Abstract
This study develops and analyzes a deterministic compartmental model to understand the transmission dynamics of Neisseria meningitidis under endemic conditions. Compartment profile plots illustrate the progression of the disease to an endemic level over time. Explicit expressions for the endemic equilibrium points and the basic reproductive number are derived. The basic reproductive number is decomposed into three interpretable components: contributions from (i) carriers, (ii) carriers progressing to symptomatic cases, and (iii) direct symptomatic cases (i.e., infectives bypassing carriership), highlighting the role of both carriers and infectives in long-term disease persistence. Sensitivity analysis identifies the transmission rate, the odds in favor of carrier infectivity, and the rate of loss of carriership as critical parameters influencing the basic reproductive number. Phase-plane plots (S–C and S–I) illustrate how variation in the basic reproductive number shapes the trajectory of convergence to the endemic equilibrium. Stability analysis shows that the endemic equilibrium is unstable when the product of the basic reproductive number and the susceptible population at equilibrium exceeds unity. Using Lyapunov functions and LaSalle’s invariance principle, we further show that the long-term dynamics are fully determined by this threshold: the disease-free equilibrium is globally asymptotically stable when the basic reproductive number is less than one, whereas a unique endemic equilibrium emerges and is globally stable when it is greater than one, demonstrating the persistence or elimination of meningitis depending on the threshold.
Supplementary Information
The online version contains supplementary material available at 10.1186/s12879-026-13589-9.
Keywords: Endemic meningitis, Stability analysis, Endemic equilibrium, Sensitivity analysis, Phase plots, Basic reproductive number, Next generation matrix
Introduction
Meningitis, an acute inflammation of the meninges, remains a serious global public health concern due to its high mortality rates and severe neurological sequelae among survivors [1]. The disease can be caused by a range of pathogens, including bacteria, viruses, and fungi, with bacterial meningitis posing the greatest threat due to its rapid progression and epidemic potential [2, 3]. Among bacterial agents, Neisseria meningitidis is a leading cause of meningitis epidemics, especially in the African meningitis belt, a region stretching from Senegal to Ethiopia that experiences recurrent seasonal outbreaks [4, 5].
The dynamics of meningitis transmission are influenced by a complex interplay of climatic, demographic, and behavioral factors. Seasonal variations in temperature, humidity, and dust levels have been strongly associated with increased disease incidence, particularly during the dry season [6, 7]. Furthermore, the disease’s epidemiology is modulated by age-specific susceptibility, with the highest burden observed among children under five years, adolescents, and young adults [8]. In endemic settings, asymptomatic carriage plays a critical role in maintaining transmission, with carriers serving as reservoirs for the spread of pathogenic strains [9–11].
Given the significant public health burden of meningitis, mathematical models have emerged as vital tools for understanding the disease’s dynamics and informing control strategies. Mathematical models allow for the exploration of key epidemiological factors, the prediction of disease outcomes under various intervention scenarios and the identification of critical thresholds for disease control [12, 13].
Endemic models are particularly important for characterizing the conditions under which meningitis persists within a population and for evaluating the effectiveness of long-term intervention strategies. Deterministic models have been widely employed to study meningitis dynamics in endemic settings [14–19]. Most of these studies focus on assessing the impact of vaccination strategies, often incorporating age structure into the modeling framework. More recently, Mustapha et al. [20] integrated the effects of carrier diagnosis and prophylactic treatment on meningitis transmission dynamics.
In contrast, stochastic and deterministic models developed under epidemic settings in [11, 21] excluded demographic turnover and long-term endemic behavior. These works extended the epidemic framework of Irving et al. [17] by incorporating two disease stages (infection with and without complications), allowing a fraction of susceptible individuals to transition directly to the infectious class without passing through the carrier state, and accounting for differential transmission rates between carriers and infectives.
In this study, we develop a deterministic model for meningitis endemicity, building on the framework of [11]. The primary objective is to elucidate the key mechanisms governing the transmission dynamics, stability, and long-term persistence of meningitis under endemic conditions. Specifically, we aim to:
-
(i)
establish conditions for the local and global stability of the endemic equilibrium,
-
(ii)
characterize the long-term persistence of meningitis in the population, and
-
(iii)
compare two model formulations to assess the impact of alternative infection pathways on equilibrium behavior and the convergence of solution trajectories.
