Dynamics and density function for a stochastic anthrax epidemic model

Bing Zhao; Shuting Lyu; Qimin Zhang; Bing Zhao; Shuting Lyu; Qimin Zhang

doi:10.3934/era.2024072

Electronic Research Archive

2024, Volume 32, Issue 3: 1574-1617. doi: 10.3934/era.2024072

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Research article

Dynamics and density function for a stochastic anthrax epidemic model

Bing Zhao ¹,
Shuting Lyu ^{1
,
,},
Qimin Zhang ^{1,2
,
,}

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2.
School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China

Received: 20 November 2023 Revised: 29 January 2024 Accepted: 02 February 2024 Published: 19 February 2024

In response to the pressing need to understand anthrax biology, this paper focused on the dynamical behavior of the anthrax model under environmental influence. We defined the threshold parameter $R^s$ , when $R^s > 1$ ; the disease was almost certainly present and the model exists a unique ergodic stationary distribution. Subsequently, statistical features were employed to analyze the dynamic behavior of the disease. The exact representation of the probability density function in the vicinity of the quasi-equilibrium point was determined by the Fokker-Planck equation. Finally, some numerical simulations validated our theoretical results.

Keywords:

Citation: Bing Zhao, Shuting Lyu, Qimin Zhang. Dynamics and density function for a stochastic anthrax epidemic model[J]. Electronic Research Archive, 2024, 32(3): 1574-1617. doi: 10.3934/era.2024072

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Abstract

1. Introduction and preliminaries

Let $W$ be a set and $H:W\longrightarrow W$ be a mapping. A point $w\in W$ is called a fixed point of $H\$ if $w = Hw.$ Fixed point theory plays a fundamental role in functional analysis (see ^[15]). Shoaib ^[17] introduced the concept of $\alpha$ -dominated mapping and obtained some fixed point results (see also ^[1,2]). George et al. ^[11] introduced a new space and called it rectangular $b$ -metric space ( $r.b.m.$ space). The triangle inequality in the $b$ -metric space was replaced by rectangle inequality. Useful results on $r.b.m.$ spaces can be seen in (^[5,6,8,9,10]). Ćirić introduced new types of contraction and proved some metrical fixed point results (see ^[4]). In this article, we introduce Ćirić type rational contractions for $\alpha$ -dominated mappings in $r.b.m.$ spaces and proved some metrical fixed point results. New interesting results in metric spaces, rectangular metric spaces and $b$ -metric spaces can be obtained as applications of our results.

Definition 1.1. ^[11] Let $U$ be a nonempty set. A function $d_{lb}:U\times U\rightarrow \lbrack 0, \infty)$ is said to be a rectangular $b$ -metric if there exists $b\geq 1$ such that

(ⅰ) $d_{lb}(\theta, \nu) = d_{lb}(\nu, \theta$ );

(ⅱ) $d_{lb}(\theta, \nu) = 0$ if and only if $\theta = \nu;$

(ⅲ) $d_{lb}(\theta, \nu)\leq b[d_{lb}(\theta, q)+d_{lb}(q, l)+d_{lb}(l, \nu)]$ for all $\theta, \nu \in U$ and all distinct points $q, l\in U \backslash \{ \theta, \nu \}.$

The pair $(U, d_{lb})$ is said a rectangular $b$ -metric space (in short, $r.b.m.$ space) with coefficient $b$ .

Definition 1.2. ^[11] Let $(U, d_{lb})$ be an $r.b.m.$ space with coefficient $b$ .

(ⅰ) A sequence $\{ \theta_{n}\}$ in $(U, d_{lb})$ is said to be Cauchy sequence if for each $\varepsilon > 0$ , there corresponds $n_{0}\in \mathbb{N}$ such that for all $n, m\geq n_{0}$ we have $d_{lb}(\theta_{m}, \theta_{n}) < \varepsilon$ or $\lim_{n, m\rightarrow +\infty }d_{lb}(\theta_{n}, \theta_{m}) = 0.$

(ⅱ) A sequence $\{ \theta_{n}\}$ is rectangular $b$ -convergent (for short, ( $d_{lb}$ )-converges) to $\theta$ if $\lim_{n\rightarrow +\infty }d_{lb}(\theta _{n}, \theta) = 0.$ In this case $\theta$ is called a ( $d_{lb}$ )-limit of $\{ \theta _{n}\}.$

(ⅲ) $(U, d_{lb})$ is complete if every Cauchy sequence in $U \; d_{lb}$ -converges to a point $\theta \in U$ .

Let $\varpi _{b}$ , where $b\geq 1,$ denote the family of all nondecreasing functions $\delta _{b}:[0, +\infty)\rightarrow \lbrack 0, +\infty)$ such that $\sum_{k = 1}^{+\infty }b^{k}\delta _{b}^{k}(t) < +\infty$ and $b\delta _{b}(t) < t$ for all $t > 0,$ where $\delta _{b}^{k}$ is the $k^{th}$ iterate of $\delta _{b}.$ Also $b^{n+1}\delta _{b}^{n+1}(t) = b^{n}b\delta _{b}(\delta _{b}^{n}(t)) < b^{n}\delta _{b}^{n}(t).$

Example 1.3. ^[11] Let $U \; = \mathbb{N}.$ Define $d_{lb}:U\times U\rightarrow \mathbb{R} ^{+}\cup \{0\}$ such that $d_{lb}(u, v) = d_{lb}(v, u)$ for all $u, v\in U$ and $\alpha > 0$

$\begin{equation*} d_{lb}(u,v) = \left\{ \begin{array}{c} 0,\text{ if }u = v; \\ 10\alpha ,\text{ if }u = 1,\text{ }v = 2; \\ \alpha ,\text{ if }u\in \{1,2\}\text{ and }v\in \{3\}; \\ 2\alpha ,\text{ if }u\in \{1,2,3\}\text{ and }v\in \{4\}; \\ 3\alpha ,\text{ if }u\text{ or }v\notin \{1,2,3,4\}\text{ and }u\neq v. \end{array} \right. \end{equation*}$

Then $(U, d_{lb})$ is an $r.b.m.$ space with $b = 2 > 1.$ Note that

$\begin{equation*} d(1,4)+d(4,3)+d(3,2) = 5\alpha \lt 10\alpha = d(1,2). \end{equation*}$

Thus $d_{lb}$ is not a rectangular metric.

