Positivity of solutions
In this section, the basic properties of model (1) will be explored. Since model (1) describes both human and rodents populations during the course of a Lassa fever epidemic, it will only be epidemiologically meaningful if all its state variables are nonnegative for all time \(t\ge 0\). In other words, solutions of the model system (1) with nonnegative initial data will remain nonnegative for all time \(t>0\).
Lemma 1
The solutions \(S_{h}(t), E_{h}(t), I_{h}(t), R_{h}(t), S_{r}(t)\), and \(I_{r}\) of the model system (1) with nonnegative initial conditions \(S_{h}(0); E_{h}(0); I_{h}(0); R_{h}(0); S_{r}(0); I_{r}(0)\) will remain nonnegative for all time \(t>0\).
Proof
Let \(t_{1}=\sup \{t>0: S_{h}(t)>0, E_{h}(t)>0, I_{h}(t)>0, R_{h}(t)>0, S_{r}(t)>0, I_{r}(t)>0 \in [0,t] \}\). Thus, \(t_{1}>0\). It follows from the first equation of system (1), that
$$\begin{aligned} \frac{dS_{h}}{dt}= & {} \Lambda _{h} +\tau _{h} R_{h}-\beta _{1}S_{h}-\mu _{h}S_{h}\ge \Lambda _{h}-\beta _{1}S_{h}-\mu _{h}S_{h} \end{aligned}$$
(2)
Employing the integrating factor method, this can be written as:
$$\begin{aligned} \frac{d}{dt}\left( S_{h}(t)exp\left[ \mu _{h}t+\int _{0}^{t}\beta _{1}(x)dx\right] \right) \ge \Lambda _{h}exp\left[ \mu _{h}t+\int _{0}^{t}\beta _{1}(x)dx\right] \end{aligned}$$
Hence,
$$\begin{aligned} S_{h}(t_{1})exp\left[ \mu _{h}t_{1}+\int _{0}^{t_{1}}\beta _{1}(x)dx\right] -S_{h}(0)\ge \int _{0}^{t_{1}}\Lambda _{h}\left( exp\left[ \mu _{h}y+\int _{0}^{y}\beta _{1}(x)dx\right] \right) dy \end{aligned}$$
so that,
$$\begin{aligned} S_{h}(t_{1})\ge & {} S_{h}(0) exp\left[ -\mu _{h}t_{1}-\int _{0}^{t_{1}}\beta _{1}(x)dx\right] \\&+ exp\left[ -\mu _{h}t_{1}-\int _{0}^{t_{1}}\beta _{1}(x)dx\right] \times \int _{0}^{t_{1}}\Lambda _{h}\left( exp\left[ \mu _{h}y+\int _{0}^{y}\beta _{1}(x)dx\right] \right) dy >0. \end{aligned}$$
Similarly, it can be shown that \(E_{h}(t)\ge 0\), \(I_{h}(t)\ge 0\), \(R_{h}(t)\ge 0\), \(S_{r}(t)>0\), and \(I_{r}(t)\ge 0\) for all time \(t>0\). Therefore, all the solutions of model (1) remain positive for all nonnegative initial conditions. \(\square\)
Invariant region
In this section, model (1) will be analyzed in a biologically feasible region as follows. Consider the biologically feasible region consisting of \(\Omega =\Omega _{h}\times \Omega _{r} \in {\mathcal {R}}_{+}^{4}\times {\mathcal {R}}_{+}^{2}\) with
$$\begin{aligned} \Omega _{h}=\left\{ S_{h}, E_{h}, I_{h}, R_{h}\in {\mathcal {R}}_{+}^{4}: N_{h}\le \frac{\Lambda _{h}}{\mu _{h}}\right\} \end{aligned}$$
and
$$\begin{aligned} \Omega _{r}=\left\{ S_{r}, I_{r} \in {\mathcal {R}}_{+}^{2}: N_{r}\le \frac{\Lambda _{r}}{\mu _{r}} \right\} \end{aligned}$$
It can be shown that the set \(\Omega\) is a positively invariant set and global attractor of this system. This implies any phase trajectory initiated anywhere in the nonnegative region \({\mathcal {R}}_{+}^{6}\) enters the feasible region \(\Omega\) and remains in \(\Omega\) thereafter.
Lemma 2
The biological feasible region \(\Omega =\Omega _{h}\cup \Omega _{r} \subset {\mathcal {R}}_{+}^{4}\times {\mathcal {R}}_{+}^{2}\) of the Lassa fever model (1) is positively invariant with nonnegative initial conditions in \({\mathcal {R}}_{+}^{6}\).