The model
Mathematical models are essential for elucidating disease transmission mechanisms, enabling the description, prediction, and analysis of epidemic behavior and long-term dynamics. This study employs a compartmental framework to model the dynamics of meningococcal infection, balancing mathematical simplicity and analytical tractability. As depicted in Fig. 1, the population is divided into six epidemiological compartments: susceptible (
), asymptomatic carrier (
), symptomatic infective (
), infective with complications (
), recovered/immune (
), and meningitis-related deaths (
). Disease transmission is driven by the interactions between the 
and 
compartments, as individuals in 
are incapacitated due to disease severity. Susceptible individuals (
) acquire infection through infective contact with carriers (
) or infectious individuals (
) at a transmission rate 
. The force of infection, 
, is given by:
 |
1 |
Fig. 1.
Compartmental diagram for endemic meningitis transmission process
In this model, 
represents the odds ratio favoring the infectious potential of carriers compared to symptomatic individuals, while 
denotes the total population size. Demographic parameters of birth (
) and natural death (
) rates are introduced to reflect the endemic nature of the disease. The compartmental diagram and the equations governing the dynamics are provided below. Birth and death rates are set equal (
) to ensure a constant population size 
throughout time,t, under consideration.
 |
2 |
where
 |
Dividing the equations by 
converts them into proportions and the system of equations are given below.
 |
3 |
where
 |
Table 1 presents the definitions, values, and sources of all model parameters.
Table 1.
Description of parameter values used in the model and their possible range of values
| S/No. | Parameter | Description | Values / Range ( ) |
Source |
|---|---|---|---|---|
| 1 |  |
Transmission rate | 25–200 | Irving et al. [17] |
| 2 |  |
Odds in favor of carrier infecting a susceptible over an infective | 0.5–10 | [11] |
| 3 |  |
Proportion of  moving to 
|
0–1 | [17] |
| 4 |  |
Rate of loss of carriership | 1–52 | [17] |
| 5 |  |
Rate at which a carrier becomes stage one infective |  |
[17] |
| 6 |  |
Rate at which stage one infective develops complication |  |
[1], |
| 7 |  |
Recovery rate of stage one infective | 20–52 | [17] |
| 8 |  |
Recovery rate of individuals with complication | 20–52 | Assumed |
| 9 |  |
Rate at which recovered individuals lose immunity | 0.04–2 | [17] |
| 10 |  |
Disease-induced death rate |  |
[17] |
Two special cases of model Eq. (2) are of interest here: Model I (
) and Model II (
). Model profile plot for proportions in each of the compartment for various values of demographic and epidemiological parameters for models I and II are given in Fig. 2.
Fig. 2.
Proportions of individuals in each compartment under varying demographic parameter rates (
and 
). The rows correspond to 
(top), 
(second), 
(third), and 
(bottom). Initial conditions: 
Figure 2 indicates that the introduction of the demographic parameters has allowed the progression of the transmission of the disease to endemic level over time, irrespective of the variation in the birth (=death) rates and the two models considered. The drift to equilibrium level is particular obvious for Susceptible, Carrier, Infective and Infective stage one profiles. These prompt a suspicion of convergence to an equilibrium point. It must be noted however that the convergence to equilibrium point is not sufficient for asymptotic stability [22].
There are obvious differences in the trajectory of the profiles exhibited for models I and II. These are listed below.
-
i.
Time to equilibrium is very much shorter for Model II than Model I
-
ii.
Model I approaches equilibrium with noticeable damped oscillation not present in model II.
-
iii.
Values at equilibrium point are remarkably higher for model II in the recovered and death compartments and slightly so for carrier, infective and infective stage one compartments.
-
iv.
For the susceptible compartment, it is the values for model I that are remarkably higher.
These are the underlying differences in the profiles of Model I and Model II that should be noted particularly when Model II is intended to be used as an approximating system to model I for inferential purpose.