Definition 1.4. ^[17] Let $(U, d_{lb})$ be an $r.b.m.$ space with coefficient $b$ . Let $S:U\rightarrow U$ be a mapping and $\alpha :U\times U\rightarrow \lbrack 0, +\infty)$ . If $A\subseteq U,$ we say that the $S$ is $\alpha$ -dominated on $A,$ whenever $\alpha (i, Si)\geq 1$ for all $i\in A.$ If $A = U$ , we say that $S$ is $\alpha$ -dominated $.$

For $\theta, \nu \in U$ , $a > 0,$ we define $D_{lb}(\theta, \nu)$ as

$\begin{equation*} D_{lb}( \theta,\nu ) = \max \{d_{lb}( \theta,\nu ),\frac{d_{lb}\left( \theta,S \theta\right) .d_{lb}\left( \nu ,S\nu \right) }{ a+d_{lb}\left( \theta,\nu \right) },d_{lb}( \theta,S \theta ),d_{lb}(\nu ,S\nu )\}. \end{equation*}$

2. Main result

Now, we present our main result.

Theorem 2.1. Let $(U, d_{lb})$ be a complete $r.b.m.$ space with coefficient $b$ , $\alpha :U\times U\rightarrow \lbrack 0, \infty), \; S:U\rightarrow U$ , $\{ \theta_{n}\}$ be a Picard sequence and $S$ be a $\alpha$ -dominated mapping on $\{ \theta_{n}\}.$ Suppose that, for some $\delta _{b}\in \varpi _{b}$ , we have

$\begin{equation} d_{lb}(S \theta,S\nu )\leq \delta _{b}(D_{lb}( \theta,\nu )), \end{equation}$

(2.1)

for all $\theta, \nu \in \{ \theta_{n}\}$ with $\alpha (\theta, \nu)\geq 1$ . Then $\{ \theta_{n}\}$ converges to $\theta^{\ast }\in U.$ Also, if (2.1) holds for $\theta^{\ast }$ and $\alpha (\theta_{n}, \theta^{\ast })\geq 1$ for all $n\in \mathbb{N} \cup \{0\}$ , then $S$ has a fixed point $\theta^{\ast }$ in $U.$

Proof. Let $\theta_{0}\in U$ be arbitrary. Define the sequence $\{ \theta _{n}\}$ by $\theta_{n+1} = S \theta_{n}$ for all $n\in \mathbb{N} \cup \{0\}.$ We shall show that $\{ \theta_{n}\}$ is a Cauchy sequence. If $\theta _{n} = \theta_{n+1},$ for some $n\in \mathbb{N},$ then $\theta_{n}$ is a fixed point of $S$ . So, suppose that any two consecutive terms of the sequence are not equal. Since $S:U\rightarrow U$ be an $\alpha$ -dominated mapping on $\{ \theta_{n}\}$ , $\alpha (\theta _{n}, S \theta_{n})\geq 1$ for all $n\in \mathbb{N} \cup \{0\}$ and then $\alpha (\theta_{n}, \theta_{n+1})\geq 1$ for all $n\in \mathbb{N} \cup \{0\}.$ Now by using inequality (2.1), we obtain

$\begin{eqnarray*} d_{lb}( \theta_{n+1}, \theta_{n+2}) & = &d_{lb}(S \theta_{n},S \theta_{n+1})\leq \delta _{b}(D_{lb}( \theta_{n}, \theta_{n+1})) \\ &\leq &\delta _{b}(\max \{d_{lb}( \theta_{n}, \theta_{n+1}),\frac{ d_{lb}\left( \theta_{n}, \theta_{n+1}\right) .d_{lb}\left( \theta_{n+1}, \theta_{n+2}\right) }{a+d_{lb}\left( \theta_{n}, \theta_{n+1}\right) }, \\ &&d_{lb}( \theta_{n}, \theta_{n+1}),d_{lb}( \theta_{n+1}, \theta_{n+2})\}) \\ &\leq &\delta _{b}(\max \{d_{lb}( \theta_{n}, \theta_{n+1}),d_{lb}( \theta_{n+1}, \theta_{n+2})\}). \end{eqnarray*}$

If $\max \{d_{lb}(\theta_{n}, \theta_{n+1}), d_{lb}(\theta_{n+1}, \theta _{n+2})\} = d_{lb}(\theta_{n+1}, \theta_{n+2}),$ then

$\begin{eqnarray*} d_{lb}( \theta_{n+1}, \theta_{n+2}) &\leq &\delta _{b}(d_{lb}( \theta_{n+1}, \theta_{n+2})) \\ &\leq &b\delta _{b}(d_{lb}( \theta_{n+1}, \theta_{n+2})). \end{eqnarray*}$

This is the contradiction to the fact that $b\delta _{b}(t) < t$ for all $t > 0.$ So

$\begin{equation*} \max \{d_{lb}( \theta_{n}, \theta_{n+1}),d_{lb}( \theta_{n+1}, \theta_{n+2})\} = d_{lb}( \theta_{n}, \theta_{n+1}). \end{equation*}$

Hence, we obtain

$\begin{equation*} d_{lb}( \theta_{n+1}, \theta_{n+2})\leq \delta _{b}(d_{lb}( \theta_{n}, \theta_{n+1}))\leq \delta _{b}^{2}(d_{lb}( \theta_{n-1}, \theta_{n})) \end{equation*}$

Continuing in this way, we obtain

$\begin{equation} d_{lb}( \theta_{n+1}, \theta_{n+2})\leq \delta _{b}^{n+1}(d_{lb}( \theta_{0}, \theta_{1})). \end{equation}$

(2.2)

Suppose for some $n, m\in \mathbb{N}$ with $m > n,$ we have $\theta_{n} = \theta_{m}$ . Then by (2.2)

$\begin{eqnarray*} d_{lb}( \theta_{n}, \theta_{n+1}) & = &d_{lb}( \theta_{n},S \theta_{n}) = d_{lb}( \theta_{m},S \theta_{m}) = d_{lb}( \theta_{m}, \theta_{m+1}) \\ &\leq &\delta _{b}^{m-n}\left( d_{lb}( \theta_{n}, \theta _{n+1})\right) \lt b\delta _{b}\left( d_{lb}( \theta_{n}, \theta_{n+1})\right) \end{eqnarray*}$

