Effect of awareness program on diabetes mellitus: deterministic and stochastic approach

Abstract

Diabetes mellitus is a silent killer and major public health problem all over the world, but knowledge and awareness about diabetes are insufficient in middle and low-level socioeconomic countries. Awareness plays a vital role in understanding about causes factors of diabetes and its prevention. The world is not completely deterministic as there are biological fluctuations present within the population. With this motivation, we propose and analyze diabetes awareness models with human beings suffering from diabetes mellitus and introducing awareness programs driven by the media in deterministic as well as the stochastic environment. In this work, we investigate the effect of awareness programs on the prevalence of epidemiology of diabetes mellitus. The local stability analysis around the biologically feasible equilibrium point of both the model systems are investigated. The analytical results of the models are verified numerically by taking a set of biologically feasible parameter values. Our study reveals that the prevention of diabetes mellitus in humans may be ensured through an awareness program at the community level.

Appendices

Appendix A

Proof of the Theorem 1

The characteristic equation of the system (2.4) at the equilibrium point ${\overline{E}}_{*}({\overline{X}}_{*}, {\overline{S}}_{A*},{\overline{N}}_{*})$ is

$$\rho ^{3}+\alpha _{1}\rho ^{2}+\alpha _{2}\rho +\alpha _{3}=0.$$

By the Routh Hurwitz stability criterion, the equilibrium point ${\overline{E}}_{*}({\overline{X}}_{*}, {\overline{S}}_{A*},{\overline{N}}_{*})$ is asymptotically stable if $\alpha _{1}>0, \ \alpha _{3}>0$ and $\alpha _{1}\alpha _{2}-\alpha _{3}>0$.

Now $\alpha _{1}=-(a_{11}+a_{22}+a_{33})>0$, using the values of $\{a_{ii}{:}\,i=1,2,3\}$ from (3.3);

$$\begin{aligned} \alpha _{3}= & {} -a_{11}a_{22}a_{33}+a_{12}a_{21}a_{33}-a_{12}a_{31}a_{23}+a_{13}a_{31}a_{22}\\= & {} -a_{11}a_{22}a_{33}+a_{12}a_{21}a_{33}-a_{31}(a_{12}a_{23}-a_{13}a_{22}) \\= & {} -a_{11}a_{22}a_{33}+a_{12}a_{21}a_{33}-a_{31}(\beta \beta _{1}\frac{\lambda {\overline{X}}_{*}}{1+{\overline{X}}_{*}}+\beta \beta _{1}+d\beta )>0, \end{aligned}$$

using the values of $\{a_{ij}{:}\,i,j=1,2,3\}$ from (3.3).

Here

$$\begin{aligned} \alpha _{1}\alpha _{2}-\alpha _{3}= & {} a_{11}a_{22}(-a_{11}-a_{23})+a_{22}a_{22}(-a_{11}+a_{23})+a_{22}a_{33}(-a_{11}-a_{23})\\&\quad +\,\left[ -a_{11}a_{11}a_{33}+a_{11}a_{12}a_{21}+a_{11}a_{13}a_{31}\right. \\&\left. \quad +\,a_{12}a_{21}a_{22}-a_{11}a_{33}a_{33}+a_{13}a_{31}a_{33}+a_{12}a_{31}a_{23}\right] \\ \end{aligned}$$

Therefore $-a_{11}-a_{23}=(\beta +d+e-\frac{\lambda {\overline{X}}_{*}}{1+{\overline{X}}_{*}}) >\beta +d+e-\lambda$, as $0<\frac{\lambda {\overline{X}}_{*}}{1+{\overline{X}}_{*}}<\lambda$,

From the condition $\beta \beta _{1}+d > \lambda$ and $0<\beta <1$ imply that $\beta +d+e-\lambda >0$

Hence $-a_{11}-a_{23}>0.$ Again $a_{11}a_{12}a_{21}=(\beta +d+e)\beta (1-\beta _{1})\frac{(\beta \beta _{1}+d){\overline{S}}_{A*}-\lambda {\overline{X}}_{*}^{2}}{{\overline{X}}_{*}(1+{\overline{X}}_{*})}$ $>0$, using given conditions.

Similarly, using these conditions and the values of $\{a_{ij}{:}\,i,j=1,2,3\}$ from (3.3), the other terms within the square bracket are greater than zero.