The basic reproductive number
The basic reproductive number, 
, is a fundamental threshold parameter in epidemiology, determining the conditions for endemic stability and the potential for epidemic outbreaks. It is defined as the expected number of secondary cases generated by a single infected individual in a fully susceptible population. The derivation of 
follows the next-generation matrix framework proposed by Diekmann et al. [23], which offers a robust mathematical approach for compartmental models. This method involves decomposing the system into a transmission matrix (
), representing the rates of new infections, and a transition matrix (
), describing the progression through infection compartments. Here these matrices are expressed as follows:
 |
4 |
and
 |
5 |
The next generation matrix, 
, is obtained by calculating 
, where 
is the inverse of the transition matrix. The matrix 
contains the rate at which new infections are produced and their transitions through different states. To calculate 
, we take the spectral radius (the largest eigenvalue) of the matrix 
.The inverse of the transition matrix 
is given by
 |
and
 |
6 |
The spectral radius of 
, which corresponds to the basic reproductive number 
, is given by
 |
7 |
This formulation highlights 
as the sum of three components, each representing a distinct pathway contributing to the transmission of infection within the population. The expression can be rewritten to emphasize specific transmission pathways, as
 |
8 |
where:
 |
Notice the similarity between the expression for 
here and that for the epidemic case given (equation 2b) in Yaga et al. [11] as
 |
9 |
Interpretation of the components of 
-
i)
: This is the number of secondary infections produced by a carrier during the average duration time of carriership. -
ii)
: This is the number of secondary symptomatic infections that passed through carriership that are produced by a symptomatic infective during an average duration of an infectious period. -
iii)
: This is the number of secondary symptomatic infectives that become infective directly, without passing through carriership produced by a symptomatic infective during an average duration of an infectious period.
When 
, the disease persists and establishes endemicity. The components of 
quantify contributions from different stages of infection to overall transmission dynamics in moving towards an established endemicity level. The rate of this progression is clearly determined by contributions from carriers and symptomatic infectives. This understanding can provide an informed decision in the design of any containment strategies aimed at reducing the spread of meningitis. This will inevitably entail making 
.
Sensitivity analysis
It must be mentioned here that the focus is on the influence of the parameters in the model on 
. While a discussion of 
would provide a clearer understanding of how its components are influenced, this is not the primary concern here. From Table 2, it is clear that 
or any of its components is not sensitive to 
(birth/death rate). This is a clear indication that the epidemic (Eq. (9), [11]) and the endemic (Eq. (8)) 
, provide similar conclusions about the understanding of the transmission of meningitis.
Table 2.
Sensitivity analysis for parameters of 
and its components 
, 
, and 
.This sensitivity indices are derived using the normalized forward sensitivity analysis method
| Param |  |
 |
 |
 |
 |
 |
 |
 |
|---|---|---|---|---|---|---|---|---|
| Response |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
— | — |  |
 |
 |
— |  |
 |
 |
 |
 |
 |
 |
 |
— | — | — |  |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
The parameters with the highest indices for endemic 
are 
and 
with 
and 
each having positive values, while the value for 
is negative. This implies that when 
and 
are respectively increased, endemic 
will increase, while it will decrease when 
is increased. The epidemic 
on the other hand has only 
and 
with positive indices. Notice that there is an additional sensitive parameter, 
, for the endemic 
over that for the epidemic. Consequently the distinction between the two 
have been clearly identified.
The values of endemic 
for various values of 
and 
are presented in Table 3. It is noticed that as the value of 
decreases the value of 
increases irrespective of 
considered. In particular, for 
for all values of 
considered. That is, in this cases the disease will persist and progress to endemicity.
Table 3.
Effect of 
and 
on 
, for 
, 
, 
, 
, 
, 
, 
 |
|||||
|---|---|---|---|---|---|
 |
0.5 | 1 | 2 | 3 | 4 |
| 0.5 |  |
 |
 |
 |
 |
| 0.5 |  |
 |
 |
 |
 |
| 10 |  |
 |
 |
 |
 |
| 15 |  |
 |
 |
 |
 |
| 20 |  |
 |
 |
 |
 |
| 25 |  |
 |
 |
 |
 |
| 30 |  |
 |
 |
 |
 |
As 
denote the rate of carrier conversion to a susceptible state and 
denotes the odd in favor of a carrier transmitting infection, Table 3 clearly establishes the significant role carrier state plays in driving the disease transmission to endemicity as was also noted in the epidemic case [11] This is not surprising as carriers serve as reservoirs for the spread of pathogenic strains. Consequently the carrier state must be approximately targeted for any control intervention to be successful.
