An SIS sex-structured influenza A model with positive case fatality in an open population with varying size

Muntaser Safan; Bayan Humadi; Muntaser Safan; Bayan Humadi

doi:10.3934/mbe.2024306

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 8: 6975-7011. doi: 10.3934/mbe.2024306

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Research article

An SIS sex-structured influenza A model with positive case fatality in an open population with varying size

Muntaser Safan ^{1,2
,
,},
Bayan Humadi ¹

1.
Mathematics Department, Faculty of Science, Umm Al-Qura University, Makkah 21955, KSA
2.
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Academic Editor: Yang Kuang

Received: 17 May 2024 Revised: 20 July 2024 Accepted: 09 August 2024 Published: 27 August 2024

This work aims to study the role of sex disparities on the overall outcome of influenza A disease. Therefore, the classical Susceptible-Infected-Susceptible (SIS) endemic model was extended to include the impact of sex disparities on the overall dynamics of influenza A infection which spreads in an open population with a varying size, and took the potential lethality of the infection. The model was mathematically analyzed, where the equilibrium and bifurcation analyses were established. The model was shown to undergo a backward bifurcation at $\mathcal{R}_0 = 1$ , for certain range of the model parameters, where $\mathcal{R}_0$ is the basic reproduction number of the model. The asymptotic stability of the equilibria was numerically investigated, and the effective threshold was determined. The differences in susceptibility, transmissibility and case fatality (of females with respect to males) are shown to remarkably affect the disease outcomes. Simulations were performed to illustrate the theoretical results.

Keywords:

Citation: Muntaser Safan, Bayan Humadi. An SIS sex-structured influenza A model with positive case fatality in an open population with varying size[J]. Mathematical Biosciences and Engineering, 2024, 21(8): 6975-7011. doi: 10.3934/mbe.2024306

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Abstract

1. Introduction and definitions

To understand in a clear way the notions used in our main results, we need to add here some basic literature of Geometric function theory. For this we start first with the notation $\mathcal{A}$ which denotes the class of holomorphic or analytic functions in the region $\mathfrak{D} = \left \{ z\in \mathbb{C} :\left \vert z\right \vert < 1\right \}$ and if a function $g\in \mathcal{A}$ , then the relations $g\left(0\right) = g^{\prime }\left(0\right) -1 = 0$ must hold. Also, all univalent functions will be in a subfamily $\mathcal{S}$ of $\mathcal{A}$ . Next we consider to define the idea of subordinations between analytic functions $g_1$ and $g_2,$ indicated by $g_1\left(z\right) \prec g_2\left(z\right),$ as; the functions $g_1, g_2\in \mathcal{A}$ are connected by the relation of subordination, if there exists an analytic function $w$ with the restrictions $w\left(0\right) = 0$ and $\left \vert w\left(z\right) \right \vert < 1$ such that $g_1(z) = g_2(w(z)).$ Moreover, if the function $g_2\in \mathcal{S}$ in $\mathfrak{D},$ then we obtain:

$\begin{equation*} g_1(z)\prec g_2(z)\Leftrightarrow \left[ g_1(0) = g_2(0)\ \ \text{& }\ g_1(\mathfrak{D} )\subset g_1(\mathfrak{D}\mathbb{})\right] . \end{equation*}$

In 1992, Ma and Minda ^[16] considered a holomorphic function $\varphi$ normalized by the conditions $\varphi (0) = 1$ and $\varphi ^{\prime }(0) > 0$ with $Re\varphi > 0$ in $\mathfrak{D}$ . The function $\varphi$ maps the disc $\mathfrak{D}$ onto region which is star-shaped about $1$ and symmetric along the real axis. In particular, the function $\varphi (z) = (1+Az)/(1+Bz), \ (-1\leq B < A\leq 1)$ maps $\mathfrak{D}$ onto the disc on the right-half plane with centre on the real axis and diameter end points $\frac{1-A}{1-B}$ and $\frac{1+A}{1+B}$ . This interesting familiar function is named as Janowski function ^[10]. The image of the function $\varphi (z) = \sqrt{1+z}$ shows that the image domain is bounded by the right-half of the Bernoulli lemniscate given by $|w^{2}-1| < 1, \$ ^[25]. The function $\varphi (z) = 1+\frac{4}{3}z+\frac{2}{3}z^{2}$ maps $\mathfrak{D}$ into the image set bounded by the cardioid given by $(9x^{2}+9y^{2}-18x+5)^{2}-16(9x^{2}+9y^{2}-6x+1) = 0,$ ^[21] and further studied in ^[23]. The function $\varphi (z) = 1+\sin z$ was examined by Cho and his coauthors in ^[3] while $\varphi (z) = e^{z}$ is recently studied in ^[17] and ^[24]. Further, by choosing particular $\varphi, \$ several subclasses of starlike functions have been studied. See the details in ^{[2,4,5,11,12,14,19]}.

