of the carriage-acquiring pool. The spectral radius of $F{\mathcal{V}}^{-1}$ is the $(1,1)$ entry, giving the following.

Proposition 2 

(${\mathcal{R}}_0$ is a carriage quantity). The basic reproduction number of (2) is

$${\mathcal{R}}_0={\displaystyle \frac{{\beta }_0\Phi }{K_C}}={\displaystyle \frac{{\beta }_0\left[(\mu +\theta )+(1-\sigma )\phi \right]}{(\mu +\theta +\phi )(\rho +\alpha +\mu )}}.$$

(7)

Because invasion is rare and carriage is short-lived, $\rho \ll \alpha$, so $K_C=\alpha +\mu +\rho \approx \alpha +\mu$ and ${\mathcal{R}}_0\approx {\beta }_0\Phi /(\alpha +\mu )$. The invasion rate ρ enters ${\mathcal{R}}_0$ only as a negligible perturbation to the carriage-clearance rate; the disease compartment I does not appear at all.

Remark 2. 

Proposition 2 has a direct empirical consequence. ${\mathcal{R}}_0$ is determined by the duration of carriage and the contact rate, essentially independently of disease incidence. Any reproduction number inferred from notified invasive cases therefore estimates a quantity distinct from the transmission potential that governs elimination. Surveillance aimed at estimating ${\mathcal{R}}_0$ must target carriage, not disease.

Setting ${\mathcal{R}}_0=1$ and solving (7) for $\phi$ gives the critical routine coverage

$${\phi }_c={\displaystyle \frac{(\mu +\theta )\left({\beta }_0-\alpha -\mu -\rho \right)}{(\rho +\alpha +\mu )-{\beta }_0(1-\sigma )}},$$

(8)

valid when the denominator is positive; when ${\beta }_0(1-\sigma )\ge K_C$ the leaky vaccine cannot drive ${\mathcal{R}}_0$ below unity at any coverage, a first indication that $\sigma$ and $\epsilon$ control feasibility.

Theorem 1 

(Local stability of the DFE). The disease-free equilibrium $E_0$ is locally asymptotically stable when ${\mathcal{R}}_0<1$ and unstable when ${\mathcal{R}}_0>1$.

Proof. 

The Jacobian of (2) at $E_0$ is block triangular with respect to the ordering $(C;S,V,I,R)$: the C row decouples with eigenvalue ${\beta }_0\Phi -K_C=K_C({\mathcal{R}}_0-1)$, while the remaining $4\times 4$ block is that of the linearised vaccination subsystem, whose eigenvalues $-(\mu +\phi )$, $-(\mu +\theta )$, $-d$, $-(\omega +\mu )$ are all negative. Hence every eigenvalue has negative real part iff ${\mathcal{R}}_0<1$. □

5. Backward Bifurcation

We now determine the direction of bifurcation at ${\mathcal{R}}_0=1$ using the centre manifold theorem in the form of Castillo-Chavez and Song [3]. To avoid a clash of notation—the invasion rate and the Castillo-Chavez–Song coefficient are both conventionally denoted a—we write the invasion rate as $\rho$ throughout and reserve $a,b$ for the bifurcation coefficients. Let $(x_1,\dots ,x_5)=(S,V,C,I,R)$ and take $\beta$ as the bifurcation parameter with critical value ${\beta }^{*}=K_C/\Phi$ (so that ${\mathcal{R}}_0=1$).

Eigenvectors. 

At $(E_0,{\beta }^{*})$ the Jacobian has a simple zero eigenvalue. Writing $J_0$ for this Jacobian, the structure of the new-infection process—new carriers are the only new infections, and they enter compartment $C=x_3$—forces the left eigenvector to collapse onto $e_3$:

$$\mathbf{v}=(0,0,1,0,0).$$

(9)

Indeed, $\mathbf{v}{J}_0=0$ requires $v_1=v_2=v_4=v_5=0$ from columns $1,2,4,5$, leaving $v_3$ free. The right eigenvector $\mathbf{w}$ with $J_0\mathbf{w}=0$, normalised by $w_3=1$, is

$$w_4={\displaystyle \frac{\rho }{d}},w_5={\displaystyle \frac{\alpha }{\omega +\mu }},w_1=-{\displaystyle \frac{(\mu +\theta )P+\theta Q}{D_2}},w_2=-{\displaystyle \frac{(\mu +\phi )Q+\phi P}{D_2}},$$

(10)

where $D_2=\mu (\mu +\theta +\phi )>0$, $P={\beta }^{*}{S}^0-\omega \alpha /(\omega +\mu )$, and $Q=(1-\sigma ){\beta }^{*}{V}^0$.

Bifurcation coefficients. 