Finally $\alpha _{1}\alpha _{2}-\alpha _{3}>0.$

Hence the equilibrium point ${\overline{E}}_{*}({\overline{X}}_{*}, {\overline{S}}_{A*},{\overline{N}}_{*})$ is asymptotically stable if $\beta \beta _{1}+d>\lambda$ and ${\overline{S}}_{A*}>{\overline{X}}_{*}^2$. $\square$

Appendix B

Proof of the Theorem 3

Let us consider the following positive definite Lyapunov function

$$\begin{aligned}&V(U(t),t)=\frac{1}{2}[\omega _{1}u_{1}^{2}+u_{2}^{2}+\omega _{3}u_{3}^{2} +2\omega _{4}u_{1}u_{3}], \end{aligned}$$

(7.1)

where $\omega _{i} \ (i=1, 2, 3)$ are real positive constants to be chosen later. It is easy to check that inequalities (5.4) hold true for the Lyapunov function defined in (7.1) with $\alpha =2$. Furthermore,

$$\begin{aligned} LV(U,t)= & {} (a_{11}u_{1}+a_{12}u_{2}+a_{13}u_{3})\omega _{1}u_{1}+(a_{21}u_{1} +a_{22}u_{2}+a_{23}u_{3})u_{2}\nonumber \\&\quad +(a_{31}u_{1}+a_{32}u_{2}+a_{33}u_{3})\omega _{3}u_{3}\nonumber \\&\quad +(a_{11}u_{1}+a_{12}u_{2}+a_{13}u_{3})\omega _{4}u_{3}+(a_{31}u_{1}+a_{32}u_{2} +a_{33}u_{3})\omega _{4}u_{1}\nonumber \\&\quad + \frac{1}{2}Tr\left[ g^{T}(U)\frac{\partial ^{2}V(U,t)}{\partial U^{2}}g(U)\right] \nonumber \\= & {} \left[ -(\beta +d+e)u_{1}-\beta (1-\beta _{1})u_{2}+\beta u_{3}\right] \omega _{1}u_{1}\nonumber \\&\quad +\left[ \left\{ \frac{(\beta \beta _{1}+d){\overline{S}}_{A*}-\lambda {\overline{X}}_{*}^{2}}{{\overline{X}}_{*}(1+{\overline{X}}_{*})}\right\} u_{1}\nonumber \right. \\&\left. \quad -\left\{ \frac{\lambda {\overline{X}}_{*}}{1+{\overline{X}}_{*}}+ \beta \beta _{1} +d \right\} u_{2}+\frac{\lambda {\overline{X}}_{*}}{1+{\overline{X}}_{*}}u_{3}\right] u_{2} +[-eu_{1}-du_{3}]\omega _{3}u_{3}\nonumber \\&\quad +\left[ -(\beta +d+e)u_{1}-\beta (1-\beta _{1})u_{2}\right. \nonumber \\&\left. \quad +\,\beta u_{3}\right] \omega _{4}u_{3}+[-eu_{1}-du_{3}]\omega _{4}u_{1} +\frac{1}{2}Tr\left[ g^{T}(U)\frac{\partial ^{2}V(U,t)}{\partial U^{2}}g(U)\right] .\nonumber \\ \end{aligned}$$

(7.2)

Now, we find that $\frac{\partial ^{2}V}{\partial U^{2}}\equiv$ $\left[ \begin{array}{ccc} \omega _{1} &{}\quad 0 &{}\quad \omega _{4}\\ 0 &{}\quad 1 &{}\quad 0\\ \omega _{4} &{}\quad 0 &{}\quad \omega _{3} \\ \end{array} \right]$.

Therefore, $g(U(t))^{T}\frac{\partial ^{2}V}{\partial U^{2}}g(U(t))\equiv$ $\left[ \begin{array}{ccc} \omega _{1}\sigma ^{2}_{1}u^{2}_{1} &{}\quad 0 &{}\quad \omega _{4}\sigma _{1}\sigma _{3}u_{1}u_{3}\\ 0 &{}\quad \sigma ^{2}_{2}u^{2}_{2} &{}\quad 0\\ \omega _{4}\sigma _{1}\sigma _{3}u_{1}u_{3} &{}\quad 0 &{}\quad \omega _{3}\sigma ^{2}_{3}u^{2}_{3} \\ \end{array} \right]$ and hence, $\frac{1}{2}Tr[g^{T}(U)\frac{\partial ^{2}V(U,t)}{\partial U^{2}}g(U)]=\frac{1}{2}[\omega _{1}\sigma ^{2}_{1}u^{2}_{1}+\sigma ^{2}_{2}u^{2}_{2}+\omega _{3}\sigma ^{2}_{3}u^{2}_{3}]$.

