# Nonstandard Finite Difference Schemes for the Study of the Dynamics of the Babesiosis Disease

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

- (i)
- System (2) has the disease-free equilibrium point ${F}_{1}^{*}=({S}_{B1}^{*},{I}_{B1}^{*},{I}_{T1}^{*})=(1,0,0)$ for all values of the parameters in this system, whereas, only if ${\mathcal{R}}_{0}>1$, there is a (unique) endemic equilibrium point ${F}_{2}^{*}=({S}_{B2}^{*},{I}_{B2}^{*},{I}_{T2}^{*})$ in the interior of $\mathsf{\Omega}$ given by$$\left\{\begin{array}{cc}{S}_{B2}^{*}\hfill & ={\displaystyle \frac{{\beta}_{T}{\lambda}_{B}({\alpha}_{B}+{\mu}_{B})+p{\lambda}_{B}({\alpha}_{B}+{\lambda}_{B}+{\mu}_{B}){\mu}_{T}}{{\beta}_{T}\left[{\alpha}_{B}({\beta}_{B}+{\lambda}_{B})+{\lambda}_{B}+{\mu}_{B}+{\beta}_{B}({\lambda}_{B}+{\mu}_{B})\right]}},\hfill \\ {I}_{B2}^{*}\hfill & ={\displaystyle \frac{({\alpha}_{B}+{\mu}_{B})({\beta}_{B}{\beta}_{T}-p{\lambda}_{B}{\mu}_{T})}{{\beta}_{T}\left[{\alpha}_{B}({\beta}_{B}+{\lambda}_{B})+{\lambda}_{B}+{\mu}_{B}+{\beta}_{B}({\lambda}_{B}+{\mu}_{B})\right]}},\hfill \\ {I}_{T2}^{*}\hfill & ={\displaystyle \frac{({\alpha}_{B}+{\mu}_{B})({\beta}_{B}{\beta}_{T}-p{\lambda}_{B}{\mu}_{T})}{{\beta}_{T}{\beta}_{B}({\alpha}_{B}+{\mu}_{B})+p{\beta}_{B}({\alpha}_{B}+{\lambda}_{B}+{\mu}_{B}){\mu}_{T}}}.\hfill \end{array}\right.$$
- (ii)
- If ${\mathcal{R}}_{0}\le 1$, then the disease-free point ${F}_{1}^{*}$ is globally asymptotically stable; otherwise, the disease-free point ${F}_{1}^{*}$ is unstable.
- (iii)
- If ${\mathcal{R}}_{0}>1$, then the endemic point ${F}_{2}^{*}$ is shown to be locally asymptotically stable by numerical simulations.

## 3. Nonstandard Finite Difference Schemes for System (2)

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**2.**

- (i)
- If${\mathcal{R}}_{0}\le 1$, then the disease-free equilibrium point${F}_{1}^{*}$is globally asymptotically stable.
- (ii)
- If${\mathcal{R}}_{0}>1$, then the disease-free equilibrium point${F}_{1}^{*}$is unstable.