Recently, Ali et al. ^[1] have obtained sufficient conditions on $\alpha$ such that

$\begin{equation*} \begin{array}{llll} 1+zg^{\prime }(z)/g^{n}(z)\prec \sqrt{1+z} & \Rightarrow & g\left( z\right) \prec \sqrt{1+z}, & \text{for }n = 0,1,2. \end{array} \end{equation*}$

Similar implications have been studied by various authors, for example see the works of Halim and Omar ^[6], Haq et al ^[7], Kumar et al ^[13,15], Paprocki and Sokól ^[18], Raza et al ^[20] and Sharma et al ^[22].

In 1994, Hayman ^[8] studied multivalent ( $p$ -valent) functions which is a generalization of univalent functions and is defined as: an analytic function $g$ in an arbitrary domain $\mathcal{D}\subset \mathbb{C}$ is said to be $p$ -valent, if for every complex number $\omega$ , the equation $g(z) = \omega$ has maximum $p$ roots in $\mathcal{D}$ and for a complex number $\omega _{0}$ the equation $g(z) = \omega _{0}$ has exactly $p$ roots in $\mathcal{D}$ . Let $\mathcal{A}_{p}$ $\left(p\in \mathbb{N} = \left \{ 1, 2, \ldots \right \} \right)$ denote the class of functions, say $g\in \mathcal{A}_{p}$ , that are multivalent holomorphic in the unit disc $\mathfrak{D}$ and which have the following series expansion:

$\begin{equation} g(z) = z^{p}+\sum \limits_{k = p+1}^{\infty }a_{k}z^{k},\text{ }\left( z\in \mathfrak{D}\right) . \end{equation}$

(1.1)

Using the idea of multivalent functions, we now introduce the class $\mathcal{SL}_{p}^{\ast }$ of multivalent starlike functions associated with lemniscate of Bernoulli and as given below:

$\begin{equation*} \mathcal{SL}_{p}^{\ast } = \left \{ g(z)\in \mathcal{A}_{p}:\frac{zg^{\prime }\left( z\right) }{pg(z)}\prec \sqrt{1+z},\text{ }\left( z\in \mathfrak{D} \right) \right \} . \end{equation*}$

In this article, we determine conditions on $\alpha$ such that for each

$\begin{equation*} \begin{array}{lllll} 1+\alpha \frac{z^{2+p\left( j-1\right) }g^{\prime }\left( z\right) }{ pg^{j}\left( z\right) },\text{ for each }j = 0,1,2,3, \end{array} \end{equation*}$

are subordinated to Janowski functions implies $\frac{g\left(z\right) }{ z^{p}}\prec \sqrt{1+z}, \ \left(z\in \mathfrak{D}\right)$ . These results are then utilized to show that $g$ are in the class $\mathcal{SL}_{p}^{\ast }.$

1.1. Lemma

Let $w$ be analytic non-constant function in $\mathfrak{D}$ with $w\left(0\right) = 0.$ If

$\begin{equation*} \left \vert w\left( z_{0}\right) \right \vert = \max \left \{ \left \vert w\left( z\right) \right \vert ,\ \ \ \left \vert z\right \vert \leq \left \vert z_{0}\right \vert \right \} ,\ \ z\in \mathfrak{D}\text{,} \end{equation*}$

then there exists a real number $m\ \left(m\geq 1\right)$ such that $z_{0}w^{\prime }\left(z_{0}\right) = mw\left(z_{0}\right).$

This Lemma is known as Jack's Lemma and it has been proved in ^[9].

2. Main results

2.1. Theorem

Let $g\in \mathcal{A}_{p}$ and satisfying

$\begin{equation*} 1+\frac{\alpha z^{1-p}g^{\prime }\left( z\right) }{p}\prec \frac{1+Az}{1+Bz}, \end{equation*}$

with the restriction on $\alpha$ is

$\begin{equation} \left \vert \alpha \right \vert \geq \frac{2^{\frac{3}{2}}p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) } . \end{equation}$

(2.1)

Then

$\begin{equation*} \frac{g\left( z\right) }{z^{p}}\prec \sqrt{1+z}. \end{equation*}$

Proof

Let us define a function

$\begin{equation} p\left( z\right) = 1+\frac{\alpha z^{1-p}g^{\prime }\left( z\right) }{p}, \end{equation}$

(2.2)

where the function $p$ is analytic in $\mathfrak{D}$ with $p(0) = 1.$ Also consider

$\begin{equation} \frac{g\left( z\right) }{z^{p}} = \sqrt{1+w\left( z\right) }. \end{equation}$

(2.3)

Now to prove our result we will only require to prove that $\left \vert w\left(z\right) \right \vert < 1.$ Logarithmically differentiating $\left(2.3\right)$ and then using $\left(2.2\right),$ we get

