In this section, we critically analyze model (1) by determining the existence of the steadystate solutions. This includes the existence of the diseasefree equilibrium (henceforth called Lassa feverfree equilibrium) and the endemic equilibrium. We further investigate the local and global stability of the equilibria. Furthermore, we investigate the nature of bifurcation the model exhibit.
Existence and stability of Lassa feverfree equilibrium
Lassa feverfree equilibrium points are the steadystate solution in the absence of Lassa fever infection. Thus, the Lassa feverfree equilibrium point for model (1) implies that \(E_{h}=I_{h}=I_{r}=0\). Hence, by solving the systems of equations simultaneously (1), the Lassa feverfree equilibrium denoted by \({\mathcal {E}}_{0}\), is obtained as
$$\begin{aligned} {\mathcal {E}}_{0}=(S_{h}^{*}, E_{h}^{*}, I_{h}^{*}, R_{h}^{*}, S_{r}^{*}, I_{r}^{*})&= \left( \frac{\pi _{h}}{\mu _{h}}, 0, 0, 0, \frac{\pi _{r}}{\mu _{r}}, 0 \right) \end{aligned}$$
(4)
To investigate the local stability of the Lassa feverfree equilibrium, we compute the basic reproduction number \({\mathcal {R}}_{0}\) by using the next generation operator method on the model system (1). Following the approach in [22, 40], the Jacobian matrices F and V, for the new infection terms and the remaining transfer terms are given by
$$\begin{aligned} F=\begin{pmatrix} 0&\quad &\quad \beta _{h} &\quad &\quad \beta _{rh}\\ \\ 0 &\quad &\quad 0&\quad &\quad 0\\ \\ 0 &\quad &\quad \frac{\beta _{hr}S_{r}^{*}}{S_{h}^{*}}&\quad &\quad \beta _{r} \end{pmatrix} \qquad \qquad and \qquad \qquad V=\begin{pmatrix} k_{1}&\quad &\quad 0 &\quad &\quad 0\\ \\ \sigma _{h}&\quad &\quad k_{2} &\quad &\quad 0\\ \\ 0 &\quad &\quad 0 &\quad &\quad \mu _{r} \end{pmatrix} \end{aligned}$$
where \(k_{1}=\sigma _{h}+\mu _{h}\), and \(k_{2}=\tau _{h}+\mu _{h}+\delta _{h}\). The next generation matrix (NGM) with large domain \(K_{L}\) is given below as
$$\begin{aligned} K_{L}=FV^{1}=\begin{pmatrix} \frac{\beta _{h}\sigma _{h}}{k_{1}k_{2}}&\quad &\quad \frac{\beta _{h}}{k_{2}} &\quad &\quad \frac{\beta _{rh}}{\mu _{r}}\\ \\ 0 &\quad &\quad 0&\quad &\quad 0\\ \\ \frac{\beta _{hr}S_{r}^{*}\sigma _{h}}{S_{h}^{*}k_{1}k_{2}} &\quad &\quad \frac{\beta _{hr}S_{r}^{*}}{S_{h}^{*}k_{2}}&\quad &\quad \frac{\beta _{r}}{\mu _{r}} \end{pmatrix} \end{aligned}$$
(5)
It can be seen from the model that, among the three infected states, there are only two that are statesatinfection. This can also be seen by looking at matrix F and observing that the entire second row contains zeros. Hence, the NGM K for the small domain is therefore twodimensional. Thus, using the approach of [41] with an auxiliary matrix E, the NGM K is obtained as
$$\begin{aligned} K=E^{T}K_{L}E=E^{T}FV^{1}E=\begin{pmatrix} \frac{\beta _{h}\sigma _{h}}{k_{1}k_{2}} &\quad &\quad \frac{\beta _{rh}}{\mu _{r}}\\ \\ \frac{\beta _{hr}S_{r}^{*}\sigma _{h}}{S_{h}^{*}k_{1}k_{2}}&\quad &\quad \frac{\beta _{r} }{\mu _{r}} \end{pmatrix}= \begin{pmatrix} {\mathcal {R}}_{h} &\quad &\quad {\mathcal {R}}_{rh}\\ \\ {\mathcal {R}}_{hr}&\quad &\quad {\mathcal {R}}_{r} \end{pmatrix} \end{aligned}$$
(6)
Thus, the characteristic polynomial of the matrix K is obtained as
$$\begin{aligned} \lambda ^{2}({\mathcal {R}}_{h}+{\mathcal {R}}_{r})\lambda +({\mathcal {R}}_{h}{\mathcal {R}}_{r}{\mathcal {R}}_{hr}{\mathcal {R}}_{rh})=0 \end{aligned}$$