As $d_{lb}(\theta_{n}, \theta_{n+1}) > 0,$ so this is not true, because $b\delta _{b}(t) < t$ for all $t > 0.$ Therefore, $\theta_{n}\neq \theta_{m}$ for any $n, m\in \mathbb{N}$ . Since $\sum_{k = 1}^{+\infty }b^{k}\delta _{b}^{k}(t) < +\infty,$ for some $\nu \in \mathbb{N},$ the series $\sum_{k = 1}^{+\infty }b^{k}\delta _{b}^{k}(\delta _{b}^{\nu -1}(d_{lb}(\theta_{0}, \theta_{1})))$ converges. As $b\delta _{b}(t) < t,$ so

$\begin{equation*} b^{n+1}\delta _{b}^{n+1}(\delta _{b}^{\nu -1}(d_{lb}( \theta_{0}, \theta_{1}))) \lt b^{n}\delta _{b}^{n}(\delta _{b}^{\nu -1}(d_{lb}( \theta _{0}, \theta_{1}))), \text{ for all }n\in \mathbb{N}. \end{equation*}$

Fix $\varepsilon > 0.$ Then $\frac{\varepsilon }{2} = \varepsilon ^{\prime } > 0.$ For $\varepsilon ^{\prime },$ there exists $\nu (\varepsilon ^{\prime })\in \mathbb{N}$ such that

$\begin{equation} b\delta _{b}(\delta _{b}^{\nu (\varepsilon ^{\prime })-1}(d_{lb}( \theta _{0}, \theta_{1})))+b^{2}\delta _{b}^{2}(\delta _{b}^{\nu (\varepsilon ^{\prime })-1}(d_{lb}( \theta_{0}, \theta_{1})))+\cdots \lt \varepsilon ^{\prime } \end{equation}$

(2.3)

Now, we suppose that any two terms of the sequence $\{ \theta_{n}\}$ are not equal. Let $n, m\in \mathbb{N}$ with $m > n > \nu (\varepsilon ^{\prime }).$ Now, if $m > n+2$ ,

$\begin{eqnarray*} d_{lb}( \theta_{n}, \theta_{m}) &\leq &b[d_{lb}( \theta_{n}, \theta_{n+1})+d_{lb}( \theta_{n+1}, \theta_{n+2})+d_{lb}( \theta_{n+2}, \theta_{m})] \\ &\leq &b[d_{lb}( \theta_{n}, \theta_{n+1})+d_{lb}( \theta_{n+1}, \theta_{n+2})]+b^{2}[d_{lb}( \theta_{n+2}, \theta_{n+3}) \\ &&+d_{lb}( \theta_{n+3}, \theta_{n+4})+d_{lb}( \theta_{n+4}, \theta_{m})] \\ &\leq &b[\delta _{b}^{n}\left( d_{lb}( \theta_{0}, \theta _{1})\right) +\delta _{b}^{n+1}\left( d_{lb}( \theta_{0}, \theta_{1})\right) ]+b^{2}[\delta _{b}^{n+2}\left( d_{lb}( \theta_{0}, \theta_{1})\right) \\ &&+\delta _{b}^{n+3}\left( d_{lb}( \theta_{0}, \theta_{1})\right) ]+b^{3}[\delta _{b}^{n+4}\left( d_{lb}( \theta_{0}, \theta _{1})\right) +\delta _{b}^{n+5}\left( d_{lb}( \theta_{0}, \theta_{1})\right) ]+\cdots \\ &\leq &b\delta _{b}^{n}(d_{lb}( \theta_{0}, \theta_{1}))+b^{2}\delta _{b}^{n+1}(d_{lb}( \theta_{0}, \theta_{1}))+b^{3}\delta _{b}^{n+2}\left( d_{lb}( \theta_{0}, \theta_{1})\right) +\cdots \\ & = &b\delta _{b}(\delta _{b}^{n-1}(d_{lb}( \theta_{0}, \theta_{1})))+b^{2}\delta _{b}^{2}(\delta _{b}^{n-1}(d_{lb}( \theta_{0}, \theta_{1})))+\cdots . \end{eqnarray*}$

By using (2.3), we have

$\begin{eqnarray*} &&d_{lb}( \theta_{n}, \theta_{m}) \\ & \lt &b\delta _{b}(\delta _{b}^{\nu (\varepsilon ^{\prime })-1}(d_{lb}( \theta_{0}, \theta_{1})))+b^{2}\delta _{b}^{2}(\delta _{b}^{\nu (\varepsilon ^{\prime })-1}(d_{lb}( \theta_{0}, \theta_{1})))+\cdots \lt \varepsilon ^{\prime } \lt \varepsilon . \end{eqnarray*}$