**Proof.**

- (i)
- We will use an extension for the discrete case (see in [28], Theorem 3.3) of the Lyapunov stability theorem [29] to prove this part. For this purpose, consider a function $V:\mathsf{\Omega}\to {\mathbb{R}}_{+}$ defined by$$V({S}_{B}^{k},{I}_{B}^{k},{I}_{T}^{k}):={\beta}_{T}{I}_{B}^{k}+{\lambda}_{B}{I}_{T}^{k}.$$Clearly, V is continuous, $V\left({S}_{B}^{k},{I}_{B}^{k},{I}_{T}^{k}\right)\ge 0$ for all $\left({S}_{B}^{k},{I}_{B}^{k},{I}_{T}^{k}\right)\in \mathsf{\Omega}$, and $V\left({F}_{1}^{*}\right)=0$.From (6), we have$$\begin{array}{cc}\hfill \mathsf{\Delta}V({S}_{B}^{k},\phantom{\rule{0.166667em}{0ex}}{I}_{B}^{k},\phantom{\rule{0.166667em}{0ex}}{I}_{T}^{k})& :=V({S}_{B}^{k+1},{I}_{B}^{k+1},{I}_{T}^{k+1})-V({S}_{B}^{k},{I}_{B}^{k},{I}_{T}^{k})={\beta}_{T}({I}_{B}^{k+1}-{I}_{B}^{k})+{\lambda}_{B}({I}_{T}^{k+1}-{I}_{T}^{k}),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\phi {\beta}_{T}\left({\beta}_{B}{S}_{B}^{k}{I}_{T}^{k}-{\lambda}_{B}{I}_{B}^{k}\right)+\phi {\lambda}_{B}\left[{\beta}_{T}\left(1-{I}_{T}^{k}\right){I}_{B}^{k}-{\mu}_{T}p{I}_{T}^{k}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \phi {\beta}_{T}\left({\beta}_{B}{I}_{T}^{k}-{\lambda}_{B}{I}_{B}^{k}\right)+\phi {\lambda}_{B}\left[{\beta}_{T}\left(1-{I}_{T}^{k}\right){I}_{B}^{k}-{\mu}_{T}p{I}_{T}^{k}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\phi \left[({\beta}_{T}{\beta}_{B}-{\lambda}_{B}{\mu}_{T}p){I}_{T}^{k}-{\lambda}_{B}{\beta}_{T}{I}_{B}^{k}{I}_{T}^{k}\right]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \le \phi ({\beta}_{T}{\beta}_{B}-{\lambda}_{B}{\mu}_{T}p){I}_{T}^{k}=\phi {\lambda}_{B}{\mu}_{T}p({\mathcal{R}}_{0}-1){I}_{T}^{k},\hfill \end{array}$$Let ${G}^{*}$ be the largest positively invariant set contained in$$G:=\left\{\left({S}_{B}^{k},{I}_{B}^{k},{I}_{T}^{k}\right)\in \mathsf{\Omega}|\mathsf{\Delta}V=0\right\}.$$Then, by using (9) we have that$${G}^{*}=\left\{\begin{array}{c}\left\{{F}_{1}^{*}\right\}\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{1.em}{0ex}}{\mathcal{R}}_{0}<1,\hfill \\ \\ \left\{(0,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}{I}_{T}^{k})|{I}_{T}^{k}\ge 0\right\}\phantom{\rule{1.em}{0ex}}\mathrm{if}\phantom{\rule{1.em}{0ex}}{\mathcal{R}}_{0}=1.\hfill \end{array}\right.$$Consequently, it is easy to verify that ${F}_{1}^{*}$ is ${G}^{*}$-globally asymptotically stable if ${\mathcal{R}}_{0}\le 1$.
- (ii)
- Computing the Jacobian matrix of system (6) evaluated at the disease free point, one obtains$$J(1,0,0)=\left(\begin{array}{ccc}1-\phi ({\alpha}_{B}+{\mu}_{B})& -\phi ({\alpha}_{B}+{\mu}_{B})& -\phi {\beta}_{B},\\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& \\ 0& 1-\phi {\lambda}_{B}& \phi {\beta}_{B}\\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& \\ 0& \phi {\beta}_{T}& 1-\phi {\mu}_{T}p\end{array}\right).$$Consequently, the eigenvalues of $J\left({F}_{1}^{*}\right)$ are ${\mathsf{\Lambda}}_{1}=1-\phi ({\alpha}_{B}+{\mu}_{B})$ and ${\mathsf{\Lambda}}_{2},{\mathsf{\Lambda}}_{3}$, where ${\mathsf{\Lambda}}_{2},{\mathsf{\Lambda}}_{3}$ are the eigenvalues of$${J}_{1}=\left(\begin{array}{cc}1-\phi {\lambda}_{B}& \phi {\beta}_{B}\\ \phantom{\rule{1.em}{0ex}}& \phantom{\rule{1.em}{0ex}}& \\ \phi {\beta}_{T}& 1-\phi {\mu}_{T}p\end{array}\right).$$We have$$det\left({J}_{1}\right)=1-\phi \phantom{\rule{0.166667em}{0ex}}({\lambda}_{B}+{\mu}_{T}p)-{\phi}^{2}{\lambda}_{B}{\mu}_{T}p({\mathcal{R}}_{0}-1),\phantom{\rule{2.em}{0ex}}Tr\left({J}_{1}\right)=2-\phi \phantom{\rule{0.166667em}{0ex}}({\lambda}_{B}+{\mu}_{T}p).$$Thus, if ${\mathcal{R}}_{0}>1$, then $1-Tr\left({J}_{1}\right)+det\left({J}_{1}\right)=-{\phi}^{2}{\lambda}_{B}{\mu}_{T}p({\mathcal{R}}_{0}-1)<0$. By Theorem 1.3.7 in [30] and Theorem 2.10 in [31], we can conclude that if ${\mathcal{R}}_{0}>1$ then the disease-free equilibrium point ${F}_{1}^{*}$ is unstable.Thus, the theorem is proved. □

