Role of incentives on the dynamics of infectious diseases: implications from a mathematical model

Abstract

In the present study, we assess the impact of incentives provided by the government on the dynamics of infectious diseases that spread due to the direct contacts of susceptibles with the infectives and also through the environmental contamination of those diseases. To this, we develop a mathematical model which comprises susceptible individuals, infected individuals, environmental contamination and the incentive provided by the government healthcare officials as dynamic variables. The proposed epidemic model is analyzed mathematically as well as numerically. System’s dynamics have been mainly studied about the disease-free and interior equilibrium points. We perform some sensitivity tests to find out model parameters that can have major roles in regulating the epidemic pattern. Our findings show that the disease could be effectively controlled by reducing the contact rate of susceptibles with the infected individuals and also their exposure to the environmental contamination. This could be possible by raising awareness among the public and using disinfectants of high quality for removing contaminants from the environment. These results suggest to increase the incentive amount in the areas facing rapid incline in the number of infected cases and the level of environmental contamination.

Appendices

Appendix A

Let $X=[S,I,E_c,M]$, then system (1) can also be written as follows:

$$\begin{aligned} \dfrac{{\text {d}}X}{{\text {d}}t}& = {} EX+F, \end{aligned}$$

where $F=[\Lambda ,0,0,0]^T$ is a nonnegative vector and

$$\begin{aligned} E=\begin{bmatrix} -E_{11} &{} \nu &{} 0 &{} 0 \\ E_{21} &{} -E_{22} &{} 0 &{} 0 \\ 0 &{} s_1 &{} -E_{33} &{} 0 \\ 0 &{} 0 &{} 0 &{} E_{44} \end{bmatrix} \end{aligned}$$

with

$$\begin{aligned}{} & {} E_{11}=\left( \beta -\beta _1\dfrac{k_1M}{p+k_1 M}\right) I+\psi _e\dfrac{E_c}{L+E_c}+d, \ E_{21}=\left( \beta -\beta _1\frac{k_1M}{p+k_1 M}\right) I+\psi _e\dfrac{E_c}{L+E_c },\\{} & {} E_{22}=\nu +\alpha +d, \ E_{33}=s_0+\eta \dfrac{(1-k_1)M}{q+(1-k_1)M}, \ E_{44}=r\left( 1-\dfrac{M}{K}\right) +\theta _1I+\theta _2E_c. \end{aligned}$$

Apparently, all the off-diagonal entries of matrix E(X) are nonnegative ensuring that the matrix is Metzler for all $X\in {\mathbb {R}}_+^4$. Thus, system (1) is positively invariant in ${\mathbb {R}}_+^4$ [38], i.e., all the solution trajectories of the system starting from an initial state in ${\mathbb {R}}_+^4$ confine there for all $t\ge 0$.

We obtain the following inequality from the second equation of system (3):

$$\begin{aligned} \frac{{\text {d}}N}{{\text {d}}t}\le \Lambda -dN\Rightarrow \frac{{\text {d}}}{{\text {d}}t}(Ne^{dt}) \le \Lambda e^{dt}. \end{aligned}$$

On integrating the above inequality from 0 to t, we get $\displaystyle N(t)\le N_0e^{-dt}+\frac{\Lambda }{d}.$ This yields that $\displaystyle \limsup _{t\rightarrow \infty } N(t)\le \dfrac{\Lambda }{d}.$ Thus, $\displaystyle N(t)\le \dfrac{\Lambda }{d}$ for large $t>0$. As $I(t)\le N(t)$ at any time, so $I(t) \le \dfrac{\Lambda }{d}$ for any large value of t.

Now, from the third equation of system (3), we have

$$\begin{aligned} \dfrac{{\text {d}}E_c}{{\text {d}}t}\le s_1 \dfrac{\Lambda }{d}-s_0 E_c. \end{aligned}$$

Using the theory of differential inequality [39], we get $\displaystyle \limsup _{t\rightarrow \infty} E_c(t)\le \dfrac{s_1 \Lambda }{ds_0}.$ This implies that $\displaystyle E_c(t)\le \dfrac{s_1 \Lambda }{ds_0}$ for large $t>0$.

