## 1. Introduction

Over the decades, quarantine (of individuals suspected of being exposed to a communicable disease) and isolation (of individuals with disease symptoms) have been widely used to control the spread of numerous communicable diseases, such as pandemic influenza, cholera, Ebola, Severe Acute Respiratory Syndrome (SARS), and most recently swine influenza pandemic [

1,

2,

3,

4,

5,

6,

7,

8,

9]. Numerous mathematical models have been studying the effect of quarantine and isolation in combatting the spread of the diseases (see, for instance, refs. [

1,

2,

4,

5,

6,

7,

8,

10,

11,

12,

13] and the references therein). In the aforementioned studies, mass action or standard incidence functions were used in the modeling of the transmission dynamics of the diseases. In this study, another nonlinear incidence function (called the Holling type II incidence function) will be used in the modeling of the transmission dynamics of a general disease. The Holling type II incidence function is given by

$g\left(I\right)=\frac{\beta I}{1+\alpha I},$ with

$\alpha >0$, where

I is the number of infectious individuals and

$\beta $ is the effective contact rate (the average number of contacts sufficient for transmitting infection). The incidence function

$g\left(I\right)$ was first used in the study of the cholera epidemic in Bari, Italy by Capasso and Serio [

14]. The reason for using the Holling type II incidence functional comes from the information that the number of effective contacts between susceptible individuals and infective individuals may saturate at very high levels due to behavioral changes or due to crowding of infective people taken by the people in reaction to the severity of the disease [

15,

16]. It is well known that some infectious diseases, such as influenza [

17] and HIV [

18], have multiple disease (infection) stages in their transmission dynamics.

The main purpose of this study is to offer a deep qualitative analysis of a new two-stage model for the transmission dynamics of a disease that can be controlled by using quarantine and isolation, where the Holling type II incidence function is used.

The paper is organized as follows. The formulation of the model is given in

Section 2. The local and global asymptotic stability of the disease-free equilibrium (DFE) is analyzed in

Section 3. The existence of the endemic equilibrium is provided in

Section 4. Global stability proof for the endemic equilibrium for the special case is also analyzed using a nonlinear Lyapunov function.

## 2. Model Formulation

The total population at time

t, denoted by

$N\left(t\right)$ is sub-divided into ten compartments of susceptible

$\left(S\right(t\left)\right),$ exposed (with two stages

$({E}_{1}\left(t\right){E}_{2}\left(t\right)$), infectious individuals (with two stages

$({I}_{1}\left(t\right){I}_{2}\left(t\right)$), Isolated individuals (with two stages

${H}_{1}\left(t\right){H}_{2}\left(t\right)$), and recovered

$\left(R\right(t\left)\right)$ individuals, so that

The model is given by the following system of nonlinear differential equations

where

$\lambda \left(t\right)$ is the infection rate given by

In (

2),

$\beta $ represents the effective contact rate, where

$0<\eta <1$ is a parameter that accounts for the reduction in disease transmission given by infectious individuals (

${I}_{1}$) in comparison to infectious individuals in the

${I}_{2}$ stage.

Susceptible people $\left(S\right)$ is increased by the recruitment of individuals into the population, at a rate $\mathsf{\Pi}$. This class is decreased by infection (with the rate of $\lambda $). Furthermore, this population is decreased by natural death (at a rate $\mu $; populations in all classes are assumed to have the same natural death rate).

Exposed individuals in stage 1 (${E}_{1}$) are generated with the rate of $\lambda $ and reduced by progression to the next exposed stage (${E}_{2}$; at a rate ${a}_{1}$) and quarantine (at a rate ${b}_{1}$). Exposed individuals in stage 2 are generated at the rate ${a}_{1}$. This population is decreased by the development of clinical symptoms of the disease (at a rate ${a}_{2}$) and quarantine (at a rate ${b}_{2}$).