(7)
where
$$\begin{aligned} {\mathcal {R}}_{h}= \frac{\beta _{h}\sigma _{h}}{k_{1}k_{2}}, \quad {\mathcal {R}}_{r}= \frac{\beta _{r}}{\mu _{r}},\quad {\mathcal {R}}_{hr}= \frac{\beta _{hr}S_{r}^{*}\sigma _{h}}{S_{h}^{*}k_{1}k_{2}}, \quad {\mathcal {R}}_{rh}= \frac{\beta _{rh}}{\mu _{r}}. \end{aligned}$$
It follows that the basic reproduction number for the model (1), which is the spectral radius of K given by \({\mathcal {R}}_{0}=\rho (K)\), is obtained as
$$\begin{aligned} {\mathcal {R}}_{0}=\frac{1}{2}\left\{ ({\mathcal {R}}_{h}+{\mathcal {R}}_{r}) + \sqrt{({\mathcal {R}}_{h}+{\mathcal {R}}_{r})^{2}  4({\mathcal {R}}_{h}{\mathcal {R}}_{r}  {\mathcal {R}}_{hr}{\mathcal {R}}_{rh})} \right\} \end{aligned}$$
(8)
Further simplification of (8) result to
$$\begin{aligned} {\mathcal {R}}_{0}=\frac{1}{2}\left\{ ({\mathcal {R}}_{h}+{\mathcal {R}}_{r}) + \sqrt{({\mathcal {R}}_{h}{\mathcal {R}}_{r})^{2} + 4{\mathcal {R}}_{\Delta }^{2}} \right\} \end{aligned}$$
(9)
where \({\mathcal {R}}_{h}\), \({\mathcal {R}}_{r}\), and \({\mathcal {R}}_{\Delta } = \sqrt{{\mathcal {R}}_{hr}{\mathcal {R}}_{rh}}\) are the reproduction numbers for humantohuman, rodenttorodent transmission and vectorial transmission respectively.
The basic reproduction number is a threshold quantity that measures the spread potential of disease in a given population. Epidemiologically, it measures the average number of secondary infections a single infected individual can generate in a population that is completely susceptible. In other words, the threshold quantity \({\mathcal {R}}_{0}\) given in (9) measures the average number of Lassa fever infections that a Lassa fever infected individual can generate in an entirely susceptible population. It is imperative to mention that, the reproduction number for the model (1) is a composition of the reproduction number of humantohuman transmission \({\mathcal {R}}_{h}\), rodenttorodent transmission \({\mathcal {R}}_{r}\), and vectorial transmission \({\mathcal {R}}_{hr}, {\mathcal {R}}_{rh}\) because the model includes the biological possibilities of infection transfer between the two interacting host. Hence, epidemiologically, \({\mathcal {R}}_{h}\) measure the average number of secondary infections a single infectious human can produce during an infectious period. Similarly, \({\mathcal {R}}_{r}\) measure the average number of secondary infections a single infectious rodent can generate during an infectious period. Since \(\beta _{hr}\), and \(\beta _{rh}\) are the transmission probability from humantorodent, and rodenttohuman respectively, then \({\mathcal {R}}_{hr}\) measure the average number of secondary infection of rodents a single infectious human can generate over its infectious period, while \({\mathcal {R}}_{rh}\) measure the average number of secondary infection of humans a single infectious rodent can generate during the infection period. In general, an increase in any of the reproduction number can upsurge the risk of Lassa fever occurrence in the human population, since the growth of any of the infectious host (either humans or rodents) can increase the spread of infection in the human populace if adequate and effective control mechanism is not utilized by the population. Next, we shall investigate the stability of the Lassa feverfree equilibrium \({\mathcal {E}}_{0}\).