$\begin{equation*} p\left( z\right) = 1+\frac{\alpha zw^{\prime }\left( z\right) }{2p\sqrt{ 1+w\left( z\right) }}+\alpha \sqrt{1+w\left( z\right) }, \end{equation*}$

and so

$\begin{eqnarray*} \left \vert \frac{p\left( z\right) -1}{A-Bp\left( z\right) }\right \vert & = &\left \vert \frac{\frac{\alpha zw^{\prime }\left( z\right) }{2p\sqrt{ 1+w\left( z\right) }}+\alpha \sqrt{1+w\left( z\right) }}{A-B\left( 1+\frac{ \alpha zw^{\prime }\left( z\right) }{2p\sqrt{1+w\left( z\right) }}+\alpha \sqrt{1+w\left( z\right) }\right) }\right \vert \\ & = &\left \vert \frac{\alpha zw^{\prime }\left( z\right) +2p\alpha \left( 1+w\left( z\right) \right) }{2p\left( A-B\right) \sqrt{1+w\left( z\right) } -B\left( \alpha zw^{\prime }\left( z\right) +2p\alpha \left( 1+w\left( z\right) \right) \right) }\right \vert . \end{eqnarray*}$

Now, we suppose that a point $z_{0}\in \mathfrak{D}$ occurs such that

$\begin{equation*} \underset{\left \vert z\right \vert \leq \left \vert z_{0}\right \vert }{ \max }\left \vert w\left( z\right) \right \vert = \left \vert w\left( z_{0}\right) \right \vert = 1. \end{equation*}$

Also by Lemma 1.1, a number $m\geq 1$ exists with $z_{0}w^{\prime }\left(z_{0}\right) = mw\left(z_{0}\right)$ . In addition, we also suppose that $w\left(z_{0}\right) = e^{i\theta }$ for $\theta \in \left[ -\pi, \pi \right].$ Then we have

$\begin{eqnarray*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert & = &\left \vert \frac{\alpha mw\left( z_{0}\right) -2p\alpha \left( 1+w\left( z_{0}\right) \right) }{2p\left( A-B\right) \sqrt{1+w\left( z_{0}\right) }-B\left( \alpha mw\left( z_{0}\right) +2p\alpha \left( 1+w\left( z_{0}\right) \right) \right) }\right \vert , \\ &\geq &\frac{\left \vert \alpha \right \vert m-2p\left \vert \alpha \right \vert \left( \left \vert 1+e^{i\theta }\right \vert \right) }{2p\left( A-B\right) \sqrt{\left \vert 1+e^{i\theta }\right \vert }+\left \vert B\right \vert \left( \left \vert \alpha \right \vert m+2p\alpha \left \vert 1+e^{i\theta }\right \vert \right) }, \\ &\geq &\frac{\left \vert \alpha \right \vert m-4p\left \vert \alpha \right \vert }{2^{\frac{3}{2}}p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \left( m+4p\right) }. \end{eqnarray*}$

Now if

$\begin{equation*} \phi \left( m\right) = \frac{\left \vert \alpha \right \vert \left( m-4p\right) }{2^{\frac{3}{2}}p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \left( m+4p\right) }, \end{equation*}$

then

$\begin{equation*} \phi ^{\prime }\left( m\right) = \frac{2^{\frac{3}{2}}p\left( A-B\right) \left \vert \alpha \right \vert +8\left \vert \alpha \right \vert ^{2}p\left \vert B\right \vert }{\left( 2^{\frac{3}{2}}p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \left( m+4p\right) \right) ^{2} } \gt 0, \end{equation*}$

which illustrates that the function $\phi \left(m\right)$ is increasing and hence $\phi \left(m\right) \geq \phi \left(1\right)$ for $m\geq 1,$ so

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq \frac{\left \vert \alpha \right \vert \left( 1-4p\right) }{2^{ \frac{3}{2}}p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \left( 1+4p\right) }. \end{equation*}$

Now, by using $\left(2.1\right),$ we have

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq 1 \end{equation*}$

which contradicts the fact that $\begin{array}{c} p\left(z\right) \prec \frac{1+Az}{1+Bz}. \end{array}$ Thus $\left \vert w\left(z\right) \right \vert < 1$ and so we get the desired result.

Taking $g\left(z\right) = \frac{z^{p+1}f^{\prime }\left(z\right) }{pf(z)}$ in the last result, we obtain the following Corollary:

2.2. Corollary

Let $f\in \mathcal{A}_{p}$ and satisfying

$\begin{equation} 1+\frac{\alpha zf^{\prime }\left( z\right) }{p^{2}f\left( z\right) }\left( p+1+\frac{zf^{\prime \prime }\left( z\right) }{f^{\prime }\left( z\right) }- \frac{zf^{\prime }\left( z\right) }{f\left( z\right) }\right) \prec \frac{ 1+Az}{1+Bz}, \end{equation}$

(2.4)

with the condition on $\alpha$ is

$\begin{equation*} \left \vert \alpha \right \vert \geq \frac{2^{\frac{3}{2}}p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) } . \end{equation*}$