Now, if $m = n+2$ , then we obtain

$\begin{eqnarray*} &&d_{lb}( \theta_{n}, \theta_{n+2}) \\ &\leq &b[d_{lb}( \theta_{n}, \theta_{n+1})+d_{lb}( \theta_{n+1}, \theta_{n+3})+d_{lb}( \theta_{n+3}, \theta_{n+2})] \\ &\leq &b[d_{lb}( \theta_{n}, \theta_{n+1})+b[d_{lb}( \theta_{n+1}, \theta_{n+2})+d_{lb}( \theta_{n+2}, \theta_{n+4})+d_{lb}( \theta_{n+4}, \theta_{n+3})] \\ &&+d_{lb}( \theta_{n+3}, \theta_{n+2})] \\ &\leq &bd_{lb}( \theta_{n}, \theta_{n+1})+b^{2}d_{lb}( \theta_{n+1}, \theta_{n+2})+bd_{lb}( \theta_{n+2}, \theta_{n+3})+b^{2}d_{lb}( \theta_{n+3}, \theta_{n+4}) \\ &&+b^{3}[d_{lb}( \theta_{n+2}, \theta_{n+3})+d_{lb}( \theta_{n+3}, \theta_{n+5})+d_{lb}( \theta_{n+5}, \theta_{n+4})] \\ &\leq &bd_{lb}( \theta_{n}, \theta_{n+1})+b^{2}d_{lb}( \theta_{n+1}, \theta_{n+2})+\left( b+b^{3}\right) d_{lb}( \theta_{n+2}, \theta_{n+3})+b^{2}d_{lb}( \theta_{n+3}, \theta_{n+4}) \\ &&+b^{3}d_{lb}( \theta_{n+5}, \theta_{n+4})+b^{4}[d_{lb}( \theta_{n+3}, \theta_{n+4})+d_{lb}( \theta_{n+4}, \theta_{n+6})+d_{lb}( \theta_{n+6}, \theta_{n+5})] \\ &\leq &bd_{lb}( \theta_{n}, \theta_{n+1})+b^{2}d_{lb}( \theta_{n+1}, \theta_{n+2})+\left( b+b^{3}\right) d_{lb}( \theta_{n+2}, \theta_{n+3}) \\ &&+\left( b^{2}+b^{4}\right) d_{lb}( \theta_{n+3}, \theta_{n+4})+b^{3}d_{lb}( \theta_{n+5}, \theta_{n+4})+b^{4}d_{lb}( \theta_{n+6}, \theta_{n+5}) \\ &&+b^{5}[d_{lb}( \theta_{n+4}, \theta_{n+5})+d_{lb}( \theta_{n+5}, \theta_{n+7})+d_{lb}( \theta_{n+7}, \theta_{n+6})] \\ &\leq &bd_{lb}( \theta_{n}, \theta_{n+1})+b^{2}d_{lb}( \theta_{n+1}, \theta_{n+2})+\left( b+b^{3}\right) d_{lb}( \theta_{n+2}, \theta_{n+3}) \\ &&+\left( b^{2}+b^{4}\right) d_{lb}( \theta_{n+3}, \theta_{n+4})+\left( b^{3}+b^{5}\right) d_{lb}( \theta_{n+4}, \theta_{n+5})+\cdots \\ & \lt &2[bd_{lb}( \theta_{n}, \theta_{n+1})+b^{2}d_{lb}( \theta_{n+1}, \theta_{n+2})+b^{3}d_{lb}( \theta_{n+2}, \theta_{n+3}) \\ &&+b^{4}d_{lb}( \theta_{n+3}, \theta_{n+4})+b^{5}d_{lb}( \theta_{n+4}, \theta_{n+5})+\cdots ] \\ &\leq &2[b\delta _{b}^{n}(d_{lb}( \theta_{0}, \theta_{1}))+b^{2}\delta _{b}^{n+1}(d_{lb}( \theta_{0}, \theta_{1}))+b^{3}\delta _{b}^{n+2}\left( d_{lb}( \theta_{0}, \theta_{1})\right) +\cdots ] \\ & \lt &2[b\delta _{b}(\delta _{b}^{\nu (\varepsilon ^{\prime })-1}(d_{lb}( \theta_{0}, \theta_{1})))+b^{2}\delta _{b}^{2}(\delta _{b}^{\nu (\varepsilon ^{\prime })-1}(d_{lb}( \theta_{0}, \theta_{1})))+\cdots ] \lt 2\varepsilon ^{\prime } = \varepsilon . \end{eqnarray*}$

It follows that

$\begin{equation} \lim\limits_{n,m\rightarrow +\infty }d_{lb}( \theta_{n}, \theta_{m}) = 0. \end{equation}$

(2.4)

Thus $\{ \theta_{n}\}$ is a Cauchy sequence in $(U, d_{lb})$ . As $(U, d_{lb})$ is complete, so there exists $\theta^{\ast }$ in $U$ such that $\{ \theta_{n}\}$ converges to $\theta^{\ast },$ that is,

$\begin{equation} \lim\limits_{n\rightarrow +\infty }d_{lb}( \theta_{n}, \theta^{\ast }) = 0. \end{equation}$

(2.5)

Now, suppose that $d_{lb}(\theta^{\ast }, S \theta^{\ast }) > 0$ . Then

$\begin{eqnarray*} d_{lb}( \theta^{\ast },S \theta^{\ast }) &\leq &b[d_{lb}( \theta^{\ast }, \theta_{n})+d_{lb}( \theta_{n}, \theta_{n+1})+d_{lb}( \theta_{n+1},S \theta^{\ast }) \\ &\leq &b[d_{lb}( \theta^{\ast }, \theta_{n+1})+d_{lb}( \theta_{n}, \theta_{n+1})+d_{lb}(S \theta_{n},S \theta^{\ast }). \end{eqnarray*}$

Since $\alpha (\theta_{n}, \theta^{\ast })\geq 1,$ we obtain

$\begin{eqnarray*} d_{lb}( \theta^{\ast },S \theta^{\ast }) &\leq &bd_{lb}( \theta^{\ast }, \theta_{n+1})+bd_{lb}( \theta_{n}, \theta_{n+1})+b\delta _{b}(\max \{d_{lb}( \theta_{n}, \theta^{\ast }), \\ &&\frac{d_{lb}\left( \theta^{\ast },S \theta^{\ast }\right) .d_{lb}\left( \theta_{n}, \theta_{n+1}\right) }{a+d_{lb}\left( \theta_{n}, \theta^{\ast }\right) },\text{ }d_{lb}( \theta_{n}, \theta_{n+1})\text{ }d_{lb}( \theta^{\ast },S \theta^{\ast })\}). \end{eqnarray*}$

Letting $n\rightarrow +\infty$ , and using the inequalities (2.4) and (2.5), we obtain $d_{lb}(\theta^{\ast }, S \theta^{\ast })\leq b\delta _{b}(d_{lb}(\theta^{\ast }, S \theta^{\ast })).$ This is not true, because $b\delta _{b}(t) < t$ for all $t > 0$ and hence $d_{lb}(\theta^{\ast }, S \theta^{\ast }) = 0$ or $\theta^{\ast } = S \theta^{\ast }$ . Hence $S$ has a fixed point $\theta^{\ast }$ in $U.$

Remark 2.2. By taking fourteen different proper subsets of $D_{lb}(\theta, \nu),$ we can obtainvnew results as corollaries of our result in a complete $r.b.m.$ space with coefficient $b.$

We have the following result without using $\alpha$ -dominated mapping.