Local stability of Lassa feverfree equilibrium
We analyze the local stability of Lassa feverfree equilibrium of the model system (1) by using the basic reproduction number \({\mathcal {R}}_{0}\) in the following theorem as described in [38]. The proof is provided in “Appendix Proof of Theorem 1”.
Theorem 1
The Lassa feverfree equilibrium \({\mathcal {E}}_{0}\), of the model (1) is locally asymptotically stable in the biological feasible region \({\mathcal {D}}\) if \({\mathcal {R}}_{0}<1\) and unstable if \({\mathcal {R}}_{0}>1\).
Global stability of Lassa feverfree equilibrium
Here, we further investigate the global stability of the Lassa feverfree equilibrium \({\mathcal {E}}_{0}\) of the model system (1), by using the technique implemented in [42]. Firstly, we rewrite the Lassa fever model (1) in the form
$$\begin{aligned} \frac{{\text{d}}X}{{\text{d}}t}&= F(X, Z) \nonumber \\ \frac{{\text{d}}Z}{{\text{d}}t}&= G(X, Z), \quad G(X, 0)=0 \end{aligned}$$
(10)
where \(X=(S_{h}, R_{h}, S_{r})\) is the uninfected population, and \(Z=(E_{h}, I_{h}, I_{r})\) is the infected population with the component of \((X, Z) \in {\mathcal {R}}^{3}\). The Lassa feverfree equilibrium is obtained as
$$\begin{aligned} {\mathcal {E}}_{0}^{*}&= (X^{*}, 0)=\left( \frac{\pi _{h}}{\mu _{h}}, 0, \frac{\pi _{r}}{\mu _{r}}\right) \end{aligned}$$
(11)
For the point \({\mathcal {E}}_{0}^{*}=(X^{*}, 0)\) to be globally asymptotically stable, the following conditions must be satisfied

(C1) : For \(\frac{{\text{d}}X}{{\text{d}}t}=F(X, 0)\), \(X^{*}\) is globally asymptotically stable (GAS),

(C2) : \(G(X, Z)=QZ{\hat{G}}(X, Z)\) with \({\hat{G}}(X, Z)\ge 0\) for \((X, Z)\in {\mathcal {D}}\)
where \(Q=B_{Z}G(X^{*}, 0)\) is an Mmatrix (the offdiagonal elements of B are nonnegative) and \({\mathcal {D}}\) is the feasible region where the model makes biological sense. If the model system (1) satisfies the conditions given above, then the following result holds. The proof is provided in “Appendix Proof of Theorem 2”.
Theorem 2
The fixed point \({\mathcal {E}}_{0}^{*}=(X^{*}, 0)\) is globally asymptotically stable (GAS) equilibrium of model system (1), if \({\mathcal {R}}_{0}<1\) (locally asymptotically stable) and the conditions (C1) and (C2) are satisfied.
The above result infers that, regardless of the initial sizes of the subpopulations of the system, Lassa fever eradication is possible whenever the reproduction number is less than unity. We illustrate this theorem numerically in Fig. 10.