Then $f\in \mathcal{SL}_{p}^{\ast }.$

2.3. Theorem

If $g\in \mathcal{A}_{p}$ such that

$\begin{equation} 1+\frac{\alpha }{p}\frac{zg^{\prime }\left( z\right) }{g\left( z\right) } \prec \frac{1+Az}{1+Bz}, \end{equation}$

(2.5)

with

$\begin{equation} \left \vert \alpha \right \vert \geq \frac{8p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) }, \end{equation}$

(2.6)

then

$\begin{equation*} \frac{g\left( z\right) }{z^{p}}\prec \sqrt{1+z}. \end{equation*}$

Proof

Let us choose a function $p$ by

$\begin{equation*} p\left( z\right) = 1+\alpha \frac{zg^{\prime }\left( z\right) }{pg\left( z\right) }, \end{equation*}$

in such a way that $p$ is analytic in $\mathfrak{D}$ with $p(0) = 1.$ Also consider

$\begin{equation*} \frac{g\left( z\right) }{z^{p}} = \sqrt{1+w\left( z\right) }. \end{equation*}$

Using some simple calculations, we obtain

$\begin{equation*} p\left( z\right) = 1+\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) }+\alpha , \end{equation*}$

and so

$\begin{eqnarray*} \left \vert \frac{p\left( z\right) -1}{A-Bp\left( z\right) }\right \vert & = &\left \vert \frac{\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) }+\alpha }{A-B\left( 1+\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) }+\alpha \right) } \right \vert \\ & = &\left \vert \frac{\alpha zw^{\prime }\left( z\right) +2p\alpha \left( 1+w\left( z\right) \right) }{2p\left( A-B\right) \left( 1+w\left( z\right) \right) -B\left( \alpha zw^{\prime }\left( z\right) +2p\alpha \left( 1+w\left( z\right) \right) \right) }\right \vert . \end{eqnarray*}$

Let a point $z_{0}\in \mathfrak{D}$ exists in such a way

Then, by virtue of Lemma 1.1, a number $m\geq 1$ occurs such that $\ z_{0}w^{\prime }\left(z_{0}\right) = mw\left(z_{0}\right)$ . In addition, we set $w\left(z_{0}\right) = e^{i\theta },$ so we have

$\begin{eqnarray*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert & = &\left \vert \frac{\alpha mw\left( z_{0}\right) +2p\alpha \left( 1+w\left( z_{0}\right) \right) }{2p\left( A-B\right) \left( 1+w\left( z_{0}\right) \right) -B\left( \alpha mw\left( z_{0}\right) +2p\alpha \left( 1+w\left( z_{0}\right) \right) \right) }\right \vert , \\ &\geq &\frac{\left \vert \alpha \right \vert m-2p\left \vert \alpha \right \vert \left \vert 1+e^{i\theta }\right \vert }{2\left( A-B\right) \left \vert 1+e^{i\theta }\right \vert +\left \vert B\right \vert \left \vert \alpha \right \vert m+2p\left \vert B\right \vert \left \vert \alpha \right \vert \left \vert 1+e^{i\theta }\right \vert }, \\ & = &\frac{\left \vert \alpha \right \vert m-2p\left \vert \alpha \right \vert \sqrt{2+2\cos \theta }}{2\left( 2\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \right) p\sqrt{2+2\cos \theta }+\left \vert B\right \vert \left \vert \alpha \right \vert m}, \\ &\geq &\frac{\left \vert \alpha \right \vert \left( m-4p\right) }{4p\left( 2\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \right) +\left \vert B\right \vert \left \vert \alpha \right \vert m}. \end{eqnarray*}$

Now let

$\begin{equation*} \phi \left( m\right) = \frac{\left \vert \alpha \right \vert \left( m-4p\right) }{4p\left( 2\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \right) +\left \vert B\right \vert \left \vert \alpha \right \vert m}, \end{equation*}$

it implies

$\begin{equation*} \phi ^{\prime }\left( m\right) = \frac{\left \vert \alpha \right \vert 8p\left( \left( A-B\right) +\left \vert \alpha \right \vert \left \vert B\right \vert \right) }{\left( 4p\left( 2\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \right) +\left \vert B\right \vert \left \vert \alpha \right \vert m\right) ^{2}} \gt 0, \end{equation*}$

which illustrates that the function $\phi \left(m\right)$ is increasing and so $\phi \left(m\right) \geq \phi \left(1\right)$ for $m\geq 1,$ hence

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq \frac{\left \vert \alpha \right \vert \left( 1-4p\right) }{ 4p\left( 2\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert \right) +\left \vert B\right \vert \left \vert \alpha \right \vert }. \end{equation*}$

Now, by using $\left(2.6\right),$ we have

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq 1, \end{equation*}$

which contradicts $\left(2.5\right).$ Thus $\left \vert w\left(z\right) \right \vert < 1$ and so the desired proof is completed.