Proof
The following steps are followed to establish the positive invariance of \(\Omega\) (i.e., solutions in \(\Omega\) remain in \(\Omega\) for all \(t>0\)). The rate of change of the total human and rodent populations \(N_{h}\) and \(N_{r}\) , respectively, are obtained by adding the respective components of model (1) which result to
$$\begin{aligned} \frac{dN_{h}(t)}{dt}= & {} \Lambda _{h}-\mu _{h}N_{h}(t)-\delta _{h}I_{h}(t) \\ \frac{dN_{r}(t)}{dt}= & {} \Lambda _{r}-\mu _{r}N_{r}(t) \end{aligned}$$
so that,
$$\begin{aligned} \frac{dN_{h}(t)}{dt} \le \Lambda _{h}-\mu _{h}N_{h}(t), \quad and \qquad \frac{dN_{r}(t)}{dt}= \Lambda _{r}-\mu _{r}N_{r}(t) \end{aligned}$$
(3)
Hence, \(N_{h}(t)\le N_{h}(0)e^{-\mu _{h}t} +\frac{\Lambda _{h}}{\mu _{h}}\left( 1-e^{-\mu _{h}t}\right)\) and \(N_{r}(t)= N_{r}(0)e^{-\mu _{r}t} +\frac{\Lambda _{r}}{\mu _{r}}\left( 1-e^{-\mu _{r}t}\right)\). In particular, \(N_{h}(t)\le \frac{\Lambda _{h}}{\mu _{h}}\) and \(N_{r}(t)\le \frac{\Lambda _{r}}{\mu _{r}}\) if the total human population and rodent population at the initial instant of time, \(N_{h}(0)\le \frac{\Lambda _{h}}{\mu _{h}}\) and \(N_{r}(0)\le \frac{\Lambda _{r}}{\mu _{r}}\) , respectively. So, the region \(\Omega\) is positively invariant. Thus, it is consequently adequate to consider the dynamics of Lassa fever governed by model (1) in the biological feasible region \(\Omega\), where the model is considered to be epidemiologically and mathematically well posed [13, 14]. \(\square\)
Existence and Stability of Lassa fever free equilibrium (LFFE)
The Lassa fever free equilibrium of model (1) denoted by \({\mathcal {E}}_{0}\) is given by
$$\begin{aligned} {\mathcal {E}}_{0}=(S_{h}^{*}, E_{h}^{*}, I_{h}^{*}, R_{h}^{*}, S_{r}^{*}, I_{r}^{*})= & {} \left( \frac{\Lambda _{h}}{\mu _{h}}, 0, 0, 0, \frac{\Lambda _{r}}{\mu _{r}}, 0 \right) \end{aligned}$$
(4)
The next-generation matrix method is used on system (1) for determining the reproduction number \({\mathcal {R}}_{0}\). The epidemiological quantity \({\mathcal {R}}_{0}\), called the reproduction number, measures the typical number of Lassa fever cases that a Lassa fever-infected individual can generate in a human population that is completely susceptible [13, 15]. The \({\mathcal {R}}_{0}\) is used in investigating the local asymptotic stability of the Lassa fever free equilibrium \({\mathcal {E}}_{0}\). By using the infected compartments (\(E_{h}^{*}, I_{h}^{*}, I_{r}^{*}\)) at the LFFE, and following the notation in [16, 17], the Jacobian matrices F and V for the new infection terms and the remaining transfer terms are, respectively, given by
$$\begin{aligned} F=\begin{pmatrix} 0&{}&{}\frac{\beta _{h} S_{h}^{*}}{N_{h}^{*}}&{}&{}\frac{\beta _{r} S_{h}^{*}}{N_{h}^{*}}\\ \\ 0 &{}&{} 0&{}&{}0\\ \\ 0 &{}&{} 0&{}&{}\frac{\beta _{r} S_{h}^{*}}{N_{h}^{*}} \end{pmatrix} \qquad \qquad and \qquad \qquad V=\begin{pmatrix} k_{1}&{}&{} 0 &{}&{}0\\ \\ -\sigma _{h}&{}&{} k_{2} &{}&{} 0\\ \\ 0 &{}&{} 0 &{}&{} \mu _{r} \end{pmatrix} \end{aligned}$$
It follows that the basic reproduction number of model (1) is given by \({\mathcal {R}}_{0}=\rho (FV^{-1})\), where \(\rho\) is the spectral radius of the matrix. Hence,
$$\begin{aligned} {\mathcal {R}}_{0}={\mathcal {R}}_{h}+{\mathcal {R}}_{r}= & {} \frac{\beta _{h}\mu _{r}\sigma _{h}+\beta _{r}k_{1}k_{2}}{k_{1}k_{2}\mu _{r}} \end{aligned}$$
(5)
where \({\mathcal {R}}_{h}= \frac{\beta _{h}\sigma _{h}}{k_{1}k_{2}}\), \({\mathcal {R}}_{r}= \frac{\beta _{r}}{\mu _{r}}\), \(k_{1}=\sigma _{h}+\mu _{h}\), and \(k_{2}=\mu _{h}+\delta _{h}+\phi _{h}\). From the threshold quantity \({\mathcal {R}}_{0}\) given above in (5), the quantity \({\mathcal {R}}_{h}\) measures the contribution of Lassa fever risk caused by human in the population, while the quantity \({\mathcal {R}}_{r}\) measures the quantity of Lassa fever risk caused by rodent in the population. It must be noted that the increase in any of the threshold quantity will directly upsurge the risk of Lassa fever in the population. The following result is established.