Endemic equilibrium
Endemic models typically incorporate demographic processes, capturing disease persistence in a long-term within populations. To analyze system behavior at equilibrium, we set the equations in (2) to zero and solve for the state variables satisfying these conditions. These variables define the endemic equilibrium points. For the disease-free equilibrium, the system yields a trivial solution [13]. The disease-free equilibrium point for the system in (2) is denoted by
 |
10 |
To establish the endemic-equilibrium point is not a trivial exercise. Setting the system of Eq. (2) to zero we have
 |
11 |
For simplicity, let 
, 
and 
. Due to the complexity of the system of equations we use Maple Version 18 to solve the system of equation in (11). This gives the endemic equilibrium point as
 |
The solution reveals the existence of an endemic equilibrium point. This point is characterised using S-C and S-I phase plots presented in Fig. 3. A comparison of these plots indicate no observable distinction between the two for a given 
. However, notable differences emerge in the pattern of the spirals as the basic reproduction number 
approaches one. Specifically, when 
, the phase plots display damped oscillations that gradually converge to the endemic equilibrium. In contrast, when 
, the convergence to the endemic equilibrium occurs with very little observable oscillatory behavior. These characteristics are depicted clearly in Fig. 3.
Fig. 3.
Phase portraits in the proportion of susceptible and carrier (
–
) and susceptible and infective (
–
) planes for varying values of the basic reproduction number 
and 
. The trajectories illustrate convergence toward a unique endemic equilibrium. For 
, the system exhibits damped oscillatory behavior before settling at equilibrium, whereas for 
, convergence occurs with minimal oscillations, indicating a stronger attracting endemic state as transmission intensity increases
Figure 4 shows the plot of profiles for S, C, and I compartments for varying values of 
. In Table 4 we are able to extract estimates from Fig. 4 that provide a good epidemiological insight on the endemic equilibrium point. It is clear from the table that the endemic equilibrium point is reached faster as 
is greater than one. It is noticeable that for a high 
(=3.9575) equilibrium point is reached only after ten years with only 15% of the susceptible population remaining. This proportion of susceptible population left is not sufficient to start an epidemic. However, with the influx of new susceptible from birth it can trigger persistence of the disease at low endemic level.
Fig. 4.
Time profiles of the proportion of susceptible (
), carrier (
), and infectious (i) compartments for different values of the basic reproduction number 
and 
. Increasing values of 
lead to faster convergence toward the endemic equilibrium and a lower equilibrium proportion of susceptibles, reflecting enhanced transmission and sustained endemic persistence
Table 4.
Epidemiological characteristics of the endemic equilibrium for different values of the basic reproduction number 
, including equilibrium time 
and compartmental proportions at equilibrium
 |
 |
 |
 |
 |
 |
 |
 |
|---|---|---|---|---|---|---|---|
| 0.7659 | 50 | 0.8984077 | 0.000274101 | 0.00016677 | 0.00019358 | 0.046189537 | 0.10019298 |
| 1.0000 | 40 | 0.70346581 | 0.001241955 | 0.00055723 | 0.00064660 | 0.152043380 | 0.23011395 |
| 1.2166 | 20 | 0.39537551 | 0.003878285 | 0.00143057 | 0.00165958 | 0.39077402 | 0.26891620 |
| 3.9575 | 10 | 0.14469764 | 0.016824061 | 0.00230515 | 0.002675064 | 0.63674375 | 0.23544029 |
Stability analysis
In theoretical epidemiology, stability analysis plays a crucial role in understanding the long-term behavior of disease dynamics. It focuses on whether solutions to dynamical system models converge to equilibrium points, indicating whether a disease will die out, persist, or reach a stable endemic state. Smooth convergence suggests a stable balance between disease transmission and control measures, while oscillatory behavior may signal periodic disease outbreaks influenced by complex population interactions or external factors. Analyzing these dynamics provide valuable insights into epidemic control strategies and the conditions required for disease eradication or sustained management.