Putting $g\left(z\right) = \frac{z^{p+1}f^{\prime }\left(z\right) }{pf(z)}$ in last Theorem, we get the following Corollary:

2.4. Corollary

If $f\in \mathcal{A}_{p}$ and satisfying

$\begin{equation*} 1+\frac{\alpha }{p}\left( p+1+\frac{zf^{\prime \prime }\left( z\right) }{ f^{\prime }\left( z\right) }-\frac{zf^{\prime }\left( z\right) }{f\left( z\right) }\right) \prec \frac{1+Az}{1+Bz}, \end{equation*}$

with

$\begin{equation*} \left \vert \alpha \right \vert \geq \frac{8p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) }, \end{equation*}$

then $f\in \mathcal{SL}_{p}^{\ast }.$

2.5. Theorem

If $g\in \mathcal{A}_{p}$ and satisfy the subordination relation

$\begin{equation} 1+\alpha \frac{z^{1-p}g^{\prime }\left( z\right) }{p\left( g\left( z\right) \right) ^{2}}\prec \frac{1+Az}{1+Bz}, \end{equation}$

(2.7)

with the condition on $\alpha$

$\begin{equation} \left \vert \alpha \right \vert \geq \frac{2^{\frac{5}{2}}p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) } \end{equation}$

(2.8)

is true, then

$\begin{equation*} \frac{g\left( z\right) }{z^{p}}\prec \sqrt{1+z}. \end{equation*}$

Proof

Let us define a function

$\begin{equation*} p\left( z\right) = 1+\alpha \frac{z^{1-p}g^{\prime }\left( z\right) }{p\left( g\left( z\right) \right) ^{2}}. \end{equation*}$

Then $p$ is analytic in $\mathfrak{D}$ with $p(0) = 1.$ Also let us consider

$\begin{equation*} \frac{g\left( z\right) }{z^{p}} = \sqrt{1+w\left( z\right) }. \end{equation*}$

Using some simplification, we obtain

$\begin{equation*} p\left( z\right) = 1+\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) ^{\frac{3}{2}}}+\frac{\alpha }{\sqrt{1+w\left( z\right) }}, \end{equation*}$

and so

$\begin{eqnarray*} \left \vert \frac{p\left( z\right) -1}{A-Bp\left( z\right) }\right \vert & = &\left \vert \frac{\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) ^{\frac{3}{2}}}+\frac{\alpha }{\sqrt{1+w\left( z\right) }}}{A-B\left( 1+\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) ^{\frac{3}{2}}}+\frac{\alpha }{\sqrt{1+w\left( z\right) }}\right) }\right \vert \\ & = &\left \vert \frac{\alpha zw^{\prime }\left( z\right) +2p\alpha \left( 1+w\left( z\right) \right) }{2p\left( A-B\right) \left( 1+w\left( z\right) \right) ^{\frac{3}{2}}-B\alpha zw^{\prime }\left( z\right) -2p\alpha B\left( 1+w\left( z\right) \right) }\right \vert . \end{eqnarray*}$

Let us choose a point $z_{0}\in \mathfrak{D}$ such a way that

Then, by the consequences of Lemma 1.1, a number $m\geq 1$ occurs such that $z_{0}w^{\prime }\left(z_{0}\right) = mw\left(z_{0}\right)$ and also put $w\left(z_{0}\right) = e^{i\theta },$ for $\theta \in \left[ -\pi, \pi \right],$ we have

$\begin{eqnarray*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert & = &\left \vert \frac{\alpha mw\left( z_{0}\right) +2p\alpha \left( 1+w\left( z_{0}\right) \right) }{2p\left( A-B\right) \left( 1+w\left( z_{0}\right) \right) ^{\frac{3}{2}}-B\alpha mw\left( z_{0}\right) -2p\alpha B\left( 1+w\left( z_{0}\right) \right) }\right \vert , \\ &\geq &\frac{\left \vert \alpha \right \vert m-2p\left \vert \alpha \right \vert \left \vert 1+e^{i\theta }\right \vert }{2p\left( A-B\right) \left \vert 1+e^{i\theta }\right \vert ^{\frac{3}{2}}+\left \vert B\right \vert \left \vert \alpha \right \vert m+2p\left \vert \alpha \right \vert \left \vert B\right \vert \left \vert 1+e^{i\theta }\right \vert }, \\ & = &\frac{\left \vert \alpha \right \vert m-4p\left \vert \alpha \right \vert }{2^{\frac{5}{2}}p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert m+4p\left \vert \alpha \right \vert \left \vert B\right \vert }, \\ &\geq &\frac{\left \vert \alpha \right \vert \left( m-4p\right) }{2^{\frac{5 }{2}}p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert m+4p\left \vert \alpha \right \vert \left \vert B\right \vert } = \phi \left( m\right) \text{ (say).} \end{eqnarray*}$

