Positivity and boundednessIn this section, we show the positivity and boundedness of the solution and the existence of feasible region within which the model is epidemiologically and mathematically well posed. The following theorems will provide these results;
Theorem 3.1
Let \(\Omega _0=(S_0,E_0,A_0,I_0,R_0,V_0,L_0)\in {\mathbb {R}}_+ ^7 \cup \{0\}\) be the initial conditions of Eq. (1), then all the solutions of Eq. (1) are non-negative for all \(t \ge 0\).
Proof
Let \(t_1=\sup \{t> 0 ; S(t_0)> 0,E(t_0)>0,A(t_0)>0,I(t_0)>0,R(t_0)>0,V(t_0)>0,L(t_0)>0, \forall t_0 \in [0,t]\}\). From the first equation of Eq. (1) it follows that$$\begin{aligned} \frac{dS}{dt}+(\lambda (t)+\tau +\mu )S=(1-\rho )\Lambda -bqI+\gamma R+ \eta V : \end{aligned}$$
(2)
where, \(\lambda =\frac{\beta _L L}{K+L}+\beta _A A + \beta _I I\). Re-writing (2) using integrating factor gives,$$\begin{aligned} \frac{d}{dt}\left[ e^{\int _{0}^{t_1} (\lambda (z)+\tau +\mu )dz}S\right] = \left[ (1-\rho )\Lambda -bqI + \gamma R+ \eta V\right] e^{\int _{0}^{t_1} (\lambda (z)+\tau +\mu )dz}. \end{aligned}$$
(3)
Hence,$$\begin{aligned} S(t_1)=e^{-\int _{0}^{t_1} (\lambda (z)+\tau +\mu )dz} \left[ S_0 + {\int _{0}^{t_1}{ \left[ (1-\rho )\Lambda -bqI(\xi )) + \gamma R(\xi ) + \eta V(\xi )\right] e^{\int _{0}^{\xi } (\lambda (z)+\tau + \mu )dz}d\xi }}\right] \ge 0. \end{aligned}$$
(4)
Also from vaccination equation of model (1) it follows that$$\begin{aligned} V(t_1)=e^{-\int _{0}^{t_1} ((1-r)\lambda (z)+\mu +\eta )dz} \left[ V_0 + {\int _{0}^{t_1}{ \left[ \rho \Lambda +\tau S(\xi )\right] e^{\int _{0}^{\xi } ((1-r)\lambda (z) + \mu +\eta )dz}d\xi }}\right] \ge 0. \end{aligned}$$
(5)
Similarly, for other state variables$$\begin{aligned} \frac{dE}{dt}\ge & -(\alpha +\mu )E, \qquad E(t)\ge E_0 e^ {-(\alpha +\mu )t}\ge 0, \forall t,\\ \frac{dA}{dt}\ge & -(\sigma +\mu )A, \qquad A(t)\ge A_0 e^ {-(\sigma +\mu )t}\ge 0, \forall t, \\ \frac{dI}{dt}\ge & -(\delta +\mu +d-bq)I, \qquad I(t) \ge I_0 e^ {-(\delta +\mu + d-b q)t}\ge 0, \forall t, \\ \frac{dR}{dt}\ge & -(\gamma +\mu )R, \qquad R(t)\ge R_0 e^ {-(\gamma +\mu )t}\ge 0, \forall t, \\ \frac{dL}{dt}\ge & -\varepsilon L, \qquad L(t)\ge L_0 e^ {-\varepsilon t}\ge 0, \forall t. \end{aligned}$$Thus, all solutions of Eq. (1) are non-negative. \(\square \)
Theorem 3.2
All solutions of the model Eq. (1) are uniformly bounded and contained in the feasible region defined by \(\Omega = \{(S(t), E(t), I(t),A(t), R(t),V(t), L(t))\in {\mathbb {R}}_+^7 ;~ 0\le N(t) \le \frac{\Lambda }{\mu } ,~~ 0 \le L(t)\le \frac{\Lambda (\omega _A+\omega _I)}{\varepsilon \mu }\}\).
