1. Introduction

Meningitis is an inflammation of the meninges, that are the membranes surrounding the brain and spinal cord [35,36]. Various different agents can cause meningitis [30, 37,38]. The most common ones are either bacterial or viral organisms. The presence of these organisms in the sub-arachnoid space (the space between the pia mater and the arachnoid mater occupied by the cerebrospinal fluid) causes an inflammatory reaction which involves all the structures that lie within or, are adjacent to these areas. Almost any bacteria gaining entrance to the meninges may produce meningitis. Bacterial meningitis is sometimes due to infection in another part of the body (lungs, ears, nose, throat, and sinuses) that spreads into the meninges. About 10% of the population carry the germs for days, weeks, or even months without becoming ill. Meningococcal carriage is most prevalent in teenagers and young adults, where up to 60% of the carriers are often recorded.

Neisseria meningitidis (Nm) is a leading cause of bacterial meningitis and other serious infections worldwide. Neisseria meningitidis serogroups B and C (NmB and NmC) are the most common cause of meningococcal disease in Europe, the Americas, Australia, and New Zealand. Cases tend to occur more frequently in winter and spring. Neisseria meningitidis serogroup A (NmA) is the main cause of disease in Africa and Asia. Worldwide, the highest rates of disease incidence are recorded in the ‘meningitis belt’ of sub-Saharan Africa [35,36]. This extends across the dry, savannah regions from Senegal in the west, across to Ethiopia in the east. During epidemics, this region can have an annual incidence rate of 1000 cases per 100,000 population [36]. The largest recorded outbreak of meningococcal disease in history occurred in Africa in 1996 when over 20,000 deaths were reported to the World Health Organization (WHO) [37]. In the 2009 epidemic season 14 African countries reported 78,416 suspected cases and 4053 deaths. This is the largest reported number since the 1996 epidemic [35]. More than 85% of these cases were reported in one epidemic foci, encompassing northern Nigeria and Niger, and were characterized by the predominance of NmA [38]. Understanding the epidemiology of meningococcal epidemics has long been limited by the lack of an adequate typing scheme capable of distinguishing epidemic strains of NmA.

The dynamics of NmA are complex due to a combination of various factors of societal order such as population dynamics or their susceptibility to virulent serotypes, but also climatic such as temperature and humidity and environmental factors such as the spatial distribution of vegetation type or the presence of desert aerosols (dust) [1,14,30,34]. A deep understanding of the disease dynamics would have a significant impact on an effective prevention and control strategies [1]. Mathematical modelling and numerical simulations have the potential, and offer a promising way, to achieve this [2,5,15,31]. Some efforts have been and are still being devoted to the modelling of this disease [6,11,20,22,28,29,32]. Martcheva and Cornell [20] studied an age-structured model of meningococcal meningitis with two disease-related classes: normal infective (infected who exhibit symptoms and effectively ill) and carriers (infected who have the bacteria but appear healthy). Tuckwell et al. [32] proposed a mathematical model to determine the impact of a schedule of vaccination on the time course of a certain class of diseases. The model was applied to NmC disease in France. Trotter et al. [6] proposed and developed a realistic, age-structured, dynamic model. They parameterized and fitted the model to epidemiological data from England and Wales. Mueller and Gessner [22] proposed an hypothetical explanatory model for epidemic meningococcal meningitis. In [28], deterministic compartmental models were fitted to age-structured data sets of NmC, Leptomeningeal carcinomatosis (LC) and meningitidis serogroup D (NmD). Coen et al. [11] presented a mathematical model for the dynamical transmission of Neisseria meningitidis and Neisseria lactamica. For the optimal control of meningitis, Tartof et al. [29] identified optimal vaccination strategies for Serogroup A Neisseiria meningitidis conjugate vaccine in the African meningitis belt. However, all of these models [6,11,20,22,28, 29,32] exhibit some drawbacks: they do not take into account the waning of the vaccine-induced immunity, the vaccine efficacy and the recover-induced immunity [25].