Lemma 3
The Lassa fever free equilibrium \({\mathcal {E}}_{0}\) of model (1) is locally asymptotically stable in the biological feasible region \(\Omega\) if \({\mathcal {R}}_{0}<1\) and unstable if \({\mathcal {R}}_{0}>1\).
Proof
In order to prove the lemma above, we obtain the Jacobian matrix by evaluating system (1) at Lassa fever free equilibrium \({\mathcal {E}}_{0}\) as
$$\begin{aligned} {\mathcal {J}}({\mathcal {E}}_{0})=\begin{pmatrix} -\mu _{h}&{}0&{}-\beta _{h}&{}\tau _{h}&{}0&{}-\beta _{r}\\ 0&{}-k_{1}&{}\beta _{h}&{}0&{}0&{}\beta _{r}\\ 0&{}\sigma _{h}&{}-k_{2}&{}0&{}0&{}0\\ 0&{}0&{}\phi _{h}&{}-k_{3}&{}0&{}0\\ 0&{}0&{}-0&{}0&{}-\mu _{r}&{}-\beta _{r}\\ 0&{}0&{}0&{}0&{}0&{}-\mu _{r}+\beta _{r} \end{pmatrix} \end{aligned}$$
(6)
where \(k_{1}=\sigma _{h}+\mu _{h}\), \(k_{2}=\mu _{h}+\delta _{h}+\phi _{h}\), and \(k_{3}=\mu _{h}+\tau _{h}\). From (6), it is sufficient to show that all the eigenvalues of \({\mathcal {J}}({\mathcal {E}}_{0})\) are negative. We obtain the first four eigenvalues as \(-\mu _{r}\), \(-\mu _{h}\), \(-(\mu _{r}-\beta _{r})\) and \(-k_{3}\). It must be noted that \(-(\mu _{r}-\beta _{r})\) can also be re-written as \(-\mu _{r}\left( 1-{\mathcal {R}}_{r}\right)\), where \({\mathcal {R}}_{r}=\frac{\beta _{r}}{\mu _{r}}\). The remaining eigenvalues can be obtained from the sub-matrix \({\mathcal {M}}\) which is written as
$$\begin{aligned} {\mathcal {M}}=\begin{pmatrix} -k_{1}&{}&{}\beta _{h}\\ \\ \sigma _{h}&{}&{}-k_{2} \end{pmatrix} \end{aligned}$$
(7)
According to the Routh–Hurwitz condition, all the matrix \({\mathcal {M}}\) are real and negative if
-
(i)
Trace(\({\mathcal {M}}\))\(<0\)
-
(ii)
Determinant(\({\mathcal {M}}\))\(>0\)
It can be shown that,
$$\begin{aligned} Tr({\mathcal {M}})=-(k_{1}+k_{2})<0 \end{aligned}$$
and
$$\begin{aligned} Det({\mathcal {M}})= k_{1}k_{2}-\beta _{h}\sigma _{h}=k_{1}k_{2}\left( 1-\frac{\beta _{h}\sigma _{h}}{k_{1}k_{2}}\right) =k_{1}k_{2}(1-{\mathcal {R}}_{h})>0 \quad if \quad {\mathcal {R}}_{h} \in {\mathcal {R}}_{0}<1 \end{aligned}$$
Thus, all the eigenvalues of the Jacobian matrix (6) are real and negative if \(\left\{ {\mathcal {R}}_{r}, {\mathcal {R}}_{h} \right\}\) \(\in\) \({\mathcal {R}}_{0}<1\), so that the Lassa fever free equilibrium \({\mathcal {E}}_{0}\) is locally asymptotically stable and unstable otherwise. \(\square\)
From an epidemiological perspective, Lemma 3 implies that the spread of Lassa fever can be effectively controlled in the population when \({\mathcal {R}}_{0}\) is less than unity, if the initial sizes of the subpopulations of the model system (1) are in the basin of attraction of the Lassa fever free equilibrium \({\mathcal {E}}_{0}\).