Theorem 2.3. Let $(U, d_{lb})$ be a complete $r.b.m.$ space with coefficient $b, \; S:U\rightarrow U$ , $\{ \theta_{n}\}$ be a Picard sequence. Suppose that, for some $\delta _{b}\in \varpi _{b}$ , we have

$\begin{equation} d_{lb}(S \theta,S\nu )\leq \delta _{b}(D_{lb}( \theta,\nu )) \end{equation}$

(2.6)

for all $\theta, \nu \in \{ \theta_{n}\}$ . Then $\{ \theta_{n}\}$ converges to $\theta^{\ast }\in U.$ Also, if (2.6) holds for $\theta^{\ast }$ , then $S$ has a fixed point $\theta^{\ast }$ in $U.$

We have the following result by taking $\delta _{b}\left(t\right) = ct$ , $t\in \mathbb{R} ^{+}$ with $0 < c < \frac{1}{b}$ without using $\alpha$ -dominated mapping.

Theorem 2.4. Let $(U, d_{lb})$ be a complete $r.b.m.$ space with coefficient $b$ , $S:U\rightarrow U$ , $\{ \theta_{n}\}$ be a Picard sequence. Suppose that, for some $0 < c < \frac{1}{b}$ , we have

$\begin{equation} d_{lb}(S \theta,S\nu )\leq c(D_{lb}( \theta,\nu )) \end{equation}$

(2.7)

for all $\theta, \nu \in \{ \theta_{n}\}$ . Then $\{ \theta_{n}\}$ converges to $\theta^{\ast }\in U.$ Also, if (2.7) holds for $\theta^{\ast }$ , then $S$ has a fixed point $\theta^{\ast }$ in $U.$

Ran and Reurings ^[16] gave an extension to the results in fixed point theory and obtained results in partially ordered metric spaces. Arshad et al. ^[3] introduced $\preceq$ -dominated mappings and established some results in an ordered complete dislocated metric space. We apply our result to obtain results in ordered complete $r.b.m.$ space.

Definition 2.5. $(U, \preceq, d_{lb})$ is said to be an ordered complete $r.b.m.$ space with coefficient $b$ if

(ⅰ) $(U, \preceq)$ is a partially ordered set.

(ⅱ) $(U, d_{lb})$ is an $r.b.m.$ space.

Definition 2.6. ^[3] Let $U$ be a nonempty set, $\preceq$ is a partial order on $\theta.$ A mapping $S:U\rightarrow U$ is said to be $\preceq$ -dominated on $A$ if $a\preceq Sa$ for each $a\in A\subseteq \theta.$ If $A = U,$ then $S:U\rightarrow U$ is said to be $\preceq$ -dominated.

We have the following result for $\preceq$ -dominated mappings in an ordered complete $r.b.m.$ space with coefficient $b$ .

Theorem 2.7. Let $(U, \preceq, d_{lb})$ be an ordered complete $r.b.m.$ space with coefficient $b$ , $S:U\rightarrow U, \; \{ \theta_{n}\}$ be a Picard sequence and $S$ be a $\preceq$ -dominated mapping on $\{ \theta _{n}\}.$ Suppose that, for some $\delta _{b}\in \varpi _{b}$ , we have

$\begin{equation} d_{lb}(S \theta,S\nu )\leq \delta _{b}(D_{lb}( \theta,\nu )), \end{equation}$

(2.8)

for all $\theta, \nu \in \{ \theta_{n}\}$ with $\theta \preceq \nu$ . Then $\{ \theta_{n}\}$ converges to $\theta^{\ast }\in U.$ Also, if (2.8) holds for $\theta^{\ast }$ and $\theta_{n}\preceq \theta^{\ast }$ for all $n\in \mathbb{N} \cup \{0\}$ . Then $S$ has a fixed point $\theta^{\ast }$ in $U$ .

Proof. Let $\alpha :U\times U\rightarrow \lbrack 0, +\infty)$ be a mapping defined by $\alpha (\theta, \nu) = 1$ for all $\theta, \nu \in U$ with $\theta \preceq \nu$ and $\alpha (\theta, \nu) = \frac{4}{11}$ for all other elements $\theta, \nu \in U.$ As $S$ is the dominated mappings on $\{ \theta_{n}\},$ so $\theta \preceq S \theta$ for all $\theta \in \{ \theta_{n}\}.$ This implies that $\alpha (\theta, S \theta) = 1$ for all $\theta \in \{ \theta_{n}\}.$ So $S:U\rightarrow U$ is the $\alpha$ -dominated mapping on $\{ \theta_{n}\}.$ Moreover, inequality (2.8) can be written as

$\begin{equation*} d_{lb}(S \theta,S\nu )\leq \delta _{b}(D_{lb}( \theta,\nu )) \end{equation*}$

for all elements $\theta, \nu$ in $\{ \theta_{n}\}$ with $\alpha (\theta, \nu)\geq 1.$ Then, as in Theorem 2.1, $\{ \theta_{n}\}$ converges to $\theta^{\ast }\in U.$ Now, $\theta_{n}\preceq \theta^{\ast }$ implies $\alpha (\theta_{n}, \theta^{\ast })\geq 1.$ So all the conditions of Theorem 2.1 are satisfied. Hence, by Theorem 2.1, $S$ has a fixed point $\theta^{\ast }$ in $U$ .

Now, we present an example of our main result. Note that the results of George et al. ^[11] and all other results in rectangular $b$ -metric space are not applicable to ensure the existence of the fixed point of the mapping given in the following example.