Existence and stability of endemic equilibria
Here, we investigate the existence and stability of the endemic equilibrium for the model (1). Lassa fever endemic equilibrium points are the steadystate solution where there is presence of Lassa fever infection in the population. We let \({\mathcal {E}}_{1}=(S_{h}^{**}, E_{h}^{**}, I_{h}^{**}, R_{h}^{**}, S_{r}^{**}, I_{r}^{**})\) represents the Lassa feverpresent equilibrium. Setting the righthand sides of the systems of equations in (1) to zero and solving simultaneously in terms of the associated form of infection yields
$$\begin{aligned} S_{h}^{**}&= \frac{\pi _{h}k_{1}k_{2}k_{3}}{k_{1}k_{2}k_{3}\lambda _{h}^{**} + k_{1}k_{2}k_{3}\mu _{h}  \lambda _{h}^{**}\sigma _{h}\tau _{h}\xi _{h}} \nonumber \\ E_{h}^{**}&= \frac{\lambda _{h}^{**}\pi _{h}k_{2}k_{3}}{k_{1}k_{2}k_{3}\lambda _{h}^{**} + k_{1}k_{2}k_{3}\mu _{h}  \lambda _{h}^{**}\sigma _{h}\tau _{h}\xi _{h}} \nonumber \\ I_{h}^{**}&= \frac{\lambda _{h}^{**}\pi _{h}\sigma _{h}k_{3}}{k_{1}k_{2}k_{3}\lambda _{h}^{**} + k_{1}k_{2}k_{3}\mu _{h}  \lambda _{h}^{**}\sigma _{h}\tau _{h}\xi _{h}} \nonumber \\ R_{h}^{**}&= \frac{\lambda _{h}^{**}\pi _{h}\sigma _{h}\tau _{h}}{k_{1}k_{2}k_{3}\lambda _{h}^{**} + k_{1}k_{2}k_{3}\mu _{h}  \lambda _{h}^{**}\sigma _{h}\tau _{h}\xi _{h}} \nonumber \\ S_{r}^{**}&= \frac{\pi _{r}}{\lambda _{r}^{**} + \mu _{r} }, \qquad \qquad I_{r}^{**}=\frac{\lambda _{r}^{**}\pi _{r}}{\mu _{r}(\lambda _{r}^{**} + \mu _{r})} \end{aligned}$$
(12)
where the force of infection are given as
$$\begin{aligned} \lambda _{h}^{**}=\frac{\beta _{rh}I_{r}^{**}}{N_{h}^{**}}+\frac{\beta _{h}I_{h}^{**}}{N_{h}^{**}}, \qquad and \qquad \lambda _{r}^{**}=\frac{\beta _{hr}I_{h}^{**}}{N_{h}^{**}}+\frac{\beta _{r}I_{r}^{**}}{N_{r}^{**}} \end{aligned}$$
(13)
Substituting the expression (12) into the force of infection (13) at steady state yields the following polynomial
$$\begin{aligned} \lambda _{h}^{**} \left\{ a_{1}(\lambda _{h}^{**})^{4}+a_{2}(\lambda _{h}^{**})^{3}+a_{3}(\lambda _{h}^{**})^{2}+a_{4}\lambda _{h}^{**}a_{5}\right\} =0 \end{aligned}$$
(14)
The coefficients \(a_{i}\), for \(i=1\ldots ,5\) of the polynomial are given in “Appendix Coefficients of polynomial (14)”. Clearly, \(\lambda _{h}^{**}=0\) is a solution. The coefficient \(a_{1}\) is positive while the sign of \(a_{5}\) depends on the values of respective reproduction number, such that if \(\left\{ {\mathcal {R}}_{h}, {\mathcal {R}}_{r}, {\mathcal {R}}_{hr},{\mathcal {R}}_{hr} \in {\mathcal {R}}_{0}>1 \right\}\), then \(a_{5}>0\) such that there is at least one sign change in the sequence of coefficients \(a_{1},\ldots a_{5}\). Thus, by Descartes rule of signs, there exists at least one positive real root for (14) aside from the root \(\lambda _{h}^{**}=0\), whenever \({\mathcal {R}}_{0}>1\). Therefore, the following result is established.
Theorem 3
The model system (1) has at least one endemic equilibrium whenever \({\mathcal {R}}_{0}>1\).