Because $\mathbf{v}=e_3$, only the second derivatives of the third component $f_3=\lambda [S+(1-\sigma )V+(1-\epsilon )R]-K_{C}C$ survive. The non-zero second-order partials at $E_0$ are the bilinear terms ${\partial }^2{f}_3/\partial x_1\partial x_3={\beta }^{*}$, ${\partial }^2{f}_3/\partial x_2\partial x_3=(1-\sigma ){\beta }^{*}$, and ${\partial }^2{f}_3/\partial x_5\partial x_3=(1-\epsilon ){\beta }^{*}$. The Castillo-Chavez–Song coefficients therefore reduce to

$$\begin{array}{cc}\hfill a& =\sum _{k,i,j}{v}_k{w}_i{w}_j{\displaystyle \frac{{\partial }^2{f}_k}{\partial x_i\partial x_j}}{|}_{E_0}=2{\beta }^{*}\left[w_1+(1-\sigma )w_2+(1-\epsilon )w_5\right],\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill b& =\sum _{k,i}{v}_k{w}_i{\displaystyle \frac{{\partial }^2{f}_k}{\partial x_i\partial \beta }}{|}_{E_0}=\Phi =S^0+(1-\sigma )V^0>0.\hfill \end{array}$$

(12)

Since $b>0$ unconditionally, the direction of bifurcation is governed entirely by the sign of a.

Theorem 2 

(Backward-bifurcation condition). System (2) undergoes a backward bifurcation at ${\mathcal{R}}_0=1$ if and only if $a>0$, that is, if and only if

$${\displaystyle \frac{(1-\epsilon )\alpha \mu (\mu +\theta +\phi )}{\omega +\mu }}>P\Pi +Q\Theta ,$$

(13)

where $\Pi =(\mu +\theta )+(1-\sigma )\phi$ and $\Theta =\theta +(1-\sigma )(\mu +\phi )$. Equivalently, defining the carriage-immunity threshold

$${\epsilon }^{*}=1-{\displaystyle \frac{(P\Pi +Q\Theta )(\omega +\mu )}{\alpha \mu (\mu +\theta +\phi )}},$$

(14)

the model exhibits backward bifurcation (and hence bistability, with a stable endemic equilibrium persisting for a range of ${\mathcal{R}}_0<1$) precisely when $\epsilon <{\epsilon }^{*}$.

Proof. 

By Castillo-Chavez and Song [3] (Theorem 4.1), with $b>0$ the bifurcation is backward iff $a>0$. Substituting (10) into (11) and clearing the positive factor $2{\beta }^{*}/D_2$ yields $(1-\epsilon )w_5{D}_2>(\mu +\theta )P+\theta Q+(1-\sigma )[(\mu +\phi )Q+\phi P]$. Collecting the P and Q terms gives $P\Pi +Q\Theta$ on the right and, with $w_5=\alpha /(\omega +\mu )$ and $D_2=\mu (\mu +\theta +\phi )$, the left-hand side of (13). Because $\epsilon$ enters only the left-hand side and linearly, solving the inequality for $\epsilon$ gives (14). □

Corollary 1 

(The disease compartment cannot cause bistability). Neither the invasion rate ρ, the recovery rate γ, nor the capacity-dependent fatality $\delta \left(I\right)$ appears in (13). The invasive-disease compartment I is absent from the bifurcation condition. Consequently no case-management intervention—faster treatment, expanded capacity K, reduced ${\delta }_T$—can create, remove, or shift the backward bifurcation. Such interventions alter disease burden only.

Proof. 

$\rho$ enters the model only through $K_C$ and through $w_4=\rho /d$; but $w_4$ multiplies ${\partial }^2{f}_3/\partial x_4\partial x_3=0$ and $v_4=0$, so it does not appear in a or b. The parameters $\gamma ,{\delta }_T,{\delta }_U,K$ enter only the I equation, which affects neither $\mathbf{v}=e_3$ nor $f_3$. Hence the coefficients are independent of the disease compartment. □

Corollary 1 is the structural counterpart of Proposition 2: just as the disease compartment does not enter ${\mathcal{R}}_0$, it does not enter the bifurcation condition. This rules out a family of otherwise-plausible “health-system capacity drives bistability” arguments that are valid for susceptible–infectious diseases but not for a carriage-transmitted one.

Remark 3 

(What actually drives bistability). Inspection of (13) identifies three channels. (i)Susceptible depletion(the $P\Pi$, $Q\Theta$ terms) is always stabilising and opposes backward bifurcation. (ii)Leaky carriage immunity, $\epsilon <1$, supplies the driving term: setting $\epsilon =1$ with $P>0$ forces $a<0$ and a forward bifurcation, so a merely leaky vaccine, absent reinfection, does not by itself produce bistability. (iii)Strong immunity waningenters through P: large ω drives P downward—possibly negative—which can produce bistability even at $\epsilon =1$, though ω also appears in the denominator of (14), so its net effect is non-monotone and is resolved numerically in Section 7.