Example 2.8. Let $U = A\cup B,$ where $A = \{\frac{1}{n}:n\in \{2, 3, 4, 5\}\}$ and $B = [1, \infty ].$ Define $d_{l}:U\times U\rightarrow \lbrack 0, \infty)$ such that $d_{l}(\theta, \nu) = d_{l}(\nu, \theta)$ for $\theta, \nu \in U$ and

$\begin{equation*} \left\{ \begin{array}{c} d_{l}(\frac{1}{2},\frac{1}{3}) = d_{l}(\frac{1}{4},\frac{1}{5}) = 0.03 \\ d_{l}(\frac{1}{2},\frac{1}{5}) = d_{l}(\frac{1}{3},\frac{1}{4}) = 0.02 \\ d_{l}(\frac{1}{2},\frac{1}{4}) = d_{l}(\frac{1}{5},\frac{1}{3}) = 0.6 \\ d_{l}( \theta,\nu ) = \left\vert \theta-\nu \right\vert ^{2}\rm{ \ \ \ \ otherwise} \end{array} \right. \end{equation*}$

be a complete $r.b.m.$ space with coefficient $b \; = 4 > 1$ but $\ (U, d_{l})$ is neither a metric space nor a rectangular metric space. Take $\delta _{b}(t) = \frac{t}{10},$ then $b\delta _{b}(t) < t.$ Let $S:U\rightarrow U$ be defined as

$\begin{equation*} S \theta = \left\{ \begin{array}{c} \frac{1}{5}{\rm{ \ \ \ \ \ \ \ \ if }}\; \theta \in A \\ \frac{1}{3}\rm{ \ \ \ \ \ \ \ \ if} \theta = 1 \\ 9 \theta^{100}+85\rm{\ otherwise.} \end{array} \right. \end{equation*}$

Let $\theta_{0} = 1.$ Then the Picard sequence $\{ \theta_{n}\}$ is $\{1, \frac{1}{ 3}, \frac{1}{5}, \frac{1}{5}, \frac{1}{5}, \cdots \}$ . Define

$\begin{equation*} \alpha ( \theta,\nu ) = \left\{ \begin{array}{c} \frac{8}{5}{\rm{ \ \ \ \ \ \ \ \ if}} \theta ,\nu \in \{ \theta_{n}\} \\ \frac{4}{7}\rm{\ \ \ \ \ \ \ \ \ \ \ \ otherwise.} \end{array} \right. \end{equation*}$

Then $S\$ is an $\alpha$ -dominated mapping on $\{ \theta_{n}\}.$ Now, $S$ satisfies all the conditions of Theorem 2.1. Here $\frac{1}{5}$ is the fixed point in $U.$

3. Fixed point results for graphic contractions

Jachymski ^[13] proved the contraction principle for mappings on a metric space with a graph. Let $(U, d)$ be a metric space and $\bigtriangleup$ represents the diagonal of the cartesian product $U\times U.$ Suppose that $G$ be a directed graph having the vertices set $V(G)$ along with $U,$ and the set $E(G)$ denoted the edges of $U_{{}}$ included all loops, i.e., $E(G)\supseteq \bigtriangleup.$ If $G$ has no parallel edges, then we can unify $G$ with pair $(V(G), E(G))$ . If $l$ and $m$ are the vertices in a graph $G,$ then a path in $G$ from $l$ to $m$ of length $N \; (N\in \mathbb{N})$ is a sequence $\{ \theta_{i}\}_{i = o}^{N}$ of $N+1$ vertices such that $l_{o} = l, \; l_{N} = m$ and $(l_{n-1}, l_{n})\in E(G)$ where $i = 1, 2, \cdots N$ (see for detail ^{[7,8,12,14,18,19]}). Recently, Younis et al. ^[20] introduced the notion of graphical rectangular $b$ -metric spaces (see also ^[5,6,21]). Now, we present our result in this direction.

Definition 3.1. Let $\theta$ be a nonempty set and $G = (V(G), E(G))$ be a graph such that $V(G) = U$ and $A\subseteq U$ . A mapping $S:U\rightarrow U$ is said to be graph dominated on $A$ if $(\theta, S \theta)\in E(G)$ for all $\theta \in A$ .

Theorem 3.2. Let $(U, d_{lb})$ be a complete rectangular $b$ -metric space endowed with a graph $G$ , $\{ \theta_{n}\}$ be a Picard sequence and $S:U\rightarrow U$ be a graph dominated mapping on $\{ \theta_{n}\}$ . Suppose that the following hold:

(i) there exists $\delta _{b}\in \varpi _{b}$ such that

$\begin{equation} d_{lb}(S \theta,S\nu )\leq \delta _{b}(D_{lb}( \theta,\nu )), \end{equation}$

(3.1)

for all $\theta, \nu \in \{ \theta_{n}\}$ and $(\theta_n, \nu)\in E(G)$ . Then $(\theta_{n}, \theta_{n+1})\in E(G)$ and $\{ \theta_{n}\}$ converges to $\theta^{\ast }.$ Also, if (3.1) holds for $\theta^{\ast }$ and $(\theta_{n}, \theta^{\ast })\in E(G)$ for all $n\in \mathbb{N} \cup \{0\}$ , then $S$ has a fixed point $\theta^{\ast }$ in $U.$

Proof. Define $\alpha :U\times U\rightarrow \lbrack 0, +\infty)$ by

$\begin{equation*} \alpha ( \theta,\nu ) = \left\{ \begin{array}{c} 1,{\rm{\ if }}\; \theta,\nu \in U,\text{ }( \theta,\nu )\in E(G) \\ \frac{1}{4},\rm{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise}. \end{array} \right. \end{equation*}$

Since $S$ is a graph dominated on $\{ \theta_{n}\},$ for $\theta \in \{ \theta_{n}\}, \; (\theta, S \theta)\in E(G)$ . This implies that $\alpha (\theta, S \theta) = 1$ for all $\theta \in \{ \theta_{n}\}.$ So $S:U\rightarrow U$ is an $\alpha$ -dominated mapping on $\{ \theta_{n}\}.$ Moreover, inequality (3.1) can be written as

$\begin{equation*} d_{lb}(S \theta,S\nu )\leq \delta _{b}(D_{lb}( \theta,\nu )), \end{equation*}$

for all elements $\theta, \nu$ in $\{ \theta_{n}\}$ with $\alpha (\theta, \nu)\geq 1.$ Then, by Theorem 2.1, $\{ \theta_{n}\}$ converges to $\theta^{\ast }\in U.$ Now, $(\theta_{n}, \theta^{\ast })\in E(G)$ implies that $\alpha (\theta_{n}, \theta^{\ast })\geq 1.$ So all the conditions of Theorem 2.1 are satisfied. Hence, by Theorem 2.1, $S$ has a fixed point $\theta^{\ast }$ in $U$ .