Again, the last equation of system (3) yields the following inequality:

$$\begin{aligned} \frac{{\text {d}}M}{{\text {d}}t}+\dfrac{r}{K}M^2\le \left[ r+\dfrac{\Lambda }{d}\left( \theta _1+\theta _2\frac{s_1}{s_0}\right) \right] M. \end{aligned}$$

By using the comparison theorem from [39], we get

$$\begin{aligned} \limsup _{t\rightarrow \infty }M(t)\le \dfrac{K}{r}\left[ r+\dfrac{\Lambda }{d}\left( \theta _1+\theta _2\dfrac{s_1}{s_0}\right) \right] . \end{aligned}$$

Thus, for any large value of t, we have

$$\begin{aligned} M(t)\le \dfrac{K}{r}\left[ r+\dfrac{\Lambda }{d}\left( \theta _1+\theta _2\dfrac{s_1}{s_0}\right) \right] . \end{aligned}$$

Thus, we find that the region $\Omega$ is positively invariant, i.e., all the solutions of system (3) with initial conditions in $\Omega$ remain therein for any large value of $t>0$. Also, $\Omega$ attracts all the solutions of system (3) with initial condition in ${\mathbb {R}}^4_+$. Hence, system (3) is mathematically well posed in the region $\Omega$.

Appendix B

We find the Jacobian matrix of system (3) as $J=[J_{ij}]_{4\times 4}$ with the following entries:

$$\begin{aligned} J_{11}& = {} \left( \beta -\beta _1\dfrac{k_1M}{p+k_1M}\right) (N-2I)-\psi _e\dfrac{E_c}{L+E_c}-(\nu +\alpha +d), \ J_{12}=\left( \beta -\beta _1\dfrac{k_1M}{p+k_1M}\right) I+\dfrac{\psi _e E_c}{L+E_c},\\ J_{13}& = {} \dfrac{L\psi _e}{(L+E_c)^2}(N-I), \ J_{14}=-\dfrac{p\beta _1k_1}{(p+k_1M)^2}(N-I)I,\ J_{33}=-s_0-\eta \dfrac{(1-k_1)M}{q+(1-k_1)M}, \\ J_{34}& = {} -\eta \dfrac{q(1-k_1)E_c}{(q+(1-k_1)M)^2}, \ J_{44}=r\left( 1-\dfrac{2M}{K}\right) +\theta _1I+\theta _2E_c, \ J_{31}=s_1, \ J_{21}=-\alpha , \ J_{22}=-d,\\ J_{23}& = {} J_{24}=J_{32}=J_{42}=0, \ J_{41}=\theta _1M, \ J_{43}=\theta _2M. \end{aligned}$$

On evaluating the Jacobian matrix J at the equilibrium $E_1$, we get two eigenvalues as $-d$ and r whereas the remaining two will be obtained by solving the following equation:

$$\begin{aligned} \xi ^2+\left( s_0+\nu +\alpha +d-\frac{\beta \Lambda}{d}\right) \xi +\left[ s_0(\nu +\alpha +d)-\frac{\Lambda }{d}\left( \beta s_0+\frac{s_1 \psi _e}{L}\right) \right] =0. \end{aligned}$$

Irrespective of signs of the reals parts of the roots of the above equation, the equilibrium point $E_1$ is unstable as one eigenvalue, r, is always positive. Thus, we can say that having an equilibrium situation with susceptible only (no disease and no incentive) is practically impossible. Similarly, the equilibrium $E_2$ with disease but no incentive is unconditionally unstable as one eigenvalue value of the Jacobian matrix J at this equilibrium point is $r+\theta _1I_2+\theta _2E_{c2}$ (always positive) while the remaining three are roots of the following cubic equation:

$$\begin{aligned} \xi ^3+B_1\xi ^2+B_2\xi +B_3=0, \end{aligned}$$

(24)

where

$$\begin{aligned}{} & {} B_1=s_0+\nu +\alpha +2d-\dfrac{\beta }{d}\{\Lambda -(\alpha +2d)I_2\} +\dfrac{\psi _es_1I_2}{s_1I_2+LS_0},\\{} & {} B_2=s_0d+\alpha \beta I_2+ (\alpha -s_0-d)\dfrac{\psi _es_1I_2}{s_0L+s_1I_2} -\dfrac{s_1\psi _es^2_0L\{\Lambda -(\alpha +d)I_2\}}{d(s_0L+s_1I_2)^2}\\{} &\qquad\quad {} -(s_0+d)\left\{ \frac{\beta }{d}\{\Lambda -(\alpha +2d)I_2\}-(\nu +\alpha +d)\right\} ,\\{} & {} B_3=s_0\alpha \beta I_2+ (\alpha +d)\dfrac{s_0\psi _es_1I_2}{s_0L+s_1I_2} -\dfrac{ds_1\psi _es^2_0L\{\Lambda -(\alpha +d)I_2\}}{d(s_0L+s_1I_2)^2}\\{} &\qquad\quad {} -s_0d\left\{ \frac{\beta }{d}\{\Lambda -(\alpha +2d)I_2\}-(\nu +\alpha +d)\right\} . \end{aligned}$$