Figure 2 confirms the analysis numerically. With leaky immunity ($\epsilon =0.35<{\epsilon }^{*}$) the endemic branch bends back below ${\mathcal{R}}_0=1$, producing a region of bistability in which a stable endemic state coexists with the stable DFE; with solid immunity ($\epsilon =1$) the bifurcation is forward and ${\mathcal{R}}_0<1$ guarantees elimination.

6. Seasonal Forcing and the Origin of Recurrence

We now let the transmission rate vary seasonally,

$$\beta \left(t\right)={\beta }_0\left[1+{\beta }_{1}cos(2\pi t+\psi )\right],0\le {\beta }_1<1,$$

(15)

with t in years, and ask what generates the multi-annual recurrence observed in the belt.

The reproduction number of the forced system. 

For a periodic environment the threshold quantity is not the autonomous ${\mathcal{R}}_0$ evaluated at ${\beta }_0$, but the basic reproduction number of the periodic system in the sense of Bacaër and Guernaoui [2], defined as the spectral radius of the next-infection operator on the space of 1-periodic functions. Because the carriage subsystem is scalar, this operator reduces to the integral kernel $\mathcal{K}(t,s)=\beta \left(t\right)\Phi e^{-K_C(t-s)}$ and ${\mathcal{R}}_0^{\mathrm{per}}$ is the principal eigenvalue of the associated integral equation. For the sinusoidal forcing (15) the naive time-average $\langle \beta \rangle ={\beta }_0$ systematically misestimates the threshold, because the exponential memory of carriage weights the high-transmission season unequally; the discrepancy grows with ${\beta }_1$ and is non-negligible at belt seasonal amplitudes. Stability of the disease-free periodic state is determined by the Floquet exponent $\Lambda ={\int }_0^1[\beta \left(t\right)\Phi -K_C]\mathrm{d}t=K_C({\mathcal{R}}_0-1)$ for the scalar carriage equation linearised at $E_0$; the DFE periodic orbit is stable iff ${\mathcal{R}}_0<1$, consistent with Theorem 1.

Attractor structure. 

Beyond threshold, the interaction of annual forcing with the slow immunity timescale $1/\omega$ generates a hierarchy of attractors through period-doubling and subharmonic resonance. Figure 3 contrasts two regimes at identical mean transmissibility (${\mathcal{R}}_0\approx 1.35$) and seasonal amplitude (${\beta }_1=0.55$): under fast waning the system locks to a simple annual cycle, whereas under slow waning—immunity persisting several years, as in the belt—it settles onto a multi-annual attractor with large epidemics separated by quieter years, reproducing the qualitative signature of belt surveillance.

Figure 4 maps the attractor type over the $({\beta }_1,\omega )$ plane. Multi-annual and irregular dynamics occupy a band of slow waning; for fast waning the system remains annual at every seasonal amplitude. This is the bifurcation-theoretic counterpart of the simulation result of Irving et al. [7]: annual forcing alone cannot produce multi-annual recurrence—waning immunity within a bounded band is required.

7. Vaccination Scenarios and Burden

We evaluate four programmatic scenarios against a no-vaccination baseline, holding the transmissibility ${\beta }_0$ fixed and forcing seasonally with ${\beta }_1=0.5$: (1) no vaccination; (2) routine infant immunisation at rate $\phi =0.12{\mathrm{yr}}^{-1}$, corresponding to the SAGE-recommended 9–18-month schedule [13]; (3) routine immunisation plus a one-off mass catch-up campaign reaching $60\%$ of susceptible and recovered individuals, representing the recommended 1–19-year catch-up; and (4) routine immunisation with a higher carriage efficacy ($\sigma =0.85$). Cumulative invasive cases and deaths over a 25-year horizon are computed with case fatality following (3).

The results in Figure 5 and Table 2 show three things. Routine infant immunisation alone reduces 25-year burden by roughly two-thirds relative to baseline but does not interrupt recurrence, because it protects new cohorts slowly while the carriage reservoir persists. A one-off mass catch-up campaign, by contrast, transiently drives carriage prevalence below the transmission threshold and collapses cumulative burden by more than an order of magnitude—the numerical expression of the fact that carriage, not disease, is the reservoir. Improving carriage efficacy $\sigma$ substitutes partially for a campaign. Crucially, whether sustained routine immunisation eliminates epidemics or merely lengthens the inter-epidemic interval depends on whether the programme moves the system across the carriage-efficacy threshold ${\epsilon }^{*}$ of Theorem 2: below it, bistability means that even ${\mathcal{R}}_0<1$ need not eliminate the endemic state, and a campaign is needed to reach the disease-free basin.