Then

$\begin{equation*} \phi ^{\prime }\left( m\right) = \frac{2^{\frac{5}{2}}p\left( A-B\right) +8\left \vert \alpha \right \vert ^{2}\left \vert B\right \vert p}{\left( 2^{ \frac{5}{2}}p\left( A-B\right) +B\left \vert \alpha \right \vert m+4p\left \vert \alpha \right \vert B\right) ^{2}} \gt 0, \end{equation*}$

which demonstrates that the function $\phi \left(m\right)$ is increasing and thus $\phi \left(m\right) \geq \phi \left(1\right)$ for $m\geq 1,$ hence

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq \frac{\left \vert \alpha \right \vert \left( 1-4p\right) }{2^{ \frac{5}{2}}p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert +4p\left \vert \alpha \right \vert \left \vert B\right \vert }. \end{equation*}$

Now, using $\left(2.8\right),$ we have

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq 1, \end{equation*}$

which contradicts $\left(2.7\right)$ . Thus $\left \vert w\left(z\right) \right \vert < 1$ and so we get the required proof.

If we set $g\left(z\right) = \frac{z^{p+1}f^{\prime }\left(z\right) }{pf(z)}$ in last theorem, we easily have the following Corollary:

2.6. Corollary

Assume that

$\begin{equation*} \left \vert \alpha \right \vert \geq \frac{2^{\frac{5}{2}}p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) } , \end{equation*}$

and if $f\in \mathcal{A}_{p}$ satisfy

$\begin{equation*} 1+\frac{\alpha f\left( z\right) }{z^{2p+1}f^{\prime }\left( z\right) }\left( p+1+\frac{zf^{\prime \prime }\left( z\right) }{f^{\prime }\left( z\right) }- \frac{zf^{\prime }\left( z\right) }{f\left( z\right) }\right) \prec \frac{ 1+Az}{1+Bz}, \end{equation*}$

then $f\in \mathcal{SL}_{p}^{\ast }.$

2.7. Theorem

If $g\in \mathcal{A}_{p}$ satisfy the subordination

$\begin{equation*} 1+\alpha \frac{z^{1-2p}g^{\prime }\left( z\right) }{p\left( g\left( z\right) \right) ^{3}}\prec \frac{1+Az}{1+Bz}, \end{equation*}$

with restriction on $\alpha$ is

$\begin{equation} \left \vert \alpha \right \vert \geq \frac{8p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) }. \end{equation}$

(2.9)

then

$\begin{equation*} \frac{g\left( z\right) }{z^{p}}\prec \sqrt{1+z}. \end{equation*}$

Proof. Let us define a function

$\begin{equation*} p\left( z\right) = 1+\alpha \frac{z^{1-2p}g^{\prime }\left( z\right) }{ p\left( g\left( z\right) \right) ^{3}}, \end{equation*}$

where $p$ is analytic in $\mathfrak{D}$ with $p(0)-1 = 0.$ Also let

$\begin{equation*} \frac{g\left( z\right) }{z^{p}} = \sqrt{1+w\left( z\right) }. \end{equation*}$

Using some simple calculations, we obtain

$\begin{equation*} p\left( z\right) = 1+\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) ^{2}}+\frac{\alpha }{1+w\left( z\right) }, \end{equation*}$

and so

$\begin{eqnarray*} \left \vert \frac{p\left( z\right) -1}{A-Bp\left( z\right) }\right \vert & = &\left \vert \frac{\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) ^{2}}+\frac{\alpha }{1+w\left( z\right) }}{ A-B\left( 1+\frac{\alpha zw^{\prime }\left( z\right) }{2p\left( 1+w\left( z\right) \right) ^{2}}+\frac{\alpha }{1+w\left( z\right) }\right) }\right \vert \\ & = &\left \vert \frac{\alpha zw^{\prime }\left( z\right) +2p\alpha \left( 1+w\left( z\right) \right) }{2p\left( A-B\right) \left( 1+w\left( z\right) \right) ^{2}-B\alpha zw^{\prime }\left( z\right) -2p\alpha B\left( 1+w\left( z\right) \right) }\right \vert . \end{eqnarray*}$

Let us pick a point $z_{0}\in \mathfrak{D}$ in such a way that

Then, by using Lemma 1.1 $,$ a number $m\geq 1$ exists such that $z_{0}w^{\prime }\left(z_{0}\right) = mw\left(z_{0}\right) \$ and put $w\left(z_{0}\right) = e^{i\theta },$ for $\theta \in \left[ -\pi, \pi \right]$ , we have