The unstable nature of the incentive-free equilibrium $E_2$ indicates that having such equilibrium situation is a realistic scenario in impossible, i.e., in a region of epidemic outbreak, government always provides some incentives for the prevention of the disease. On computing the matrix J at the equilibrium $E_3$, we obtain the two eigenvalues as $-r$ and $-d$ , whereas the other two are roots of the equation below:

$$\begin{aligned}{} & {} \xi ^2+\left[ s_0+\nu +\alpha +d+\frac{\eta (1-k_1)K}{q+(1-k_1)K} -\frac{\Lambda }{d}\left( \beta -\frac{\beta _1k_1K}{p+k_1K}\right) \right] \xi \nonumber \\{} & {} +(1-{\mathcal {R}}_0)(\nu +\alpha +d)\left( s_0+\frac{\eta (1-k_1)K}{q+(1-k_1)K}\right) =0. \end{aligned}$$

(25)

Clearly, if ${\mathcal {R}}_0<1$ and the condition (16) is satisfied, then the roots of above equation are either negative or have negative real parts leading to the stability of the equilibrium $E_3$. Next, we evaluate the matrix J at the equilibrium $E^*$ and obtain the associated characteristic equation as,

$$\begin{aligned} \xi ^4+A_1\xi ^3+A_2\xi ^2+A_3\xi +A_4=0, \end{aligned}$$

(26)

where

$$\begin{aligned} A_1& = {} d+\frac{r}{K}M^*-a_{11}-a_{33},\\ A_2& = {} \alpha a_{12}+a_{11}a_{33}+d\frac{r}{K}M^* -\left\{ (a_{11}+a_{33})\left( \frac{r}{K}M^*+d\right) +s_1a_{13}+\theta _1 a_{14}M^*\right\} ,\\ A_3& = {} \alpha a_{12}\frac{r}{K}M^*+\theta _1a_{14}a_{33}M^*+da_{11}a_{33}\\{} &\qquad {} -\left[ d(\theta _2a_{34}M^*+\theta _1a_{14}M^* +s_1a_{13})+\alpha a_{12}a_{33}+\theta _1a_{13}a_{34}M^* +s_1\theta _2a_{14}M^*+a_{11}a_{34}\theta _2M^*\right. \\{} &\qquad {} \left. +\frac{r}{K}M^*(s_1a_{13}+a_{11}a_{33}+da_{33}+da_{11})\right] ,\\ A_4& = {} \left[ \left( a_{33}\frac{r}{K}+\theta _2a_{34}\right) (da_{11}-\alpha a_{12})-d\left\{ a_{13}\left( s_1\frac{r}{K} +\theta _1a_{34}\right) +a_{14}(s_1\theta _2-\theta _1a_{33})\right\} \right] M^* \end{aligned}$$

with

$$\begin{aligned}{} & {} a_{11}=-\left( \beta -\beta _1\dfrac{k_1M^*}{p+k_1M^*}\right) I^* -\dfrac{\psi _eE_c^*}{L+E_c^*}\dfrac{N^*}{I^*}, \ a_{12} =(\nu +\alpha +d)\dfrac{I^*}{N^*-I^*},\\{} & {} a_{13}=\dfrac{\psi _eL}{(L+E_c^*)^2}(N^*-I^*), \ a_{14} =-\dfrac{\beta _1k_1p}{(p+k_1M^*)^2}(N^*-I^*)I^*, \ a_{21}=-\alpha , \ a_{22}=-d,\\{} & {} a_{23}=a_{24}=a_{32}=0, \ a_{31}=s_1, \ a_{33}=-s_1 \dfrac{I^*}{E_c^*}, \ a_{34}=-\dfrac{\eta q(1-k_1)E_c^*}{(q+(1-k_1)M^*)^2}, \ a_{41}=\theta _1M^*,\\{} & {} a_{42}=0, \ a_{43}=\theta _2M^*, \ a_{44}=-\dfrac{r}{K}M^*. \end{aligned}$$

The roots of equation (26) are either negative or have negative real parts if and only if all the inequalities in (17) hold; in view of the Routh-Hurwitz criterion [42], these conditions imply the local asymptotic stability of the equilibrium $E^*$. Note that the equilibrium $E^*$ will be no longer stable if any of the conditions in (17) does not hold.