This paper builds on the existing works mentioned above and fills some of the gaps observed in these works. We formulate and analyse a mathematical model for NmA that incorporates the key epidemiological and biological features of the disease such as the vaccination against NmA, the waning of the vaccine-induced immunity, the vaccine efficacy, and the waning of the recovery-induced immunity. The main interest in studying the meningitis infection is to understand the long- and short-term behaviour of the disease and to predict whether the disease will die or persist within a community. The model is completely analysed. We derive the basic reproduction number ${\mathcal{R}}_0$ that determines the outcome of the disease. We compute equilibria and study their stability. More precisely, we establish that there exists a threshold parameter ξ such that if ${\mathcal{R}}_0\le \xi <1$, the disease-free equilibrium (DFE) is globally asymptotically stable, whereas the DFE becomes unstable when ${\mathcal{R}}_0>1$. However, when $\xi \le {\mathcal{R}}_0<1$, the model may exhibit the phenomenon of backward bifurcation, where a locally asymptotically stable DFE co-exists with a locally asymptotically stable endemic equilibrium. We also show that there exists a unique endemic equilibrium which is locally asymptotically stable when ${\mathcal{R}}_0>1$ but close to 1. A way of distributing the vaccines against NmA such as random mass vaccines, with distinct levels of efficacy and durability are also investigated. The theoretical results are illustrated using numerical simulations. The sensitivity analysis of the model is carried out to identify the most influential parameters on the model output variables.

The paper is organized as follows. After the formulation of the model in Section 2, we present its quantitative and qualitative analysis in Section 3. Theoretical results are illustrated by numerical simulations. A sensitivity analysis of the model is provided in Section 4. The last section is devoted to concluding remarks on how our work fits in the literature and on possible extensions.

2. Model formulation

The dynamics of NmA is governed by the following set of biological assumptions: (i) newborns who are protected from the infection for an average period of 3 months due to maternal antibody are assumed to be susceptible to the infection; (ii) the vaccine-induced immunity acquired by vaccinated individuals is supposed to wane; (iii) vaccinated individuals whose vaccination is unsuccessful can be infected by bacteria, and (iv) the recovery does not confer a permanent immunity to infected individuals with or without symptoms.

We consider five distinct populations, according to their disease status: susceptible S (individuals who are susceptible to the disease), vaccinated individuals V (healthy individuals who have been vaccinated acquiring temporal immunity), carriers C (healthy individuals who carry NmA and are infectious), infectious I (infected individuals who show the symptoms of the infection), and recovered individuals R (self-recovered or individuals who have cleared the infection after a treatment acquiring a temporal immunity). Thus, the total population $N\left(t\right)$ at time t is (1) $N\left(t\right)=S\left(t\right)+V\left(t\right)+C\left(t\right)+I\left(t\right)+R\left(t\right).$(1) Individuals are recruited into the population at constant rate Λ. The vaccinated population increases by the recruitment of individuals who are vaccinated at a constant rate $\pi \mathrm{\Lambda }$ where π is the proportion of vaccinated recruitment, while the complementary proportion $(1-\pi )\mathrm{\Lambda }$ is not protected and enters the class of susceptible individuals S.

A mass vaccination programme may be initiated whenever there is an increase of the risk of an epidemic. For most human diseases, it is possible and more efficient, to not vaccinate those individuals who have recovered from the disease, because they are already protected. Another situation could be the introduction of a vaccine in a population living in an endemic situation. We suppose that a fraction $0\le \rho \le 1$ of the entire susceptible population is being continuously vaccinated. Thus, the population of vaccinated individuals is increased by the vaccination of susceptible individuals at constant rate ρ. Since the vaccine does not confer a total immunity to all vaccine recipients, vaccinated individuals lose their immunity and return to the susceptible class S at a constant rate θ.