Proof
From Eq. (1) we have$$\begin{aligned} \frac{dN}{dt}=\frac{dS}{dt}+\frac{dE}{dt}+\frac{dI}{dt}+\frac{dA}{dt}+\frac{dR}{dt}+\frac{dV}{dt}=\Lambda -\mu N(t)-d I(t). \end{aligned}$$
(6)
Thus, since I and d are positive,$$\begin{aligned} \frac{dN}{dt}\le \Lambda – \mu N . \end{aligned}$$
(7)
Solving (7) we have,$$\begin{aligned} N(t) \le N_0 e^{-\mu t}+\frac{\Lambda }{\mu }\left( 1-e^{-\mu t}\right) . \end{aligned}$$
(8)
As \(t \rightarrow \infty \), the total population \(N(t) \rightarrow \frac{\Lambda }{\mu }\) for \(\mu > 0\). Hence, \(0 \le N(t) \le \frac{\Lambda }{\mu } \) for \(0 \le N_0 \le \frac{\Lambda }{\mu } \). If \(N_0 > \frac{\Lambda }{\mu }\), N(t) decreases to \(\frac{\Lambda }{\mu }\), as \(t \rightarrow \infty \).
Also, from the last equation of model (1)$$\begin{aligned} \frac{dL}{dt}= & \omega _A A + \omega _I I – \varepsilon L. \end{aligned}$$
(9)
Since \(A(t)\le N(t)\) and \(I(t)\le N(t)\), we have \(\frac{dL}{dt} \le \omega _A \frac{\Lambda }{\mu } + \omega _I \frac{\Lambda }{\mu } – \varepsilon L = (\omega _A + \omega _I) \frac{\Lambda }{\mu } – \varepsilon L \), solving using integrating factor yields$$\begin{aligned} L(t)\le L_0 e^{-\varepsilon t}+ \frac{\Lambda (\omega _A+ \omega _I)}{\mu \varepsilon } (1-e^{-\varepsilon t}). \end{aligned}$$
(10)
As \(t \rightarrow \infty \), \(L(t) \rightarrow \frac{\Lambda (\omega _A+ \omega _I)}{\mu \varepsilon }\).
Hence, \(\Omega = \{(S(t), E(t), I(t),A(t), R(t),V(t), L(t))\in {\mathbb {R}}_+^7 ;~ 0\le N(t) \le \frac{\Lambda }{\mu } ,~ 0 \le L(t)\le \frac{\Lambda (\omega _A+\omega _I)}{\varepsilon \mu }\} \) is positively invariant region with respect to the system in Eq. (1). \(\square \)
Remark 1
Theorem (3.2) proves that the total population of the cattle cannot exceed the ratio of constant recruitment rate to the natural death rate. Also the total bacterial load in the environment cannot exceed the ratio of the product of recruitment rate of cattle and shedding rate of asymptomatic and symptomatic cattle to the product of bacterial decay rate and cattle natural death rate. This shows that the model is epidemiologically feasible.
Disease free equilibrium and basic reproduction numberThe disease free equilibrium (DFE) can be found by equating the right-hand side of Eq. (1) to zero by setting disease state equal to zero \((A_0^*=I_0^*=L_0^*=0)\). The remaining state variables result in \(E_0^*=0\), \(R_0^*=0\), \(V_0^*=\dfrac{\Lambda ( \rho \mu +\tau )}{\mu (\eta +\tau +\mu )} \) and \(S_0^*=\dfrac{\Lambda ((1-\rho )\mu +\eta )}{\mu (\eta +\tau +\mu )}\). Hence, the disease free equilibrium point denoted by, \({\mathscr {E}}_0\), is given by$$\begin{aligned} {\mathscr {E}}_0=\left( \dfrac{\Lambda ((1-\rho )\mu +\eta )}{\mu (\eta +\tau +\mu )},0,0,0,0,\dfrac{\Lambda (\mu \rho +\tau )}{\mu (\eta +\tau +\mu )},0\right) . \end{aligned}$$
(11)
To compute the basic reproduction number we used next generation matrix approach presented in38 as follows. The rate of appearance of new infection and rate of transfer from one compartment to another in the infectious classes gives the following:$$\begin{aligned} {\mathscr {F}}=\begin{bmatrix} (\frac{\beta _L L}{K+L}+\beta _A A + \beta _I I ) (S+(1-r)V) \\ 0\\ 0\\ 0\\ \end{bmatrix}, \qquad {\mathscr {V}}= \begin{bmatrix} (\alpha +\mu )E \\ (\sigma +\mu )A-(1-\nu )\alpha E\\ (\delta +\mu +d- bq)I-\nu \alpha E\\ \varepsilon – \omega _A A -\omega _I I\\ \end{bmatrix} . \end{aligned}$$