Appendix C

Considering only the variables I and $E_c$, we get the following subsystem of system (3):

$$\begin{aligned} \dfrac{{\text {d}}I}{{\text {d}}t}& = {} \left( \beta -\beta _1\dfrac{k_1M}{p+k_1M}\right) (N-I)I +\psi _e\dfrac{E_c}{L+E_c}(N-I)-(\nu +\alpha +d)I:=f_1,\\ \dfrac{{\text {d}}E_c}{{\text {d}}t}& = {} s_1I-s_0E_c-\eta \dfrac{(1-k_1)M}{q+(1-k_1)M}E_c:=f_2. \end{aligned}$$

Defining $\displaystyle h(I,E_c)=\dfrac{1}{IE_c},$ we have $\displaystyle \Delta _{E_3}(I,E_c)=\dfrac{\partial }{\partial I}(hf_1)+\dfrac{\partial }{\partial E_c}(hf_2).$ Obviously, $h(I,E_c)>0$ for all $I,E_c>0$. Thus,

$$\begin{aligned} \Delta _{E_3}(I,E_c)=\dfrac{\partial }{\partial I}(hf_1) +\dfrac{\partial }{\partial E_c}(hf_2)=-\left[ \dfrac{1}{E_c} \left( \beta -\beta _1\dfrac{k_1M}{p+k_1 M}\right) +\dfrac{\psi _e}{I^2}\dfrac{N}{L+E_c}+\dfrac{s_1}{E^2_c}\right] <0. \end{aligned}$$

Obviously, $\Delta _{E_3}(I,E_c)$ does not change its sign and is also not identically zero in the positive quadrant of the $I-E_c$ plane. Thus, according to the Bendixson–Dulac criterion [40], system (3) could not exhibit any limit cycle in the positive quadrant of the $I-E_c$ plane. As the disease-free equilibrium $E_3$ is locally asymptotically stable whenever ${\mathcal {R}}_0<1$, so it is globally asymptotically stable in the positive quadrant of the $I-E_c$ plane for ${\mathcal {R}}_0<1$. In the same line, one can show that the disease-free equilibrium $E_3$ of system (3) is globally asymptotically in the other planes also [40].

Appendix D

Corresponding to system (3), we consider the following positive definite function:

$$\begin{aligned} G=\left[ I-I^*-I^*\ln \left( \dfrac{I}{I^*}\right) \right] +\dfrac{1}{2}m_1(N-N^*)^2 +\dfrac{1}{2}m_2(E-E^*_c)^2+m_3\left[ M-M^*-M^*\ln \left( \dfrac{M}{M^*}\right) \right] , \end{aligned}$$

where $m_1,m_2$ and $m_3$ are some positive constants to be chosen later. On calculating the time derivative of the function G along the solutions of system (3), choosing $\displaystyle m_1=\dfrac{1}{\alpha }\left( \beta -\beta _1\dfrac{k_1M^*}{p+k_1M^*}\right)$ and rearranging the terms, we get