Most of the theory about disease evolution is based on the assumption that the host population is homogeneous. Individual hosts, however, may differ and they may constitute very different habitats. In particular, some habitats may provide more resources or be more vulnerable to bacteria exploitation [13]. The use of models with imperfect vaccines can describe better this type of human heterogeneity. The vaccination may reduce but not completely eliminate susceptibility to infection. For this reason, we consider a factor ν as the vaccine efficacy. When $\nu =1$, the vaccine is perfect while, when $\nu =0$, the vaccine has no effect at all. The value $1-\nu$ can be understood as the inefficacy level of the vaccine. Also, a majority of the available vaccines for the human population does not produce 100% success in the disease battle [12,23–25]. Usually, the vaccines are imperfect, which means that a minor percentage of cases, in spite of vaccination, are infected [12, 23].

The susceptible and vaccinated populations are decreased due to the NmA infection at rates $\lambda S$ and $(1-\nu )\lambda V$, respectively where λ is the force of infection given by (2) $\lambda =\frac{\beta (C+\epsilon I)}{N},$(2) with β the effective contact rate for NmA transmission and $0<\epsilon <1$ the modification parameter accounting the fact that infectious (those in the I class) are less infectious than carriers (those in the C class). In fact the transmission potential of the later is higher because they can freely establish contacts with susceptible individuals since they may not be aware of their disease status.

The population of carriers is increased by the infection of susceptible and vaccinated individuals at rates $\lambda S$ and $(1-\nu )\lambda V$, respectively, and is diminished by natural death, the progression to the infectious stage of the disease (moving to the infectious class I) and the recovery from the disease (moving to the recovered class R) at constant rates μ, $q(1-\gamma )$, and $\gamma q$, respectively where q is the rate moving from carriers to recover and γ the probability that carriers clear the infection. The population of infectious is increased by carriers who develop the disease at rate $q(1-\gamma )$ and is diminished by recovery from the disease (moving to the recovered class R), natural death and NmA induced mortality at constant rates α, μ, and d, respectively. The population of recovered individuals is increased by the recovery of carriers and infectious at rates $q\gamma$ and α, respectively, and is decreased as recovered individuals can lose their immunity (moving to the susceptible class S) or succumb to natural death at rates δ and μ, respectively.

A schematic model flowchart is depicted in Figure 1. From this, the NmA transmission model is described by the following system of nonlinear ordinary differential equations: (3) $\begin{array}{rl}\dot{S}& =(1-\pi )\mathrm{\Lambda }+\delta R+\theta V-(\lambda +\rho +\mu )S,\\ \dot{V}& =\pi \mathrm{\Lambda }+\rho S-\left[\right(1-\nu )\lambda +\theta +\mu ]V,\\ \dot{C}& =[S+(1-\nu \left)V\right]\lambda -(q+\mu )C,\\ \dot{I}& =q(1-\gamma )C-(\mu +d+\alpha )I,\\ \dot{R}& =q\gamma C+\alpha I-(\delta +\mu )R,\end{array}$(3) where λ is the force of infection defined as in Equation (2).

Mathematical analysis of the dynamical transmission of Neisseria meningitidis serogroup A

All authors

G. Djatcha Yaleu, S. Bowong, E. Houpa Danga & J. Kurths

https://doi.org/10.1080/00207160.2017.1283411

Published online:

08 February 2017

Figure 1. Structure of the model.

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Table 1 summarizes the model variables and parameters.

3. Mathematical analysis

3.2. The DFE and its stability

The DFE for an epidemiological model is an equilibrium such that the disease is absent in the community. Thus, if $Q_0=(S_0,V_0,C_0,I_0,R_0)$ is the DFE of model system (3), then $C_0=I_0=0$. As a consequence of model system (3), $R_0=0$ with $S_0$ and $V_0$ being solutions of the system: (7) $\begin{array}{rl}(1-\pi )\mathrm{\Lambda }+\theta V_0-(\rho +\mu )S_0& =0,\\ \pi \mathrm{\Lambda }+\rho S_0-(\theta +\mu )V_0& =0,\end{array}$(7) which has the unique solution: (8) $S_0=\frac{\mathrm{\Lambda }(\theta +(1-\pi \left)\mu \right)}{\mu (\rho +\theta +\mu )},V_0=\frac{\mathrm{\Lambda }(\pi \mu +\rho )}{\mu (\rho +\theta +\mu )}\mathrm{and}{N}_0=S_0+V_0=\frac{\mathrm{\Lambda }}{\mu }.$(8)