$\begin{eqnarray*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert & = &\left \vert \frac{\alpha mw\left( z_{0}\right) +2p\alpha \left( 1+w\left( z_{0}\right) \right) }{2p\left( A-B\right) \left( 1+w\left( z_{0}\right) \right) ^{2}-B\alpha mw\left( z_{0}\right) -2p\alpha B\left( 1+w\left( z_{0}\right) \right) }\right \vert \\ &\geq &\frac{\left \vert \alpha \right \vert m-2p\left \vert \alpha \right \vert \left \vert 1+e^{i\theta }\right \vert }{2p\left( A-B\right) \left \vert 1+e^{i\theta }\right \vert ^{2}+\left \vert B\right \vert \left \vert \alpha \right \vert m+2p\left \vert \alpha \right \vert \left \vert B\right \vert \left \vert 1+e^{i\theta }\right \vert } \\ & = &\frac{\left \vert \alpha \right \vert m-2p\left \vert \alpha \right \vert \sqrt{2+2\cos \theta }}{2p\left( A-B\right) \left( \sqrt{2+2\cos \theta } \right) ^{2}+\left \vert B\right \vert \left \vert \alpha \right \vert m+2p\left \vert \alpha \right \vert \left \vert B\right \vert \sqrt{2+2\cos \theta }} \\ &\geq &\frac{\left \vert \alpha \right \vert \left( m-4p\right) }{8p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert m+4p\left \vert \alpha \right \vert \left \vert B\right \vert }, \end{eqnarray*}$

Now let

$\begin{equation*} \phi \left( m\right) = \frac{\left \vert \alpha \right \vert \left( m-4p\right) }{8p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert m+4p\left \vert \alpha \right \vert \left \vert B\right \vert }, \end{equation*}$

then

$\begin{equation*} \phi ^{\prime }\left( m\right) = \frac{8p\left \vert \alpha \right \vert \left( A-B\right) +8\left \vert \alpha \right \vert ^{2}\left \vert B\right \vert p}{\left( 8p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert m+4p\left \vert \alpha \right \vert \left \vert B\right \vert \right) ^{2}} \gt 0 \end{equation*}$

which shows that $\phi \left(m\right)$ is an increasing function and hence it will have its minimum value at $m = 1$ , so

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq \frac{\left \vert \alpha \right \vert \left( 1-4p\right) }{ 8p\left( A-B\right) +\left \vert B\right \vert \left \vert \alpha \right \vert +4p\left \vert \alpha \right \vert \left \vert B\right \vert }. \end{equation*}$

Using $\left(2.9\right),$ we easily obtain

$\begin{equation*} \left \vert \frac{p\left( z_{0}\right) -1}{A-Bp\left( z_{0}\right) }\right \vert \geq 1, \end{equation*}$

which is a contradiction to the fact that $p\left(z\right) \prec \frac{1+Az }{1+Bz},$ and so $\left \vert w\left(z\right) \right \vert < 1.$ Hence we get the desired result.

If we put $g\left(z\right) = \frac{z^{p+1}f^{\prime }\left(z\right) }{pf(z)}$ in last Theorem, we achieve the following result:

2.8. Corollary

If $f\in \mathcal{A}_{p}$ and satisfy the condition

$\begin{equation*} \left \vert \alpha \right \vert \geq \frac{8p\left( A-B\right) }{1-\left \vert B\right \vert -4p\left( 1+\left \vert B\right \vert \right) }, \end{equation*}$

and

$\begin{equation*} 1+\alpha \frac{p\left( f\left( z\right) \right) ^{2}}{z^{3p+2}\left( f^{\prime }\left( z\right) \right) ^{2}}\left( p+1+\frac{zf^{\prime \prime }\left( z\right) }{f^{\prime }\left( z\right) }-\frac{zf^{\prime }\left( z\right) }{f\left( z\right) }\right) \prec \frac{1+Az}{1+Bz}, \end{equation*}$

then $f\in \mathcal{SL}_{p}^{\ast }.$

Conflict of interest

All authors declare no conflict of interest in this paper.