The stability of equilibrium points are determined analytically [13]) using the Jacobian of the system of differential equation in (3), given as:
 |
12 |
We first consider the stability of the endemic equilibrium point 
as that of the disease equilibrium point 
is a special case of this.The Jacobian mattrix evaluated at the endemic equilibrium point 
is given by
 |
13 |
The eigenvalues of the Jaciabn (
) are
 |
14 |
 |
15 |
 |
16 |
 |
17 |
 |
18 |
 |
19 |
We simplify 
further.
 |
where
 |
where
 |
substituting into 
and 
we have
 |
and
 |
It is clear that the eigenvalue 
is positive only if 
. Consequently, the endemic equilibrium point 
is asymptotically unstable. By extension, the disease free equilibrium point, 
is also asymptotically unstable when 
. This implies that there is disease persistence in the long term due to renewed number of susceptibles through birth but that the endemic equilibtium point is asymptotically unstable to small perturbations when 
. This is another example that shows that convergence to an equilibrium point is not sufficient for asymptotic stability as pointed out in [22]. Of course if 
the disease free equilibrium is stable because the pathogen cannot invade if each infected host passes on the infection to fewer than one other host.
Global stability analysis
Global stability of the disease-free equilibrium
Global stability is commonly established through the construction of an appropriate Lyapunov function whose time derivative along solutions of the system is negative definite [24–26]. The use of linear Lyapunov functions involving infected compartments to establish global stability of the disease-free equilibrium (DFE) is well established in epidemic modeling Hethcote [27].
Let 
be a Lyapunov function defined as a linear combination of the infected states, namely the carrier class 
and the infectious class 
, given by
 |
20 |
where 
and 
are constants to be determined. Taking the time derivative of 
along solutions of the system yields
 |
21 |
Substituting the corresponding expressions for 
and 
from the original system of Eq. (2), we obtain
 |
22 |
where 
denotes the force of infection. Rearranging terms, we separate the transmission and transition components as follows:
 |
23 |
To relate this expression to the basic reproduction number 
, we equate the coefficient of the transmission term with the numerator of 
, and the transition terms with its denominator. This leads to the system
 |
24a |
 |
24b |
 |
24c |
Solving Eq. (24a)–(24c) yields
 |
25 |
which are clearly positive. Substituting these expressions together with the force of infection 
into Eq. (23), we obtain
 |
26 |
The expression can be factored in terms of 
only when 
, corresponding to the case where carriers and infectious individuals have equal transmission potential. This assumption is therefore crucial for establishing global stability.
Under this condition, factoring out 
yields
 |
27 |
From the positivity and boundedness result established in Theorem 1, the susceptible population satisfies 
. For a closed population where birth balances natural death (
), we have 
. Consequently, the above expression reduces to
 |
28 |
It follows immediately that 
if and only if 
. By LaSalle’s Invariance Principle [28], every solution of the system (2) with initial conditions in the positively invariant region 
(see supplementary sheet) approaches the disease-free equilibrium 
as 
.
Hence, the disease-free equilibrium 
is globally asymptotically stable in 
whenever 
.
The Lyapunov function establishing this result is given by
 |
29 |
Global asymptotic stability of the disease-free equilibrium is therefore guaranteed under the assumptions that carriers and infectious individuals have equal transmission potential (
) and that the population is demographically closed (
). Under these conditions, regardless of how far the initial conditions are from the disease-free equilibrium, the system converges to 
in the long run when 
; otherwise, the disease-free equilibrium is globally unstable.