Acknowledgments

The authors would like to thank the Editor, the Associate Editor and the anonymous referees for sparing their valuable time for reviewing this article. The thoughtful comments of reviewers are very useful to improve and modify this article.

Conflict of interest

The authors declare that they have no competing interests.

References

[1]	N. A. Logan, M. Rodríguez‐Díaz, Bacillus spp. and related genera, Princ. Pract. of Clini. Bacteriol., (2006), 139–158. https://doi.org/10.1002/9780470017968.ch9
[2]	P. R. Murray, K. S. Rosenthal, M. A. Pfaller, Medical Microbiology, Elsevier Health Sciences, 2021.
[3]	P. C. Hanna, Ireland J A W, Understanding Bacillus anthracis pathogenesis, Trends. Microbiol., (1999), 180–182. https://doi.org/10.1016/S0966-842X(99)01507-3
[4]	M. E. Bales, A. L. Dannenberg, P. S. Brachman, A. F. Kaufmann, P. C. Klatsky, D. A. Ashford, Epidemiologic responses to anthrax outbreaks: a review of field investigations, 1950–2001, Emerging Infect. Dis., 8 (2002), 1163. https://doi.org/10.3201/eid0810.020223 doi: 10.3201/eid0810.020223
[5]	J. R. Ezzel, W. C. L. JW, Bacillus anthracis, Pathog. Bact. Infect. Anim., (1993), 36–43. https://doi.org/10.1137/0149110
[6]	N. A. Suverly, B. Kvasnicka, R. Torrell, Anthrax: a guide for livestock producers, University of Nevada-Reno, 2001. https://doi.org/10.1111/j.1863-2378.2008.01135.x
[7]	C. M. Saad-Roy, P. Van den Driessche, A. A. Yakubu, A mathematical model of anthrax transmission in animal populations, Bull. Math. Biol., 79 (2017), 303–324. https://doi.org/10.1007/s11538-016-0238-1 doi: 10.1007/s11538-016-0238-1
[8]	S. S. Lewerin, M. Elvander, T. Westermark, L. N. Hartzell, A. K. Norström, S. Ehrs, et al., Anthrax outbreak in a Swedish beef cattle herd-1st case in 27 years: Case report, Acta Vet. Scand., 52 (2010), 1–8. https://doi.org/10.1186/1751-0147-52-7 doi: 10.1186/1751-0147-52-7
[9]	P. R. Furniss, B. D. Hahn, A mathematical model of an anthrax epizoötic in the Kruger National Park, Appl. Math. Modell., 5 (1981), 130–136. https://doi.org/10.1016/0307-904X(81)90034-2 doi: 10.1016/0307-904X(81)90034-2
[10]	S. V. Shadomy, A. E. Idrissi, E. Raizman, Anthrax outbreaks: a warning for improved prevention, control and heightened awareness, Rome, 2016.
[11]	S. B. Clegg, P. C. B. Turnbull, C. M. Foggin, P. M. Lindeque, Massive outbreak of anthrax in wildlife in the Malilangwe Wildlife Reserve, Zimbabwe, Vet. Rec., 160 (2007), 113–118. https://doi.org/10.1136/vr.160.4.113 doi: 10.1136/vr.160.4.113
[12]	M. N. Mongoh, N. W. Dyer, C. L. Stoltenow, M. L. Khaitsa, Risk factors associated with anthrax outbreak in animals in North Dakota, 2005: A retrospective case-control study, Public Health Rep., 123 (2008), 352–359. https://doi.org/10.1177/003335490812300315 doi: 10.1177/003335490812300315
[13]	A. Chakraborty, S. U. Khan, M. A. Hasnat, S. Parveen, M. Saiful Islam, A. Mikolon, et al., Anthrax outbreaks in Bangladesh, 2009–2010, Am. J. Trop. Med. Hyg., 86 (2012), 703. https://doi.org/10.4269/ajtmh.2012.11-0234 doi: 10.4269/ajtmh.2012.11-0234
[14]	W. Beyer, P. C. B. Turnbull, Anthrax in animals, Mol. Aspects Med., 30 (2009), 481–489. https://doi.org/10.1016/j.mam.2009.08.004
[15]	B. D. Hahn, P. R. Furniss, A deterministic model of an anthrax epizootic: threshold results, Ecol. Modell., 20 (1983), 233–241. https://doi.org/10.1016/0304-3800(83)90009-1 doi: 10.1016/0304-3800(83)90009-1
[16]	A. Friedman, A. A. Yakubu, Anthrax epizootic and migration: Persistence or extinction, Math. Biosci., 241 (2013), 137–144. https://doi.org/10.1016/j.mbs.2012.10.004 doi: 10.1016/j.mbs.2012.10.004
[17]	S. Mushayabasa, T. Marijani, M. Masocha, Dynamical analysis and control strategies in modeling anthrax, Comput. Appl. Math., 36 (2017), 1333–1348. https://doi.org/10.1007/s40314-015-0297-1 doi: 10.1007/s40314-015-0297-1
[18]	X. Li, G. Song, Y. Xia, C.Yuan, Dynamical behaviors of the tumor-immune system in a stochastic environment, SIAM J. Appl. Math., 79 (2019), 2193–2217. https://doi.org/10.1137/19M1243580 doi: 10.1137/19M1243580
[19]	Y. Cai, Y. Kang, M. Banerjee, W. Wang, A stochastic epidemic model incorporating media coverage, Commun. Math. Sci., 14 (2016), 893–910. https://doi.org/10.1186/s13662-018-1925-z doi: 10.1186/s13662-018-1925-z
[20]	Y. Zhao, D. Jiang, The threshold of a stochastic SIS epidemic model with vaccination, Appl. Math. Comput., 243 (2014), 718–727. https://doi.org/10.1016/j.amc.2014.05.124 doi: 10.1016/j.amc.2014.05.124
[21]	Q. Liu, D. Jiang, T. Hayat, B. Ahmad, Analysis of a delayed vaccinated SIR epidemic model with temporary immunity and Lévy jumps, Nonlinear Anal. Hybrid Syst., 27 (2018), 29–43. https://doi.org/10.1016/j.nahs.2017.08.002 doi: 10.1016/j.nahs.2017.08.002
[22]	B. Zhou, D. Jiang, Y. Dai, T. Hayat, Stationary distribution and density function expression for a stochastic SIQRS epidemic model with temporary immunity, Nonlinerar. Dyn., 105 (2021), 931–955. https://doi.org/10.1007/s11071-020-06151-y doi: 10.1007/s11071-020-06151-y