$$\begin{aligned} \frac{{\text {d}}G}{{\text {d}}t}& = {} -\left[ \left( \beta -\beta _1\dfrac{k_1M^*}{p+k_1M^*}\right) +\dfrac{\psi _eE_cN}{(L+E^*_c)II^*}\right] (I-I^*)^2-\dfrac{d}{\alpha } \left( \beta -\beta _1\dfrac{k_1M^*}{p+k_1M^*}\right) (N-N^*)^2 \\ \vphantom{1_{1_{1_{1_{1_{1_{1}}}}}}}{} &\qquad {} -m_2\left( s_0+\eta \dfrac{(1-k_1)M^*}{q+(1-k_1)M^*}\right) (E_c-E^*_c)^2 -m_3\dfrac{r}{K}(M-M^*)^2+\dfrac{\psi _eE_c}{(L+E_c)I^*}(I-I^*)(N-N^*)\\{} &\qquad {} +\dfrac{\psi _eL(N^*-I^*)}{(L+E_c)(L+E_c^*)I^*}(I-I^*)(E_c-E^*_c)+m_2s_1(I-I^*)(E_c-E^*_c)\\{} &\qquad {} -\dfrac{\beta _1k_1p(N-I)}{(p+k_1M)(p+k_1M^*)}(I-I^*)(M-M^*)+m_3\theta _1(I-I^*)(E_c-E^*_c)\\{} &\qquad {} -m_2\left( \dfrac{\eta (1-k_1)qE_c}{(q+(1-k_1)M)(q+(1-k_1)M^*)}\right) (E_c-E^*_c)(M-M^*)+m_3\theta _2(M-M^*)(E_c-E^*_c). \end{aligned}$$

Notably, the derivative function $\displaystyle \dfrac{{\text {d}}G}{{\text {d}}t}$ will be negative definite inside $\Omega$ if the following inequalities are satisfied:

$$\begin{aligned} \left( \dfrac{\psi _es_1\Lambda }{dLs_0I^*}\right) ^2< & {} \dfrac{4}{5} \left( \dfrac{d}{\alpha }\right) \left( \beta -\beta _1\frac{k_1M^*}{p+k_1M^*}\right) ^2, \end{aligned}$$

(27)

$$\begin{aligned} \left( \frac{\psi _e(N^*-I^*)}{(L+E^*_c)I^*}\right) ^2< & {} \dfrac{m_2}{5} \left( s_0+\frac{\eta (1-k_1)M^*}{q+(1-k_1)M^*}\right) \left( \beta -\beta _1\frac{k_1M^*}{p+k_1M^*}\right) , \end{aligned}$$

(28)

$$\begin{aligned} m_2s_1^2< & {} \frac{1}{5}\left( s_0+\frac{\eta (1-k_1)M^*}{q+(1-k_1)M^*}\right) \left( \beta -\beta _1\frac{k_1M^*}{p+k_1M^*}\right) , \end{aligned}$$

(29)

$$\begin{aligned} \left( \frac{\beta _1k_1\Lambda }{d(p+k_1M^*)}\right) ^2< & {} \left( \frac{m_3r}{5K}\right) \left( \beta -\beta _1\frac{k_1M^*}{p+k_1M^*}\right) , \end{aligned}$$

(30)

$$\begin{aligned} m_3\theta ^2_1< & {} \frac{1}{5}\left( \frac{r}{K}\right) \left( \beta -\beta _1\frac{k_1M^*}{p+k_1M^*}\right) , \end{aligned}$$

(31)

$$\begin{aligned} m_2\left( \frac{\eta (1-k_1)s_1\Lambda }{ds_0(q+(1-k_1)M^*)}\right) ^2< & {} \frac{1}{4}m_3\left( \frac{r}{K}\right) \left( s_0+\eta \frac{(1-k_1)M^*}{q+(1-k_1)M^*}\right) , \end{aligned}$$

(32)

$$\begin{aligned} m_3\theta ^2_2< & {} \left( \frac{m_2}{4}\right) \left( \frac{r}{K}\right) \left( s_0+\eta \frac{(1-k_1)M^*}{q+(1-k_1)M^*}\right) . \end{aligned}$$

(33)

From inequalities (28) and (29), one can choose a positive value of $m_2$ if inequality (19) holds. Also, from inequalities (30)–(33), one can get a positive value of $m_3$ if the inequality (20) holds. Thus, one can conclude that $\displaystyle \dfrac{{\text {d}}G}{{\text {d}}t}$ will be negative definite inside $\Omega$ if inequalities (18)−(20) hold.