Using this in (5.1) and simplifying, we get

$$\begin{aligned} LV (U,t)&={} -\left[ (\beta +d+e)\omega _{1}+e\omega _{4}-\frac{\sigma ^{2}_{1}}{2} \omega _{1}\right] u^{2}_{1} \nonumber \\&\quad -\left[ \beta (1-\beta _{1})\omega _{1} -\frac{(\beta \beta _{1}+d) {\overline{S}}_{A*}-\lambda {\overline{X}}_{*}^{2}}{{\overline{X}}_{*}(1 +{\overline{X}}_{*})}\right] u_{1}u_{2} \nonumber \\&\quad -\left[ e\omega _{3}-\beta \omega _{1}+(\beta +d+e)\omega _{4}\right] u_{1}u_{3} -\left[ \frac{\lambda {\overline{X}}_{*}}{1+\lambda {\overline{X}}_{*}}+\beta \beta _{1}+d-\frac{\sigma ^{2}_{2}}{2}\right] u^{2}_{2} \nonumber \\&\quad -\left[ \beta (1-\beta _{1})\omega _{4}-\frac{\lambda {\overline{X}}_{*}}{1+\lambda {\overline{X}}_{*}}\right] u_{2}u_{3}-\left[ \left( d-\frac{\sigma ^{2}_{3}}{2}\right) \omega _{3}-\beta \omega _{4}\right] u^{2}_{3}. \nonumber \\ \end{aligned}$$

(7.3)

If we choose $\omega ^{*}_{1}$, $\omega ^{*}_{2}$ in such away that

$$\begin{aligned} \beta (1-\beta _{1})\omega _{1}-\frac{(\beta \beta _{1}+d){\overline{S}}_{A*}-\lambda {\overline{X}}_{*}^{2}}{{\overline{X}}_{*}(1+{\overline{X}}_{*})}=0 \ and \ \beta (1-\beta _{1})\omega _{4}-\frac{\lambda {\overline{X}}_{*}}{1+ {\overline{X}}_{*}}=0. \end{aligned}$$

i.e.,

$$\begin{aligned} \omega ^{*}_{1}=\frac{(\beta \beta _{1}+d){\overline{S}}_{A*}- \lambda {\overline{X}}_{*}^{2}}{\beta (1-\beta _{1}){\overline{X}}_{*}(1 +{\overline{X}}_{*})} \ \ and \ \omega ^{*}_{4}=\frac{\lambda {\overline{X}}_{*}}{\beta (1-\beta _{1})(1+{\overline{X}}_{*})}. \end{aligned}$$

Then the Eq. (7.3) becomes

$$\begin{aligned} LV(U,t)&< -\left[ (\beta +d+e)\omega ^{*}_{1}-\frac{\sigma ^{2}_{1}}{2} \omega ^{*}_{1}\right] u^{2}_{1}-\left[ e\omega _{3}-\beta \omega ^{*}_{1}+(\beta +d+e)\omega ^{*}_{4}\right] u_{1}u_{3}\nonumber \\&\quad -\left[ \frac{\lambda {\overline{X}}_{*}}{1+\lambda {\overline{X}}_{*}}+\beta \beta _{1}+d-\frac{\sigma ^{2}_{2}}{2}\right] u^{2}_{2}-\left[ \left( d-\frac{\sigma ^{2}_{3}}{2}\right) \omega _{3}-\beta \omega ^{*}_{4}\right] u^{2}_{3}.\nonumber \\ \end{aligned}$$

(7.4)

Thus, we can write

$$\begin{aligned} LV(U,t)<-u^{T}Qu. \end{aligned}$$

(7.5)

where

$$\begin{aligned} Q\equiv \left[ \begin{array}{ccc} m_{11} &{}\quad m_{12} &{}\quad m_{13}\\ m_{21} &{}\quad m_{22} &{}\quad m_{23}\\ m_{31} &{}\quad m_{32} &{}\quad m_{33}\\ \end{array} \right] \end{aligned}$$

with $m_{11}=[(\beta +d+e)\omega ^{*}_{1}-\frac{\sigma ^{2}_{1}}{2}\omega ^{*}_{1}]$; $m_{12}=m_{21}=0$; $m_{13}=m_{31}=\frac{1}{2}[e\omega _{3}-\beta \omega ^{*}_{1}+(\beta +d+e)\omega ^{*}_{4}]$; $m_{22}=[\frac{\lambda {\overline{X}}_{*}}{1+\lambda {\overline{X}}_{*}}+\beta \beta _{1}+d-\frac{\sigma ^{2}_{2}}{2}]$; $m_{23}=m_{32}=0$; $m_{33}=[(d-\frac{\sigma ^{2}_{3}}{2})\omega _{3}-\beta \omega ^{*}_{4}]$.

Thus, we have $m_{ij}\ge 0$ for ${i, j=1,2,3}$; if the conditions (i) to (iii) of the Theorem 3 are hold. Therefore Q is a real symmetric positive definite matrix and hence all the three eigenvalues $\lambda _{i}(Q)$ (say) are real positive. Let $\lambda _{m}=min\{\lambda _{i}(Q),i=1,2,3\}$, then $\lambda _{m}>0$. Therefore, from inequality (7.5), we get $LV(u(t))<-\lambda _{m}|u(t)|^{2}$.

Hence the condition (5.5) of Theorem 2 is satisfied. This complete the proof of the theorem. $\square$