In order to investigate the stability properties of the DFE $Q_0$, we need to compute the reproduction/threshold number ${\mathcal{R}}_0$ of model system (3). To this end, we apply the method in Van den Driessche and Watmough [33], with $(C,I,R)$ and $(S,V)$ being the infected and uninfected classes, respectively. We point out that the noninfected classes are the classes of individuals who cannot carry the bacteria in their body, while the infected classes are the classes of individuals who carry the bacteria in their body. Using the notations in [33], the matrices $\mathcal{F}$ and $\mathcal{V}$, for the new infection and the remaining transfer, are, respectively, given by $\mathcal{F}=\left(\begin{array}{l}[S+(1-\nu \left)V\right]\lambda \\ 0\\ 0\end{array}\right)\text{and}\mathcal{V}=\left(\begin{array}{c}(q+\mu )C\\ -q(1-\gamma )C+(\mu +d+\alpha )I\\ -q\gamma C-\alpha I+(\delta +\mu )R\end{array}\right).$ The Jacobian matrices of $\mathcal{F}$ and $\mathcal{V}$ at the DFE $Q_0$ are respectively $D\mathcal{F}\left(Q_0\right)=\left[\begin{array}{cc}F& 0\\ 0& 0\end{array}\right]\text{and}D\mathcal{V}\left(Q_0\right)=\left[\begin{array}{cc}\overline{V}& 0\\ V_1& V_2\end{array}\right],$ where $F=\left[\begin{array}{cc}\frac{\beta [S_0+(1-\nu \left)V_0\right]}{N_0}& \frac{\beta \epsilon [S_0+(1-\nu \left)V_0\right]}{N_0}\\ 0& 0\end{array}\right]\text{and}\overline{V}=\left[\begin{array}{cc}(q+\mu )& 0\\ -q(1-\gamma )& (\mu +d+\alpha )\end{array}\right].$

Then, the reproduction number ${\mathcal{R}}_0$ of model system (3) is the spectral radius of the next generation matrix $F{\tilde{V}}^{-1}$, i.e (9) ${\mathcal{R}}_0=\rho \left(F{\overline{V}}^{-1}\right)=\frac{\beta [S_0+(1-\nu \left)V_0\right](\mu +d+\alpha +q\epsilon (1-\gamma \left)\right)}{N_0(q+\mu )(\mu +d+\alpha )}.$(9) The relevance of the basic reproduction number relies on the following result established in [33].

Proposition 3.2.

The DFE $Q_0$ is locally asymptotically stable in Ω if ${\mathcal{R}}_0<1$ and unstable if ${\mathcal{R}}_0>1$.

The biological implication of Proposition 3.1 is that, a sufficiently small flow of infected individuals will not generate an outbreak of the disease unless ${\mathcal{R}}_0>1$. For a better control of the disease, the global asymptotic stability (GAS) of the DFE is needed. Actually, enlarging the basin of attraction of $Q_0$ to be a part or the entire region Ω is for the model under consideration, a more challenging task involving relatively new result [6,20], as detailed below. We use a result of Kamgang and Sallet [17] for the global stability of the DFE for a class of epidemiological models.