Appendix E

The second additive compound matrix associated with the matrix $J_{E^*}$ is obtained as,

$$\begin{aligned} J_{E^*}^{[2]}=\begin{bmatrix} a_{11}+a_{22} &{} 0 &{} 0 &{} -a_{13} &{} -a_{14} &{} 0\\ 0 &{} a_{11}+a_{33} &{} a_{34} &{} a_{12} &{} 0 &{} -a_{14}\\ 0 &{} a_{43} &{} a_{11}+a_{44} &{} 0 &{} a_{12} &{} a_{13}\\ -a_{31} &{} a_{21} &{} 0 &{} a_{22}+a_{33} &{} a_{34} &{} 0\\ -a_{41} &{} 0 &{} a_{21} &{} a_{43} &{} a_{22}+a_{44} &{} 0\\ 0 &{} -a_{41} &{} a_{31} &{} 0 &{} 0 &{} a_{33}+a_{44} \end{bmatrix}. \end{aligned}$$

Define $|Y|_\infty =\sup |Y_i|.$ The logarithmic norm $\mu _\infty (J_{E^*}^{[2]})$ of the matrix $J_{E^*}^{[2]}$ endowed with the vector norm $|Y|_\infty$ is determined by the supremum of the following quantities:

$$\begin{aligned}{} & {} a_{11}+a_{22}+|a_{13}|+|a_{14}|, \ a_{11}+a_{33}+|a_{34}|+|a_{12}|+|a_{14}|, \ |a_{43}|+a_{11}+a_{44}+|a_{12}|+|a_{13}|,\\{} & {} |a_{31}|+|a_{21}|+a_{22}+a_{33}+|a_{34}|, \ |a_{41}|+|a_{21}|+|a_{43}|+a_{22}+a_{44}, \ |a_{41}|+|a_{31}|+a_{33}+a_{44}. \end{aligned}$$

Notably, ($a_{11}+a_{22}+|a_{13}|+|a_{14}|)_{E^*}<0$ if

$$\begin{aligned} \dfrac{\psi _eL}{(L+E_c^*)^2}(N^*-I^*)+\dfrac{\beta _1k_1p}{(p+k_1M^*)^2}(N^*-I^*)I^* <\left( \beta -\beta _1\dfrac{k_1M^*}{p+k_1M^*}\right) I^*+\dfrac{\psi _eE_c^*}{L+E_c^*}\dfrac{N^*}{I^*}. \end{aligned}$$

(34)

Also, $(a_{11}+a_{33}+|a_{34}|+|a_{12}|+|a_{14}|)_{E^*}<0$ if

$$\begin{aligned}{} & {} \dfrac{\beta _1k_1p}{(p+k_1M^*)^2}(N^*-I^*)I^*+(\nu +\alpha +d)\dfrac{I^*}{N^*-I^*} +\dfrac{\beta _1k_1p}{(p+k_1M^*)^2}(N^*-I^*)I^*\nonumber \\{} & {} <\dfrac{\psi _eE_c^*}{L+E_c^*}\dfrac{N^*}{I^*} +\left( \beta -\beta _1\dfrac{k_1M^*}{p+k_1M^*}\right) I^*. \end{aligned}$$

(35)

Again, $(|a_{43}|+a_{11}+a_{44}+|a_{12}|+|a_{13}|)_{E^*}<0$ if

$$\begin{aligned}{} & {} \theta _2M^*+(\nu +\alpha +d)\dfrac{I^*}{N^*-I^*}+\dfrac{\psi _eL}{(L+E_c^*)^2}(N^*-I^*)\nonumber \\{} & {} <\left( \beta -\beta _1\dfrac{k_1M^*}{p+k_1M^*}\right) I^* +\dfrac{\psi _eE_c^*}{L+E_c^*}\dfrac{N^*}{I^*}+\dfrac{r}{K}M^*. \end{aligned}$$

(36)

Further, $(|a_{31}|+|a_{21}|+a_{22}+a_{33}+|a_{34}|)_{E^*}<0$ if

$$\begin{aligned} s_1+\alpha +\dfrac{\eta q(1-k_1)E_c^*}{(q+(1-k_1)M^*)^2}<d+s_1\dfrac{I^*}{E_c^*}. \end{aligned}$$

(37)

Furthermore, $(|a_{41}|+|a_{21}|+|a_{43}|+a_{22}+a_{44})_{E^*}<0$ if

$$\begin{aligned} \alpha +\theta _1M^*+\theta _2M^*<d+\dfrac{r}{K}M^*. \end{aligned}$$

(38)

Finally, ($|a_{41}+|+|a_{31}|+a_{33}+a_{44})_{E^*}<0$ if

$$\begin{aligned} s_1+\theta _1M^*<s_1\dfrac{I^*}{E_c^*}+\dfrac{r}{K}M^*. \end{aligned}$$

(39)

Combining the inequalities (34)−(39), one can get the required condition (23).