[23]	Y. Tan, Y. Cai, X. Sun, K. Wang, R. Yao, W. Wang, A stochastic SICA model for HIV/AIDS transmission, Chaos, Solitons Fractals, 165 (2022), 112768. https://doi.org/10.1016/j.chaos.2022.112768 doi: 10.1016/j.chaos.2022.112768
[24]	L. Imhof, S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Differ. Equations, 217 (2005), 26–53. https://doi.org/10.1016/j.jde.2005.06.017 doi: 10.1016/j.jde.2005.06.017
[25]	X. Mao, Stationary distribution of stochastic population systems, Syst. Control Lett., 60 (2011), 398–405. https://doi.org/10.1016/j.sysconle.2011.02.013 doi: 10.1016/j.sysconle.2011.02.013
[26]	A. Bahar, X. Mao, Stochastic delay lotka–volterra model, J. Math. Anal. Appl., 292 (2004), 364–380. https://doi.org/10.1016/j.jmaa.2003.12.004 doi: 10.1016/j.jmaa.2003.12.004
[27]	R. Khasminskii, Stochastic stability of differential equations, Springer Science and Business Media, 2011.
[28]	B. Zhou, D. Jiang, Y. Dai, T. Hayat, Ergodic property, extinction, and density function of an SIRI epidemic model with nonlinear incidence rate and high‐order stochastic perturbations, Math. Methods. Appl. Sci., 45 (2022), 1513–1537. https://doi.org/10.1002/mma.7870 doi: 10.1002/mma.7870
[29]	B. Zhou, X. Zhang, D. Jiang, Dynamics and density function analysis of a stochastic SVI epidemic model with half saturated incidence rate, Chaos, Solitons Fractals, 137 (2020), 109865. https://doi.org/10.1016/j.chaos.2020.109865 doi: 10.1016/j.chaos.2020.109865
[30]	C. Xu, W. Ou, Y. Pang, Q. Cui, M. U. Rahman, M. Farman, et al., Hopf bifurcation control of a fractional-order delayed turbidostat model via a novel extended hybrid controller, Match-Commun. Math. Comput. Chem., 91 (2024), 367–413. https://doi.org/10.46793/match.91-2.367X doi: 10.46793/match.91-2.367X
[31]	C. Xu, Z. Liu, Y. Pang, A. Akgül, Stochastic analysis of a COVID-19 model with effects of vaccination and different transition rates: Real data approach, Chaos, Solitons Fractals, 170 (2023), 113395. https://doi.org/10.1016/j.chaos.2023.113395 doi: 10.1016/j.chaos.2023.113395
[32]	C. Xu, Y. Pang, Z. Liu, J. Shen, M. Liao, P. Li, Insights into COVID-19 stochastic modelling with effects of various transmission rates: simulations with real statistical data from UK, Australia, Spain, and India, Phys. Scr., 99 (2024), 025218. https://doi.org/10.1088/1402-4896/ad186c doi: 10.1088/1402-4896/ad186c
[33]	C. Xu, Y. Zhao, J. Lin, Y. Pang, Z. Liu, J. Shen, et al., Mathematical exploration on control of bifurcation for a plankton–oxygen dynamical model owning delay, J. Math. Chem., (2023), 1–31. https://doi.org/10.1007/s10910-023-01543-y
[34]	W. Ou, C. Xu, Q. Cui, Y. Pang, Z. Liu, J. Shen, et al., Hopf bifurcation exploration and control technique in a predator-prey system incorporating delay, AIMS Math., 9 (2024), 1622–1651. https://doi.org/10.3934/math.2024080 doi: 10.3934/math.2024080
[35]	Q. Cui, C. Xu, W. Ou, Y. Pang, Z. Liu, P. Li, et al., Bifurcation behavior and hybrid controller design of a 2D Lotka–Volterra commensal symbiosis system accompanying delay, Mathematics, 11 (2023), 4808. https://doi.org/10.3390/math11234808 doi: 10.3390/math11234808
[36]	C. W. Gardiner, Handbook of stochastic methods, Berlin: springer, 1985.
[37]	X. Tian, C, Ren, Linear equations, superposition principle and complex exponential notation, Coll. Phys., 23 (2004), 23–25
[38]	H. Roozen, An asymptotic solution to a two-dimensional exit problem arising in population dynamics, SIAM J. Appl. Math., 49 (1989), 1793–1810. https://doi.org/10.1137/0149110 doi: 10.1137/0149110
[39]	D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525–546. https://doi.org/10.1137/S0036144500378302 doi: 10.1137/S0036144500378302
[40]	D. C. Dragon, B. T. Elkin, J. S. Nishi, T. R. Ellsworth, A review of anthrax in Canada and implications for research on the disease in northern bison, J. Appl. Microbiol., 87 (1999), 208–213. https://doi.org/10.1046/j.1365-2672.1999.00872.x doi: 10.1046/j.1365-2672.1999.00872.x
[41]	World Health Organization, International Office of Epizootics, Anthrax in humans and animals, World Health Organization, 2008.
[42]	Z. Shi, D. Jiang, X. Zhang, A. Alsaedi, A stochastic SEIRS rabies model with population dispersal: Stationary distribution and probability density function, Appl. Math. Comput., 427 (2022), 127189. https://doi.org/10.1016/j.amc.2022.127189 doi: 10.1016/j.amc.2022.127189
[43]	B. Han, B. Zhou, D. Jiang, T. Hayat, A. Alsaedi, Stationary solution, extinction and density function for a high-dimensional stochastic SEI epidemic model with general distributed delay, Appl. Math. Comput., 405 (2021), 126236. https://doi.org/10.1016/j.amc.2021.126236 doi: 10.1016/j.amc.2021.126236
[44]	H. Yang, F. Wu, P. E. Kloeden, Stationary distribution of stochastic population dynamics with infinite delay, J. Differ. Equations, 340 (2022), 205–226. https://doi.org/10.1016/j.jde.2022.08.035 doi: 10.1016/j.jde.2022.08.035

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