(12)
Derivation of Eq. (12) with respect to disease state variables and evaluating at \({\mathscr {E}}_0\),$$\begin{aligned} F=\frac{\mathscr {\partial F}_i}{\partial x_j}=\begin{bmatrix} 0& \beta _A (S_0^*+(1-r)V_0^*) & \beta _I (S_0^*+(1-r)V_0^*)& \frac{\beta _L}{K} (S_0^*+(1-r)V_0^*) \\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \end{bmatrix},\\ V=\dfrac{\mathscr {\partial V}_i}{\partial x_j}= \begin{bmatrix} (\alpha +\mu )& 0& 0& 0 \\ -(1-\nu )\alpha & (\sigma +\mu )& 0& 0\\ -\nu \alpha & 0& (\delta +\mu +d)-bq& 0\\ 0& – \omega _A & -\omega _I & \varepsilon \\ \end{bmatrix}. \end{aligned}$$Hence the basic reproduction number \({\mathscr {R}}_0\) is given by the largest spectral radius of the next generation matrix, \(FV^{-1}\). That is$$\begin{aligned} {\mathscr {R}}_0=\rho (FV^{-1})={\mathscr {R}}_A + {\mathscr {R}}_I + {\mathscr {R}}_L : \end{aligned}$$
(13)
where,\({\mathscr {R}}_A=\frac{\beta _A (S_0^*+(1-r)V_0^*) (1-\nu )\alpha }{(\alpha +\mu )(\sigma +\mu )},~ {\mathscr {R}}_I=\frac{\beta _I (S_0^*+(1-r)V_0^*) \nu \alpha }{(\alpha +\mu )[(\delta +\mu +d)-bq]}\) and \({\mathscr {R}}_L=\frac{\beta _L}{K} \frac{(S_0^*+(1-r)V_0^*) (\omega _A(1-\nu )\alpha [(\delta +\mu +d)-bq]+\omega _I\nu \alpha (\sigma +\mu ))}{\varepsilon (\alpha +\mu )(\sigma +\mu )[(\delta +\mu +d)-bq]}\),with \((\delta +\mu +d)-bq > 0\) as \({\mathscr {R}}_0\) is non-negative for epidemiological sense-full model. \({\mathscr {R}}_A \), \({\mathscr {R}}_I\) and \( {\mathscr {R}}_L\) are reproduction numbers that corresponds to transmissions from asymptomatic cattle , infected cattle and from the environment respectively. We can further subdivide \({\mathscr {R}}_L\) for suitability as \({\mathscr {R}}_L={\mathscr {R}}_L^{‘} + {\mathscr {R}}_L ^{”}\),where, \({\mathscr {R}}_L^{‘}=\frac{\beta _L}{K} \frac{(S_0^*+(1-r)V_0^*) \omega _A(1-\nu )\alpha }{\varepsilon (\alpha +\mu )(\sigma +\mu )}\) and \({\mathscr {R}}_L^{”}=\frac{\beta _L}{K} \frac{(S_0^*+(1-r)V_0^*)\omega _I\nu \alpha }{\varepsilon (\alpha +\mu )[(\delta +\mu +d)-bq]}\).
Remark 2
Epidemiologically, \({\mathscr {R}}_A\) and \({\mathscr {R}}_I\) shows a new infection generated by asymptomatic and symptomatic infected cattle respectively prior to their recovery. In particular \({\mathscr {R}}_I\) is regulated by the term \([(\delta +\mu +d)-bq]\) , which implies increasing \((\delta +\mu +d) \) and/or decreasing either of b or q will improve the basic reproduction number. Furthermore, \({\mathscr {R}}_L\) signifies the pace at which Leptospira is released into the environment by asymptomatic and symptomatic infected cattle (\(\omega _A\) and \(\omega _I\)). This factor amplifies the transfer rate of bacteria to susceptible and vaccinated cattle, potentially triggering disease outbreaks. However, implementing control measures that aim to enhance the decay rate of Leptospira in the environment (\(\varepsilon \)) will effectively minimize the rate of new infections.