Following Kamgang and Sallet [17], we write model system (3) in the following form: (10) $\begin{array}{rl}{\dot{x}}_1& =A_1\left(x\right)(x_1-x_0^1)+A_{12}\left(x\right)x_2,\\ {\dot{x}}_2& =A_2\left(x\right)x_2,\end{array}$(10) where $x_1=(S,V{)}^T$ denotes the noninfected classes (i.e. susceptible and vaccinated individuals), $x_2=(C,I,R{)}^T$ represents the infected classes (i.e. carriers, infectious, and recovered individuals), $x=(x_1,x_2{)}^T$, $x_0^1=(S_0,V_0{)}^T$ is the nonzero component of the DFE, $\begin{array}{rl}{A}_1\left(x\right)& =\left[\begin{array}{cc}-(\rho +\mu )& \theta \\ \rho & -(\theta +\mu )\end{array}\right],A_{12}\left(x\right)=\left[\begin{array}{ccc}-\frac{\beta S}{N}& -\frac{\beta \epsilon S}{N}& \delta \\ -\frac{\beta \nu V}{N}& -\frac{\beta \nu \epsilon V}{N}& 0\end{array}\right]\mathrm{and}\\ A_2\left(x\right)& =\left[\begin{array}{ccc}\frac{\beta [S+(1-\nu \left)V\right]}{N}-(q+\mu )& \frac{\beta \epsilon [S+(1-\nu \left)V\right]}{N}& 0\\ q(1-\gamma )& -(\mu +d+\alpha )& 0\\ q\gamma & \alpha & -(\mu +\delta )\end{array}\right].\end{array}$ The following conditions $H_1$– $H_5$ below must be met to guarantee the GAS of $Q_0$ [17].

1. Model system (10) is defined on a positively invariant subset $\mathcal{D}$ of Ω and is dissipative on $\mathcal{D}$.

2. The sub-system ${\dot{x}}_1=A_1(x_1,0)(x_1-x_0^1)$ is globally asymptotically stable at the equilibrium $x_0^1$ on the canonical projection of Ω on $\mathcal{D}$.

3. The matrix $A_2\left(x\right)$ is Metzler (A Metzler matrix is a matrix with off-diagonal entries non-negative  [3,16]) and irreducible for any given $x\in \mathcal{D}$.

4. There exists an upper-bound matrix ${\overline{A}}_2$ for $\mathcal{M}=\left\{A_2\right(x),x\in \mathcal{D}\}$ with the property that if $A_2\in \mathcal{M}$ (i.e. $A_2=\underset{\mathcal{D}}{max}\mathcal{M}$), then for any $\overline{x}\in \mathcal{D}$ such that ${\overline{A}}_2=A_2\left(\overline{x}\right)$, $\overline{x}\in \mathcal{D}\times \left\{0\right\}$ (i.e. the points where the maximum is realized are contained in the disease-free sub-manifold).

5. $\alpha \left({\overline{A}}_2\right)\le 0$ where $\alpha \left({\overline{A}}_2\right)$ denotes the largest real part of the eigenvalues of ${\overline{A}}_2$.

If the conditions $H_1$–$H_5$ are satisfied, then $Q_0^1$ is globally asymptotically stable in $\mathcal{D}$.