References

[1]	WHO, The burden of Influenza. Available from: https://www.who.int/news-room/feature-stories/detail/the-burden-of-influenza#: : text = Nevertheless.
[2]	J. Paget, P. Spreeuwenberg, V. Charu, R. J. Taylor, A. D. Luliano, J. Bresee, et al., Global mortality associated with seasonal influenza epidemics: New burden estimates and predictors from the GLaMOR Project, J. Glob. Health, 9 (2019), 020421. https://doi.org/10.7189/jogh.09.020421 doi: 10.7189/jogh.09.020421
[3]	S. S. Chaves, J. Nealon, K. G. Burkart, D. Modin, T. Biering-Sørensen, J. R. Ortiz, et al., Global, regional and national estimates of influenza-attributable ischemic heart disease mortality, eClinicalMedicine, 55 (2023), 101740. https://doi.org/10.1016/j.eclinm.2022.101740 doi: 10.1016/j.eclinm.2022.101740
[4]	M. Safan, Mathematical analysis of an SIR respiratory infection model with sex and gender disparity: Special reference to influenza A, Math. Biosci. Eng., 16 (2019), 2613–2649. https://doi.org/10.3934/mbe.2019131 doi: 10.3934/mbe.2019131
[5]	World Health Organization, Sex, Gender and Influenza, WHO Library Cataloguing-in-Publication Data, 2010. Available from: https://iris.who.int/bitstream/handle/10665/44401/9789241500111_eng.pdf.
[6]	S. L. Klein, A. Hodgson, D. P. Robinson, Mechanisms of sex disparities in influenza pathogenesis, J. Leukocyte Biol., 92 (2012), 67–73. https://doi.org/10.1189/jlb.0811427 doi: 10.1189/jlb.0811427
[7]	R. Casagrandi, L. Bolzoni, S. A. Levin, V. Andreasen, The SIRC model and influenza A, Math. Biosci., 200 (2006), 152–169. https://doi.org/10.1016/j.mbs.2005.12.029 doi: 10.1016/j.mbs.2005.12.029
[8]	M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel, B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503–524. https://doi.org/10.1137/030600370 doi: 10.1137/030600370
[9]	Z. Qiu, Z. Feng, Transmission dynamics of an influenza model with vaccination and antiviral treatment, Bull. Math. Biol., 72 (2010), 1–33. https://doi.org/10.1007/s11538-009-9435-5 doi: 10.1007/s11538-009-9435-5
[10]	A. L. Vivas-Barber, C. Castillo-Chavez, E. Barany, Dynamics of an "SAIQR" influenza model, Biomath, 3 (2014), 1409251. https://doi.org/10.11145/j.biomath.2014.09.251 doi: 10.11145/j.biomath.2014.09.251
[11]	M. Erdem, M. Safan, C. Castillo-Chavez, Mathematical analysis of an SIQR influenza model with imperfect quarantine, Bull. Math. Biol., 79 (2017), 1612–1636. https://doi.org/10.1007/s11538-017-0301-6 doi: 10.1007/s11538-017-0301-6
[12]	H. Manchanda, N. Seidel, A. Krumbholz, A. Sauerbrei, M. Schmidtke, R. Guthke, Within-host influenza dynamics: A small-scale mathematical modeling approach, Biosystems, 118 (2014), 51–59. https://doi.org/10.1016/j.biosystems.2014.02.004 doi: 10.1016/j.biosystems.2014.02.004
[13]	B. Emerenini, R. Williams, R. N. G. R. Grimaldo, K. Wurscher, R. Ijioma, Mathematical modeling and analysis of influenza in-host Infection dynamics, Lett. Biomath., 8 (2021), 229–253. https://doi.org/10.30707/LiB8.1.1647878866.124006 doi: 10.30707/LiB8.1.1647878866.124006
[14]	M. Samsuzzoha, M. Singh, D. Lucy, Parameter estimation of influenza epidemic model, Appl. Math. Comput., 220 (2013), 616–629. https://doi.org/10.1016/j.amc.2013.07.040 doi: 10.1016/j.amc.2013.07.040
[15]	M. Nuño, Z. Feng, M. Martcheva, C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), 964–982. https://doi.org/10.1137/S003613990343882X doi: 10.1137/S003613990343882X
[16]	M. E. Alexander, S. M. Moghadas, G. Röst, J. Wu, A delay differential model for pandemic influenza with antiviral treatment, Bull. Math. Biol., 70 (2008), 382–397. https://doi.org/10.1007/s11538-007-9257-2 doi: 10.1007/s11538-007-9257-2
[17]	P. Krishnapriya, M. Pitchaimani, T. M. Witten, Mathematical analysis of an influenza A epidemic model with discrete delay, J. Comput. Appl. Math., 324 (2017), 155–172. https://doi.org/10.1016/j.cam.2017.04.030 doi: 10.1016/j.cam.2017.04.030
[18]	L. Bailey, What determines our human carrying capacity on the planet, Population Education. Available from: https://populationeducation.org/what-determines-our-human-carrying-capacity-planet/#: : text = The.
[19]	X. Tan, L. Yuan, J. Zhou, Y. Zheng, F. Yang, Modeling the initial transmission dynamics of influenza A H1N1 in Guangdong Province, China, Int. J. Infect. Dis., 17 (2013), e479–e484. https://doi.org/10.1016/j.ijid.2012.11.018 doi: 10.1016/j.ijid.2012.11.018
[20]	P. van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6 doi: 10.1016/S0025-5564(02)00108-6
[21]	M. Safan, K. Dietz, On the eradicability of infections with partially protective vaccination in models with backward bifurcation, Math. Biosci. Eng., 6 (2009), 395–407. https://doi.org/10.3934/mbe.2009.6.395 doi: 10.3934/mbe.2009.6.395
[22]	M. Safan, M. Kretzschmar, K. P. Hadeler, Vaccination based control of infections in SIRS models with reinfection: Special reference to pertussis, J. Math. Biol., 67 (2013), 1083–1110. https://doi.org/10.1007/s00285-012-0582-1 doi: 10.1007/s00285-012-0582-1
[23]	M. Safan, H. Heesterbeek, K. Dietz, The minimum effort required to eradicate infections in models with backward bifurcation, J. Math. Biol., 53 (2006), 703–718. https://doi.org/10.1007/s00285-006-0028-8 doi: 10.1007/s00285-006-0028-8

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