Bifurcation analysis
Following Theorem 1, it is imperative to restate that, whenever the reproduction number of the model (1) is greater than unity \({\mathcal {R}}_{0} > 1\), the asymptotic local stability of the Lassa feverfree equilibrium will undergo a tradeoff with the asymptomatic local stability of the endemic equilibrium. Hence, in this section, we will investigate the criteria for the tradeoff between the asymptomatic local stability of the Lassa feverfree equilibrium and asymptomatic local stability of the endemic equilibrium, as the threshold quantity crosses unity. In other words, we will show the conditions under which model (1) undergo supercritical or subcritical (forward or backward) bifurcation. By employing the Center Manifold Theory of bifurcation analysis described in [27], we write the Lassa fever model (1) in the vector form
$$\begin{aligned} \frac{{\text{d}}X}{{\text{d}}t}=F(X) \end{aligned}$$
(15)
where \(X=\left( x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}\right) ^{T}\) and \(F=\left( f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6}\right) ^{T}\). We further modify the variables be setting
$$\begin{aligned} S_{h}=x_{1},\quad E_{h}=x_{2},\quad I_{h}=x_{3},\quad R_{h}=x_{4},\quad S_{r}=x_{5}, \quad I_{r}=x_{6} \end{aligned}$$
such that the total human and rodent populations are respectively given as
$$\begin{aligned} N_{h}=x_{1}+x_{2}+x_{3}+x_{4}, \qquad and \qquad N_{r}=x_{5}+x_{6} \end{aligned}$$
Hence, following the above transformation, the transformed model (1) is given as
$$\begin{aligned} \frac{{\text{d}}x_{1}}{{\text{d}}t}&= f_{1}=\pi _{h} + \xi _{h}x_{4} \lambda _{h}x_{1}\mu _{h}x_{1}\nonumber \\ \frac{{\text{d}}x_{2}}{{\text{d}}t}&= f_{2}=\lambda _{h}x_{1}(\sigma _{h}+\mu _{h})x_{2} \nonumber \\ \frac{{\text{d}}x_{3}}{{\text{d}}t}&= f_{3}=\sigma _{h}x_{2}(\tau _{h}+\mu _{h}+\delta _{h})x_{3} \nonumber \\ \frac{{\text{d}}x_{4}}{{\text{d}}t}&= f_{4}=\tau _{h}x_{3}(\mu _{h}+\xi _{h})x_{4} \nonumber \\ \frac{{\text{d}}x_{5}}{{\text{d}}t}&= f_{5}=\pi _{r}\lambda _{r}x_{5}\mu _{r}x_{5} \nonumber \\ \frac{{\text{d}}x_{6}}{{\text{d}}t}&= f_{6}=\lambda _{r}x_{5}\mu _{r}x_{6} \end{aligned}$$
(16)
with the associated force of infection given as
$$\begin{aligned} \lambda _{h}=\frac{\beta _{rh}x_{6}+\beta _{h}x_{3}}{x_{1}+x_{2}+x_{3}+x_{4}}, \qquad \lambda _{r}=\frac{\beta _{hr}x_{3}}{x_{1}+x_{2}+x_{3}+x_{4}}+\frac{\beta _{r}x_{6}}{x_{5}+x_{6}} \end{aligned}$$
Suppose that \(\beta _{rh}^{*}\) is chosen as the bifurcation parameter, solving (8) at \({\mathcal {R}}_{0}=1\), the parameter \(\beta _{rh}=\beta _{rh}^{*}\) is obtained as
$$\begin{aligned} \beta _{rh}:=\beta _{rh}^{*}&= \frac{\pi _{h}\mu _{r}\left\{ \mu _{r}k_{1}k_{2}(\beta _{h}\sigma _{h}\mu _{r}+\beta _{r}\beta _{h}\sigma _{h}+\beta _{r}k_{1}k_{2})\right\} }{\beta _{hr}\pi _{r}\sigma _{h}\mu _{h}} \end{aligned}$$
(17)