The result of the Kamgang–Sallet approach [17] uses the algebraic structure of model system (10), namely the fact that $A_1\left(x\right)$ and $A_2\left(x\right)$ are Metzler matrices. Since in the said approach, the matrix $A_2\left(x\right)$ is required to be irreducible, we further restrict the domain of the system to (11) $\mathcal{D}=\left\{\right(x_1,x_2)\in \mathrm{\Omega },x_1\ne 0\}.$(11) The set $\mathcal{D}$ is positively invariant because only the initial point of any trajectory can have $x_1=0$ (see Theorem 3.1). Indeed, from the first and second equations of model system (3), one has $S^{\prime }>0$ and $V^{\prime }>0$ whenever S=0 and V =0, respectively. Thus, (12) $A_2\left(x\right)\text{ is Metzler and irreducible for all}x\in \mathcal{D}.$(12) The sub-system: ${\dot{x}}_1=A_1(x_1,0)(x_1-x_0^1),$ can be expressed as (13) $\begin{array}{rl}\dot{S}& =(1-\pi )\mathrm{\Lambda }+\theta V-(\rho +\mu )S,\\ \dot{V}& =\pi \mathrm{\Lambda }+\rho S-(\theta +\mu )V.\end{array}$(13) System (13) can be written in the following compact form: (14) ${\overline{x}}^{\prime }=A_x\overline{x}+\mathrm{\Gamma },$(14) where $\overline{x}=(S,V{)}^T$, $A_x=\left(\begin{array}{cc}-(\mu +\rho )& \theta \\ \rho & -(\mu +\theta )\end{array}\right)\text{and}\mathrm{\Gamma }=\left(\begin{array}{c}(1-\pi )\mathrm{\Lambda }\\ \pi \mathrm{\Lambda }\end{array}\right).$ Resolving the above equation (14) yields (15) $\overline{x}\left(t\right)=x_1^0+Q{e}^{Dt}{Q}^{-1}\left[x\right(0)-x_0^1],$(15) where $Q=\left(\begin{array}{cc}\theta & -1\\ \rho & 1\end{array}\right),Q^{-1}=\frac{1}{(\theta +\rho )}\left(\begin{array}{cc}1& 1\\ -\rho & \theta \end{array}\right),D=\left(\begin{array}{cc}-\mu & 0\\ 0& -(\mu +\theta +\rho )\end{array}\right),$ and $Q{e}^{Dt}{Q}^{-1}=\frac{1}{\rho +\theta }\left(\begin{array}{cc}{e}^{-\mu t}[\omega +\rho e^{-(\rho +\theta )t}]& \theta e^{-\mu t}[1-e^{-(\rho +\theta )t}]\\ \rho e^{-\mu t}[1-e^{-(\rho +\theta )t}]& e^{-\mu t}[\rho +\theta e^{-(\rho +\theta )t}]\end{array}\right)>0.$ Taking the limit of Equation (15), when $t\to +\mathrm{\infty }$ yields (16) $\underset{t\to +\mathrm{\infty }}{lim}S\left(t\right)=S_0=\frac{\mathrm{\Lambda }(\theta +(1-\pi \left)\mu \right)}{\mu (\rho +\theta +\mu )}\text{and}\underset{t\to +\mathrm{\infty }}{lim}V\left(t\right)=V_0=\frac{\mathrm{\Lambda }(\pi \mu +\rho )}{\mu (\rho +\theta +\mu )}.$(16) Therefore, $x_0^1=(S_0,V_0)$ is a GAS equilibrium of the reduced system (13) on the sub-domain $\{x\in \mathcal{D},x_2=0\}$. Then, the hypothesis $H_2$ is satisfied.

The result of Kamgang and Sallet (see [17, Theorem 4.3] ) gives the GAS of the DFE of a dissipative system of the form (10) which satisfies (12) and (16) provided there exists a matrix $A_2\left(x\right)$ with the following additional properties: (17) $\begin{array}{rl}& A_2\left(x\right)\le {\overline{A}}_2,x\in \mathcal{D},\\ & \text{if}{A}_2\left(\overline{x}\right)={\overline{A}}_2\text{for some}\overline{x}=({\overline{x}}_1,{\overline{x}}_2{)}^T\in \mathcal{D}\text{then}{\overline{x}}_2=0,\\ & \alpha \left({\overline{A}}_2\right)\le 0.\end{array}$(17) Note that $S/N\le 1$ and $V/N\le 1$ in $\mathcal{D}$. With this in mind, the upper bound of the matrix $A_2$ is ${\overline{A}}_2=\left[\begin{array}{ccc}\beta (2-\nu )-(q+\mu )& \beta \epsilon (2-\nu )& \delta \\ q(1-\gamma )& -(\mu +d+\alpha )& 0\\ q\gamma & \alpha & -(\mu +\delta )\end{array}\right].$ The equality $A_2\left(x\right)={\overline{A}}_2$ is possible only when S=V =N which implies that $x_2=0$. Therefore, the first and second conditions in (17) hold. Note that ${\overline{A}}_2$ is a Metzler matrix which satisfies the stability condition of Kamgang and Sallet [17].