Local stability of disease free equilibrium
Theorem 3.3
The disease-free equilibrium point, \({\mathscr {E}}_0\), of system in Eq. (1) is locally asymptotically stable whenever \({\mathscr {R}}_0 < 1\) and unstable otherwise.
Proof
The Jacobian matrices of model Eq. (1) at disease free equilibrium (DFE), \({\mathscr {E}}_0\), is given by$$\begin{aligned} \small J=\begin{bmatrix} -k_1 & 0 & -\beta _A S_0^* & -bq -\beta _I S_0^* & \gamma & \eta & \frac{-\beta _L S_0^*}{K} \\ 0& -k_2& \beta _A (S_0^*+(1-r)V_0^*)& \beta _I (S_0^*+(1-r)V_0^*)& 0& 0& \frac{\beta _L (S_0^*+(1-r)V_0^*)}{K} \\ 0& (1-\nu )\alpha & -k_3& 0& 0& 0& 0 \\ 0& \nu \alpha & 0& -k_4& 0& 0& 0\\ 0& 0& \sigma & \delta & -k_5& 0& 0\\ \tau & 0& -\beta _A (1-r)V_0^*& -\beta _I (1-r)V_0^*& 0& -k_6& \frac{-\beta _L (1-r)V_0^*}{K}\\ 0& 0& \omega _A & \omega _I& 0& 0& -\varepsilon \end{bmatrix} \end{aligned}$$
(14)
where,
\(k_1=\tau +\mu ,~ k_2=\alpha +\mu ,~ k_3=\sigma +\mu ,~ k_4=(\delta +\mu +d)-bq,~ k_5=\gamma +\mu ,~ k_6=\eta +\mu \),
\(V_0^*=\dfrac{\Lambda ( \rho \mu +\tau )}{\mu (\eta +\tau +\mu )} \) and \(S_0^*=\dfrac{\Lambda ((1-\rho )\mu +\eta )}{\mu (\eta +\tau +\mu )}\).
The eigenvalues of Jacobian of the matrix in Eq. (14) satisfy the following characteristic polynomial equation.$$\begin{aligned} \lambda ^7+D_6 \lambda ^6 +D_5 \lambda ^5+D_4 \lambda ^4+ D_3 \lambda ^3 + D_2 \lambda ^2 + D_1 \lambda + D_0 =0 . \end{aligned}$$
(15)
Here, \(D_0 > 0\), \(D_1>0\), \(D_2>0\), \(D_3>0\), \(D_4>0\), \(D_5>0\) and \(D_6>0\) for \({\mathscr {R}}_0 < 1\) (i.e \({\mathscr {R}}_A<1\), \({\mathscr {R}}_I<1\), and \({\mathscr {R}}_L<1\)) [see appendix]. Using Descartes rule of signs39 we can conclude that the characteristic equation (15) has all roots of negative real part for \({\mathscr {R}}_0 < 1\). Hence this complete the proof of theorem (3.3). \(\square \)
Remark 3
The biological implication of Theorem (3.3) is that the flow of Leptospira infected cattle will not generate an outbreak of the disease and eventually extincts for \({\mathscr {R}}_0\) less than unity for the herd size sufficiently close to disease free equilibrium as time goes very large.
Global stability of disease free equilibriumIn this section we will present the global stability of the disease free equilibrium using the method of Castillo-Chavez and Song40,
Theorem 3.4
The Disease free equilibrium \({\mathscr {E}}_0\) of the system Eq. (1) is globally asymptotically stable for \({\mathscr {R}}_0< 1\) on its feasible region.
Proof
See Appendix\(\square \)
Remark 4
Epidemiological Theorem (3.4) implies that, independent of the initial available cattle herd size Leptospira infected cattle will not lead to an outbreak of the disease and it extincts in the long run for \({\mathscr {R}}_0\) less than unity.