The global sensitivity analysis in Figure 6 ranks the drivers of endemic carriage prevalence, and every sign accords with the model’s analytical structure. Endemic carriage is governed most strongly by the carriage-clearance rate $\alpha$ (PRCC $\approx -0.87$) and the transmission rate ${\beta }_0$ ($\approx +0.85$), the numerical signature of Proposition 2: since ${\mathcal{R}}_0\approx {\beta }_0/(\alpha +\mu )$, carriage duration and contact rate dominate burden, while the invasion and recovery parameters, acting only on the dynamically inert disease compartment in Corollary 1, do not register. Among controllable parameters, routine coverage $\phi$ ($\approx -0.60$) and vaccine efficacy against carriage $\sigma$ ($\approx -0.55$) are both strongly protective and of comparable magnitude, confirming that the conjugate vaccine acts by suppressing the carriage reservoir and that coverage and carriage efficacy are near-substitutable levers; vaccine waning $\theta$ enters positively ($\approx +0.28$), as expected. The carriage-blocking immunity $\epsilon$ has only a modest coefficient ($\approx -0.13$) yet governs the qualitative existence of the backward bifurcation (Theorem 2), so it should be read through the threshold ${\epsilon }^{*}$ rather than its PRCC; natural-immunity waning $\omega$ is near-zero, consistent with its non-monotone role in Section 5, and its importance for recurrence is better seen in Figure 4. Signs and rank order are robust, though magnitudes shift slightly with sample size and seed.

8. Discussion

We have analysed a carriage-structured model of meningococcal meningitis whose minimal structure is nonetheless faithful to two defining features of the disease: transmission is carried by asymptomatic carriage, and invasive disease is a rare dead end. Three structural conclusions follow. First, the reproduction number is a property of carriage and is decoupled from disease incidence, so reproduction numbers estimated from case data measure the wrong process. Second, the model can exhibit backward bifurcation, and we obtained the condition in closed form: it is governed by a threshold ${\epsilon }^{*}$ on carriage-blocking immunity, while the invasive-disease compartment—and therefore all case-management interventions—is provably absent from it. Third, under seasonal forcing the multi-annual recurrence characteristic of the belt arises only when immunity wanes within a bounded band, giving a bifurcation-theoretic account of the earlier simulation finding of Irving et al. [7].

These results speak directly to prevention. The parameter that the recent multivalent-vaccine modelling literature identifies as the critical unknown—vaccine effectiveness against carriage [10]—is precisely the quantity $\sigma$, and together with the immunity parameter $\epsilon$ it determines whether the disease-free state is globally attracting or merely one of two stable states. Where $\epsilon <{\epsilon }^{*}$, driving ${\mathcal{R}}_0$ below unity through routine coverage is not sufficient for elimination, and a mass campaign that carries the system into the disease-free basin becomes necessary rather than merely helpful. This gives a dynamical rationale for the SAGE recommendation that routine introduction be paired with catch-up campaigns in high-risk districts [13], and it reframes the “eliminate versus postpone” question as a question about a measurable carriage-immunity threshold.

Limitations. 

The model is deterministic and homogeneously mixing; it omits age structure, the spatial wave dynamics of belt epidemics, and serogroup competition, each of which matters for specific questions. The invasion rate $\rho$ and the carriage-immunity parameter $\epsilon$ are weakly identifiable from routine surveillance, which is syndromic and only partially serogroup-confirmed; the scenario magnitudes are therefore illustrative rather than calibrated, and should not be read as country-specific forecasts. A natural extension is a multi-strain carriage model addressing serogroup replacement after MenAfriVac and the ecological question of whether pentavalent vaccination closes the niche or selects for serogroup B or non-groupable strains. A second extension is stochastic, to capture the establishment probability of a spillover and the timing of reactive campaigns relative to the epidemic threshold.

9. Conclusions

Meningococcal recurrence in the African meningitis belt is, in this model, a resonance between annual seasonal forcing and slow immunity dynamics, sitting atop a carriage system whose elimination can be obstructed by bistability when carriage-blocking immunity is weak. Because the invasive case is a dead end, both the reproduction number and the bifurcation that determines elimination are properties of carriage alone. The practical implication is consistent across the analysis: measuring and improving vaccine efficacy against carriage, and using mass campaigns to carry the reservoir across the threshold, are the levers that determine whether conjugate-vaccine programmes end belt epidemics or only defer them.