Since $-(\mu +\delta )<0$ is an eigenvalue of ${\overline{A}}_2$, the stability of the matrix ${\overline{A}}_2$ is associated to the stability of the following sub-matrix ${\overline{A}}_2^0$: ${\overline{A}}_2^0=\left[\begin{array}{cc}\beta (2-\nu )-(q+\mu )& \beta \epsilon (2-\nu )\\ q(1-\gamma )& -(\mu +d+\alpha )\end{array}\right].$ Using Kamgang and Sallet's result [17], the sub-matrix ${\overline{A}}_2^0$ should be a stable Metzler matrix, that is $\mathrm{tr}\left({\overline{A}}_2^0\right)<0$ and $\det \left({\overline{A}}_2^0\right)\ge 0$. It is straighforward to prove that the condition $\det \left({\overline{A}}_2^0\right)\ge 0$ implies that $\mathrm{tr}\left({\overline{A}}_2^0\right)<0$. Then, using the expression of the basic reproduction number ${\mathcal{R}}_0$ defined as in Equation (9) and the condition $\det \left({\overline{A}}_2^0\right)\ge 0$, the sub-matrix ${\overline{A}}_2^0$ is a stable Metzler matrix if (18) ${\mathcal{R}}_0\le \xi ,$(18) where $\xi =\frac{S_0+(1-\nu )V_0}{(2-\nu )N_0}<1.$ We can now apply Theorem 4.3 in Kamgang and Sallet [17] and conclude that under the condition (18), the DFE $(x_0^1,0)$ of model system (3) is GAS in $\mathcal{D}$. From Equation (11), for the points of $\mathcal{D}$ where $x_1=0$, and from Equation (18) the DFE is GAS on Ω.

We have established the following result for the global stability of the DFE $Q_0$.

Theorem 3.3.

The DFE point $Q_0$ of model system (3) is GAS if ${\mathcal{R}}_0\le \xi <1$ and unstable if ${\mathcal{R}}_0>1$ in Ω. However, when $\xi \le {\mathcal{R}}_0<1,$ the backward bifurcation phenomenon occurs, i.e. the DFE may co-exists with two endemic equilibria, one asymptotically stable and one unstable.

The parameter values used for numerical simulations are given in Table 2.

Mathematical analysis of the dynamical transmission of Neisseria meningitidis serogroup A

All authors

G. Djatcha Yaleu, S. Bowong, E. Houpa Danga & J. Kurths

https://doi.org/10.1080/00207160.2017.1283411

Published online:

08 February 2017

Table 2. Numerical values for the parameters of model system (3).

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Figure 2 presents time series of model system (3) when $\beta =0.25$, $\rho =0.04574795$, and $\theta =0.0001699$ (so that $\xi =0.37733$ and ${\mathcal{R}}_0=0.23313<\xi \le 1$). It illustrates that the trajectories of model system (3) converge to the DFE. This means that meningitis disappears in the host population and the infection is controllable.

Mathematical analysis of the dynamical transmission of Neisseria meningitidis serogroup A

All authors

G. Djatcha Yaleu, S. Bowong, E. Houpa Danga & J. Kurths

https://doi.org/10.1080/00207160.2017.1283411

Published online:

08 February 2017

Figure 2. Simulation of model system (3) when $\beta =0.25$, $\rho =0.04574795$ and $\theta =0.0001699$ (so that $\xi =0.97821$ and ${\mathcal{R}}_0=0.23313<\xi$). (a) Susceptible individuals; (b) vaccinated individuals; (c) carriers ; (d) infectious and (e) recovered individuals. All other parameter values are as in Table 2.

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Figure 2. Simulation of model system (3) when $\beta =0.25$, $\rho =0.04574795$ and $\theta =0.0001699$ (so that $\xi =0.97821$ and ${\mathcal{R}}_0=0.23313<\xi$). (a) Susceptible individuals; (b) vaccinated individuals; (c) carriers ; (d) infectious and (e) recovered individuals. All other parameter values are as in Table 2.