Existence of endemic equilibriumEndemic equilibrium can be found by setting right side of Eq. (1) to zero and solving for state variables. Let \({\mathscr {E}}^*=(S^*,E^*,A^*,I^*,R^*,V^*,L^*)\) be an endemic equilibrium and expressing all state variables in terms of exposed class, \(E^*\) gives,$$\begin{aligned} A^*= & k_A E^*, \nonumber \\ I^*= & k_I E^*, \nonumber \\ R^*= & k_R E^*,\nonumber \\ L^*= & k_L E^*, \nonumber \\ S^*= & \dfrac{k_2((1-r)\lambda ^*+k_6)E^*-(1-r)\rho \Lambda \lambda ^*}{\lambda ^*((1-r)\lambda ^*+k_6+(1-r)\tau )}, \nonumber \\ V^*= & \dfrac{\rho \Lambda +\tau S^*}{(1-r)\lambda ^*+k_6}, \end{aligned}$$
(16)
where,\(\lambda ^*=\beta _A A^* + \beta _I I^*+\frac{\beta _L L^*}{K+L^*},\qquad k_A=\dfrac{(1-\nu )\alpha }{k_3},\qquad k_I=\dfrac{\nu \alpha }{k_4}, k_R=\dfrac{\sigma k_A + \delta k_I}{k_5},\qquad k_L=\dfrac{\omega _A k_A + \omega _I k_I}{\varepsilon } ,\qquad k_1=\tau +\mu ,\qquad k_2=\alpha +\mu , k_3=\sigma +\mu ,\qquad k_4=(\delta +\mu +d)-bq,\qquad k_5=\gamma +\mu ,\qquad k_6=\eta +\mu \).Substituting Eq. (16) in the first equation of Eq. (1), will give us the following characteristic equation for \(E^*\ne 0\),$$\begin{aligned} A_6 {E^*}^6+A_5 {E^*}^5+A_4 {E^*}^4+A_3 {E^*}^3+A_2 {E^*}^2 + A_1 E^* + A_0 =0 : \end{aligned}$$
(17)
where,\(A_6=(1-r)^2k_L^3(\beta _A k_A+\beta _I k_I)^3(\gamma k_R-bq k_I-k_2)\le -\mu (1-r)^2k_L^3(\beta _A k_A+\beta _I k_I)^3<0\),\(A_0=\mu k_2k_6(k_6+\tau )K^3[{\mathscr {R}}_0-1]>0\), for \({\mathscr {R}}_0>1\).Here, \(A_6<0\) and \(A_0>0 \), which shows that equation Eq. (17) has at least one positive root for \({\mathscr {R}}_0>1\) by exploiting Descartes’ rule of sign39. In fact, substituting the parameter values in Table 1 gives us \(A_6<0, A_5<0, A_4<0, A_3<0, A_2<0, A_1<0, A_0>0 \), which shows the system have unique endemic equilibrium for \({\mathscr {R}}_0>1\) since there is only one sign change for Eq. (17). Because of the complexity of analysis, stability of the unique endemic equilibrium will be discussed graphically in “Stability of disease free equilibrium and endemic equilibrium” section. Moreover the system Eq. (1) undergoes forward bifurcation at \({\mathscr {R}}_0=1\) as observed from Fig. 2.Fig. 2Sensitivity analysisIn this section we have conducted the sensitivity analysis on the basic reproduction number, \({\mathscr {R}}_0\). The measure of sensitivity index of \({\mathscr {R}}_0\) to a parameter, p, denoted by \(\Omega ^{{\mathscr {R}}_0}_p\), as cited in41, is given by$$\begin{aligned} \Omega ^{{\mathscr {R}}_0}_p=\dfrac{\partial {\mathscr {R}}_0}{\partial p}\left( \dfrac{p}{{\mathscr {R}}_0}\right) \end{aligned}$$
(18)
Accordingly, the most sensitive parameters are the vaccine efficiency, r, recruitment rate, \(\Lambda \) and contact rate from asymptomatic infected cattle, \(\beta _A\) respectively. Also the least sensitive parameter is vertical transmission rate, q. As an example a \(10\%\) increase (or decrease) in \(\beta _A\), increase (or decrease) \({\mathscr {R}}_0 \) by \(9.2\%\) and similarly \(10\%\) increase (or decrease) in \(\sigma \), decrease (or increase ) \({\mathscr {R}}_0 \) by \(6.5\%\). The sensitivity indices of \({\mathscr {R}}_0\) with respect to each parameter values are summarized in Table 2.Table 2 Sensitivity of the parameters.Table 2 shows that the dynamics of leptospirosis transmission is much affected by direct transmission from asymptomatic infected cattle than other transmission route. However vertical transmission has least effect on the dynamics.