The Jacobian of system (16), evaluated at Lassa feverfree \(({\mathcal {E}}_{0}^{\bigtriangleup }=x_{1}^{*}, 0, 0, 0, x_{5}^{*}, 0)\) with \(\beta _{rh}=\beta _{rh}^{*}\) denoted by \({\mathcal {J}}({\mathcal {E}}_{0}^{\bigtriangleup }, \beta _{rh}^{*})\) is given by
$$\begin{aligned} {\mathcal {J}}({\mathcal {E}}_{0}^{\bigtriangleup }, \beta _{rh}^{*})=\begin{pmatrix} \mu _{h}&\quad 0&\quad \beta _{h}&\quad \xi _{h}&\quad 0&\quad \beta _{rh}^{*}\\ \\ 0&\quad k_{1}&\quad \beta _{h}&\quad 0&\quad 0&\quad \beta _{rh}^{*}\\ \\ 0&\quad \sigma _{h}&\quad k_{2}&\quad 0&\quad 0&\quad 0\\ \\ 0&\quad 0&\quad \tau _{h}&\quad k_{3}&\quad 0&\quad 0\\ \\ 0&\quad 0&\quad \frac{x_{5}^{*}\beta _{hr}}{x_{1}^{*}}&\quad 0&\quad \mu _{r}&\quad \beta _{r}\\ \\ 0&\quad 0&\quad \frac{x_{5}^{*}\beta _{hr}}{x_{1}^{*}}&\quad 0&\quad 0&\quad \mu _{r}+\beta _{r} \end{pmatrix} \end{aligned}$$
(18)
The Jacobian matrix (18) has a right eigenvector (associated with the zero eigenvalues) given by \(\mathbf{w }=(w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6})^{T}\), where
$$\begin{aligned} w_{1}&= \left( \frac{x_{1}^{*}\mu _{r}(1{\mathcal {R}}_{r})(\tau _{h}\xi _{h}\beta _{h}k_{3})x_{5}^{*}\beta _{hr}}{x_{1}^{*}\mu _{r}\mu _{h}k_{3}(1{\mathcal {R}}_{r})}\right) w_{3} ; \quad w_{2}=\frac{w_{3}k_{2}}{\sigma _{h}}; \quad w_{3}=w_{3}>0 ;\\ w_{4}&= \frac{w_{3}\tau _{h}}{k_{3}} ; \qquad w_{5}= \frac{w_{3}x_{5}^{*}}{x_{1}^{*}\mu _{r}(1{\mathcal {R}}_{r})} ; \qquad w_{6}=\frac{w_{3}x_{5}^{*}\beta _{hr}}{x_{1}^{*}\mu _{r}(1{\mathcal {R}}_{r})} \end{aligned}$$
Similarly, the Jacobian matrix (18) has a left eigenvector (associated with the zero eigenvalues) given by \(\mathbf{v }=(v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6})^{T}\), where
$$\begin{aligned} v_{1}= 0; \quad v_{2}=\frac{v_{3}\sigma _{h}}{k_{1}}; \quad v_{3}=v_{3}>0; \quad v_{4}=0; \qquad v_{5}=0; \quad v_{6}=\frac{v_{3}\beta _{rh}^{*}\sigma _{h}}{k_{1}\mu _{r}(1{\mathcal {R}}_{r})} \end{aligned}$$
Computation of bifurcation coefficient a and b
The direction of the bifurcation at \({\mathcal {R}}_{0}=1\) is determined by the signs of bifurcation coefficients a and b, obtained by computing the associated nonzero partial derivative of F(X) (evaluated at the disease free equilibrium \({\mathcal {E}}_{0}^{\bigtriangleup }\)). Thus, the coefficient of a is given as
$$\begin{aligned} a&= \sum _{k,i,j=1}^{6}v_{k}w_{i}w_{j}\frac{\partial ^2f_{k}}{\partial x_{i}\partial x_{j}}(0, 0) \nonumber \\&= \frac{2(m_{1}m_{2})}{x_{1}^{*2}x_{5}^{*}} \end{aligned}$$
(19)
where
$$\begin{aligned} m_{1}&= x_{1}^{*}x_{5}^{*}\left\{ m_{3}(\beta _{h}w_{3}v_{1}+\beta _{rh}^{*}v_{1}w_{6})+\beta _{hr}w_{3}w_{5}v_{6} \right\} +\beta _{hr}v_{5}w_{3}x_{5}^{*2}(w_{1}+m_{3})+\beta _{r}x_{1}^{*2}v_{5}w_{6}^{2}\\ m_{2}&= x_{1}^{*}x_{5}^{*}\left\{ m_{3}(\beta _{h}w_{3}v_{2}+\beta _{rh}^{*}v_{2}w_{6})+\beta _{hr}w_{3}w_{5}v_{5} \right\} +\beta _{hr}v_{6}w_{3}x_{5}^{*2}(w_{1}+m_{3})+\beta _{r}x_{1}^{*2}v_{6}w_{6}^{2}\\ m_{3}&= w_{2}+w_{3}+w_{4} \end{aligned}$$
Similarly, the bifurcation coefficient b is obtained as follows