Existence of Lassa fever endemic equilibrium (EEP)
We shall investigate the existence of the Lassa fever endemic equilibrium for system (1). The endemic equilibria denoted by \({\mathcal {E}}_{1}=(S_{h}^{**}, E_{h}^{**}, I_{h}^{**}, R_{h}^{**}, S_{r}^{**}, I_{r}^{**})\) represents the steady-state solution in the presence of the disease. By setting the right-hand sides of system (1) to zero and solving simultaneously in terms of the associated force of infection, it gives
$$\begin{aligned} S_{h}^{**}= & {} \frac{\Lambda _{h}k_{1}k_{2}k_{3}}{k_{1}k_{2}k_{3}\beta _{1}^{**} + k_{1}k_{2}k_{3}\mu _{h} - \beta _{1}^{**}\sigma _{h}\tau _{h}\phi _{h}} \nonumber \\ E_{h}^{**}= & {} \frac{\beta _{1}^{**}\Lambda _{h}k_{2}k_{3}}{k_{1}k_{2}k_{3}\beta _{1}^{**} + k_{1}k_{2}k_{3}\mu _{h} - \beta _{1}^{**}\sigma _{h}\tau _{h}\phi _{h}} \nonumber \\ I_{h}^{**}= & {} \frac{\beta _{1}^{**}\Lambda _{h}\sigma _{h}k_{3}}{k_{1}k_{2}k_{3}\beta _{1}^{**} + k_{1}k_{2}k_{3}\mu _{h} - \beta _{1}^{**}\sigma _{h}\tau _{h}\phi _{h}} \nonumber \\ R_{h}^{**}= & {} \frac{\beta _{1}^{**}\Lambda _{h}\sigma _{h}\phi _{h}}{k_{1}k_{2}k_{3}\beta _{1}^{**} + k_{1}k_{2}k_{3}\mu _{h} - \beta _{1}^{**}\sigma _{h}\tau _{h}\phi _{h}} \nonumber \\ S_{r}^{**}= & {} \frac{\Lambda _{r}}{\beta _{2}^{**} + \mu _{r} }, \qquad \qquad I_{r}^{**}=\frac{\beta _{2}^{**}\Lambda _{r}}{\mu _{r}(\beta _{2}^{**} + \mu _{r})} \end{aligned}$$
(8)
where the force of infection is given as
$$\begin{aligned} \beta _{1}^{**}=\frac{\beta _{r}I_{r}^{**}+\beta _{h}I_{h}^{**}}{N_{h}^{**}}, \qquad and \qquad \beta _{2}^{**}=\frac{\beta _{r}I_{r}^{**}}{N_{r}^{**}} \end{aligned}$$
(9)
Substituting expression (8) into the force of infection (9) at steady state yields the following polynomial
$$\begin{aligned} p_{1}(\beta _{1}^{**})^{2}+p_{2}\beta _{1}^{**}-p_{3} =0 \end{aligned}$$
(10)
where the coefficients \(p_{i}\) for \(i=1\ldots ,3\) of the polynomial are given as
$$\begin{aligned} p_{1}= & {} \mu _{r}{\mathcal {R}}_{r}\left( \Lambda _{h}k_{2}k_{3}+\Lambda _{h}\sigma _{h}k_{3}+\Lambda _{h}\sigma _{h}k_{3}+\Lambda _{h}\sigma _{h}\phi _{h}\right) \\ p_{2}= & {} \mu _{r}{\mathcal {R}}_{r}\left[ \Lambda _{h}k_{1}k_{2}k_{3}+\beta _{h}\Lambda _{h}\sigma _{h}k_{3}+\mu _{r}({\mathcal {R}}_{r}-1)(k_{1}k_{2}k_{3}-\sigma _{h}\tau _{h}\phi _{h})\right] \\ p_{3}= & {} \Lambda _{r}k_{1}k_{2}k_{3} \mu _{r}^{2}{\mathcal {R}}_{r}({\mathcal {R}}_{r}-1) \end{aligned}$$
It can be seen that the coefficient \(p_{1}\) is positive while the sign of \(p_{2}\) and \(p_{3}\) depends on the values of the reproduction number. That is, if \(\left\{ {\mathcal {R}}_{h}, {\mathcal {R}}_{r} \in {\mathcal {R}}_{0}>1 \right\}\), then \(p_{2}>0\) and \(p_{3}>0\). In addition, for \(p_{2}\) to be positive, \(k_{1}k_{2}k_{3}>\sigma _{h}\tau _{h}\phi _{h}\) so that there is at least one sign change in the sequence of coefficients \(p_{1}, p_{2}, p_{3}\). Thus, by Descartes rule of signs, there exists at least one positive real root for (10) whenever \({\mathcal {R}}_{0}>1\). Therefore, the following result is established.
Lemma 4
The model system (1) has at least one endemic equilibrium whenever \({\mathcal {R}}_{0}>1\).