Global stability of disease endemic equilibrium
The global stability of the endemic equilibrium point is determined by constructing a Lyapunov function, such that the derivative of this function is less than zero. The Lyapunov function defined as 
is the Goh-Voltera-type [27, 29]. Here is given as
 |
30 |
Taking first derivative of the Lyapunov function 
, we have
 |
31 |
substituting the right hand side of the system of equations in (2) into each component in Eq. (31) we have:
 |
32 |
we substitute the value of 
with its corresponding value at the endemic equilibrium point, as 
and substitute into (31), we have
 |
where 
. At endemic equilibrium point, 
. Using the normal procedure, we obtain the following for the remaining components of (30)
 |
substituting the inequalities into Eq. (31) and simplify, we have
 |
33 |
where 
and 
is collapsed by first and second component of (31). By direct calculations,
 |
similarly, we have
 |
Here, by substituting all the corresponding expressions into the Lyapunov function, and for a suitable choice of 
and 
, the Goh-Volterra type Lyapunov function satisfies 
, with equality 
holding in the limiting case.
 |
34 |
We set 
and 
in the expression for 
, which then becomes
 |
35 |
To ensure that 
, the equality 
holds if and only if
 |
Therefore, the endemic equilibrium point 
is the only positively invariant set in 
satisfying
 |
It then follows from LaSalle’s invariance principle [28] that every solution of the system in (2) with initial conditions in the invariant manifold 
(see Supplementary sheet) converges to the endemic equilibrium point 
as 
. Hence, the positive endemic equilibrium is globally asymptotically stable when 
, since the endemic equilibrium exists only for 
.
Conclusion
In this study, a six-compartment deterministic model proposed in Yaga and Saporu (2024) was extended to investigate the transmission dynamics of meningococcal meningitis in an endemic setting. Two model variants (Model I: 
and Model II: 
) were examined to compare disease dynamics under differing assumptions on infection pathways.
Compartmental profile plots under varying birth–death rates demonstrated the system’s tendency to converge toward an endemic equilibrium, with Model II exhibiting faster convergence and higher endemic levels in the recovered and death compartments compared to Model I. Notably, the presence or absence of damped oscillations in these trajectories further distinguished the two models, indicating differences in temporal disease dynamics.
An explicit expression for the basic reproductive number was derived and decomposed into three interpretable components: 
, representing the contribution from carriers; 
, accounting for carriers progressing to symptomatic infection; and 
, corresponding to direct symptomatic infections. Sensitivity analysis revealed that the basic reproductive number is most responsive to the transmission rate (
), the odds in favour of carrier infectivity (
), and the rate of loss from the carrier state (
). Notably, unlike epidemic settings, the endemic reproductive number also exhibits sensitivity to the carrier transmission pathway, further emphasizing the critical role of carriers in sustaining long-term disease persistence.
A detailed equilibrium analysis established the existence of both disease-free and endemic equilibrium points. Numerical investigations showed that the time required to reach the endemic equilibrium decreases as the basic reproductive number increases. Importantly, even when only a small fraction of the population remains susceptible at equilibrium, the continuous influx of new susceptibles through births ensures the persistence of the disease at low endemic levels.
Local stability analysis [13, 30], based on Jacobian matrices and eigenvalue decomposition, showed that the endemic and disease-free equilibria are asymptotically unstable when 
and 
, respectively, indicating that small perturbations can lead to divergent dynamics in these regimes. As expected, when 
, the disease dies out.
Using Lyapunov functions and LaSalle’s invariance principle, we further established that the global dynamics of the model are fully governed by the basic reproductive number: the disease-free equilibrium is globally asymptotically stable when the basic reproductive number is less than one, whereas a unique endemic equilibrium emerges and is globally asymptotically stable when the basic reproductive number exceeds one. This result rigorously confirms the threshold-driven persistence or elimination of meningococcal meningitis in the population.
Overall, this work provides a rigorous mathematical foundation for understanding endemic meningitis dynamics and offers valuable insights for targeted intervention strategies, particularly those aimed at asymptomatic carriers, to achieve long-term disease control and potential elimination.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Author contributions
S.J.Y. conceived and designed the study, developed the mathematical model, and performed the theoretical analysis. F.W.O.S. provided supervision, guidance on model formulation, and critical revisions to the manuscript. S.J.Y. wrote the first draft of the manuscript. Both authors contributed to interpretation of the results, reviewed the manuscript, and approved the final version for submission.
Funding
Not applicable.
Data availability
The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.
Declarations
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The author declares no competing interests.
Footnotes
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
The datasets generated and/or analysed during the current study are available from the corresponding author on reasonable request.