**Theorem**

**3.**

**Theorem**

**4**(Dynamically consistent discrete models)

**.**

- (i)
- In the case${\mathcal{R}}_{0}\le 1$, scheme (6) preserves positivity, boundedness, and global stability of${F}_{1}^{*}$of system (2) if$$0<\phi \left(h\right)<{\phi}^{*}:=min\left\{{\displaystyle \frac{1}{{\beta}_{B}}},\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{1}{{\lambda}_{B}}},\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{1}{{\mu}_{T}p}},\phantom{\rule{1.em}{0ex}}\frac{1}{{\beta}_{T}},\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{1}{{\mu}_{B}+{\alpha}_{B}}}\right\},\phantom{\rule{2.em}{0ex}}\forall h>0.$$
- (ii)
- In the case${\mathcal{R}}_{0}>1$, scheme (6) preserves positivity, boundedness and local stability of${F}_{2}^{*}$and unstability of${F}_{1}^{*}$if$$0<\phi \left(h\right)<{\phi}^{*}:=min\left\{{\displaystyle \frac{1}{{\beta}_{B}}},\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{1}{{\lambda}_{B}}},\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{1}{{\mu}_{T}p}},\phantom{\rule{1.em}{0ex}}\frac{1}{{\beta}_{T}},\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{1}{{\mu}_{B}+{\alpha}_{B}}},\phantom{\rule{1.em}{0ex}}{\tau}^{*}\right\},\phantom{\rule{2.em}{0ex}}\forall h>0,$$

**Remark**

**1.**

- Part (i) of Theorem 2 is only appropriate when $\mathsf{\Omega}$ is a positively invariant set of (6).

## 4. Numerical Simulations

**Example**

**1**(Dynamics of standard finite difference schemes)

**.**

**Example**

**2**(Dynamics of NSFD schemes in the case ${\mathcal{R}}_{0}\le 1$)

**.**

**Example**

**3**(Dynamics of NSFD schemes in the case ${\mathcal{R}}_{0}>1$)

**.**

## 5. Conclusions

## Supplementary materials

Supplementary File 1## Author Contributions

## Funding

## Conflicts of Interest

## Data Availability

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**Figure 1.**Numerical solutions obtained by the RK4 scheme (with $h=4$), the Euler scheme (with $h=2.5$) and NSFD scheme (with $h=5$ and $\phi \left(h\right)=(1-{e}^{-2h})/2$) in Example 1.

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**MDPI and ACS Style**

Dang, Q.A.; Hoang, M.T.; Trejos, D.Y.; Valverde, J.C.
Nonstandard Finite Difference Schemes for the Study of the Dynamics of the Babesiosis Disease. *Symmetry* **2020**, *12*, 1447.
https://doi.org/10.3390/sym12091447

**AMA Style**

Dang QA, Hoang MT, Trejos DY, Valverde JC.
Nonstandard Finite Difference Schemes for the Study of the Dynamics of the Babesiosis Disease. *Symmetry*. 2020; 12(9):1447.
https://doi.org/10.3390/sym12091447

**Chicago/Turabian Style**

Dang, Quang A., Manh T. Hoang, Deccy Y. Trejos, and Jose C. Valverde.
2020. "Nonstandard Finite Difference Schemes for the Study of the Dynamics of the Babesiosis Disease" *Symmetry* 12, no. 9: 1447.
https://doi.org/10.3390/sym12091447