$$\begin{aligned} b&= \sum _{k,i=1}^{6}v_{k}w_{i}\frac{\partial ^2f_{k}}{\partial x_{i}\partial \beta _{rh}^{*}}(0, 0) \nonumber \\&= w_{6}(v_{2}v_{1})>0 \end{aligned}$$
(20)
Since all the parameters of model (1) are nonnegative and \(v_{1}=0\), it can be shown that the inequality (20) holds if \({\mathcal {R}}_{r}<1\). It follows from Theorem 4.1 in [27] that the Lassa fever model (1) will exhibit a subcritical (backward) bifurcation if the coefficient a given by (19) is positive. This implies that \(m_{1}>m_{2}\) must be satisfied. Hence, the following result will be established.
Theorem 4
The Lassa fever model (1) undergoes a subcritical (backward) bifurcation as \({\mathcal {R}}_{0}\) crosses unity, whenever the coefficient \(a>0\) and \(b>0\).
Backward bifurcation (BB) occurs when a small positive unstable equilibrium appears while the diseasefree equilibrium (DFE) and a larger positive equilibrium are locally asymptotically stable when the threshold quantity \({\mathcal {R}}_{0}\) is less than unity. In other words, BB occurs when a stable DFE and a stable endemic equilibrium coexist under some given values for which \({\mathcal {R}}_{0}\) is less than unity. The backward bifurcation phenomenon suggests that the epidemiological condition of having the reproduction number less than unity to eliminate a disease although necessary is no longer enough for the effective control of the disease in the population. Hence, the effective control of Lassa fever in the population is difficult, since disease control when \({\mathcal {R}}_{0} <1\) is dependent on the initial sizes of the subpopulations. We further explore the condition for which system (1) undergo supercritical bifurcation. It must be noted that the Lassa fever model (1) will exhibit a forward bifurcation if the coefficient a given by (19) is negative. This implies that \(m_{1}<m_{2}\) must be satisfied. Thus, the following result will be established.
Theorem 5
The Lassa fever model (1) undergoes a supercritical (forward) bifurcation as \({\mathcal {R}}_{0}\) crosses unity, whenever the coefficient \(a<0\) and \(b>0\).
A system exhibits a forward bifurcation when the diseasefree equilibrium losses its stability due to an introduction of a small positive asymptomatically stable equilibrium. Epidemiologically, the result above implies that a small inflow of individuals with Lassa fever infection into an entirely susceptible population will lead to a continuance of Lassa fever in the populace, whenever the reproduction number is less than unity. In other words, the exchange of the local asymptotic stability of the equilibria depends on the initial number of Lassa fever infectious individuals in the population. It must be noted that the transfer of the local asymptotic stability of the equilibria is independent of the initial sizes of the subpopulations. This can be proved by establishing the global asymptomatic stability of the diseasefree equilibrium (see “section Global stability of Lassa feverfree equilibrium”).