The class of quarantined individuals in stage 1 is increased by quarantine of exposed people in stage ${E}_{1}$ (at the rate ${b}_{1}$) and it is reduced by progression to the second quarantined stage (at a rate ${c}_{1}$). Similarly, quarantined people in stage 2 are increased by the quarantine of exposed people in the second stage (at the rate ${b}_{2}$) and the progression of quarantined people from the first stage into the second stage (at the rate ${c}_{1}$). It is decreased by hospitalization (at a rate ${c}_{2}$).

The infectious people in stage 1 are increased when exposed people in the second stage develop symptoms (at the rate ${a}_{2}$). It is reduced by progression to the second infectious stage (at a rate ${d}_{1}$), hospitalization (isolation) (at a rate ${e}_{1}$) and disease-induced death (at a rate ${\delta}_{1}$). The population of infectious class in the second stage is generated by progression of individuals in the first stage (at a rate ${d}_{1}$). It is reduced by isolation (at a rate ${e}_{2}$), recovery (at a rate ${\gamma}_{1}$) and disease-induced death (at a rate ${\delta}_{2}$).

The population of Isolated individuals in the first stage is increased by the hospitalization of infectious people in stage 1 (at the rate ${e}_{1}$) and quarantined individuals in the second stage (at the rate ${c}_{2}$). It is decreased by progression to the second Isolated stage (at a rate ${f}_{1}$), and disease-induced death (at a rate ${\delta}_{3}$). The population of Isolated individuals in the second stage is generated by the progression of Isolated individuals from the first stage into the second one (at the rate ${f}_{1}$). It is decreased by recovery (a rate ${\gamma}_{2}$) and disease-induced death (at a rate ${\delta}_{4}$).

Finally, the recovered individuals is increased by the recovery of infectious individuals and hospitalization individuals (at the rates

${\gamma}_{1}$ and

${\gamma}_{2}$, respectively). It is reduced by natural death (at the rate

$\mu $). (A flow diagram of the model is depicted in

Figure 1. The associated variables and parameters are described in

Table 1):

It should be noted the model (

1) is different by the basic model considered in [

19] by

- (a)
Using a Holling type incidence function to model the infection rate (the standard incidence function was used in [

19])

- (b)
Considering two stages for the infectious compartments (Exposed, infected, quarantined, and isolated compartments)

#### 2.1. Preliminaries and Basic Properties

Since the model (

1) for human populations, all its parameters are non-negative. Furthermore, the following non-negativity result holds.

**Theorem** **1.** All variables of the model (1) are non-negative for all $t>0$. This mean, the solutions of system (1) with positive initial conditions will remain positive for all time $t>0$. **Proof.** Hence,

${t}_{1}>0.$ From the first equation of the system (

1) it follows that

which gives,

hence,

In the same way, it can be shown that ${E}_{1}>0,{E}_{2}>0,{Q}_{1}>0,{Q}_{2}>0,{I}_{1}>0,{I}_{2}>0,{H}_{1}>0,$${H}_{2}>0$ and $R>0$ for all time $t>0$. □

**Lemma** **1.** The closed setis positively invariant. **Proof.** Adding all the equations of the model (

1) gives,

It follows that

$\frac{dN}{dt}\le \mathsf{\Pi}-\mu N$, thus

$\frac{dN}{dt}\le 0$ provided that

$N\ge \frac{\mathsf{\Pi}}{\mu}$. By using standard comparison theorem [

20] it can be shown that

$N\le {\displaystyle N\left(0\right){e}^{-\mu t}+\frac{\mathsf{\Pi}}{\mu}(1-{e}^{-\mu t}})$. In particular,

$N\left(t\right)\le \frac{\mathsf{\Pi}}{\mu}$ if

$N\left(0\right)\le \frac{\mathsf{\Pi}}{\mu}.$ Thus, the region

$\mathcal{D}$ is positively invariant. Furthermore, if

$N\left(0\right)>\frac{\mathsf{\Pi}}{\mu},$ then either the solution enters

$\mathcal{D}$ in finite time, or

$N\left(t\right)$ approaches

$\frac{\mathsf{\Pi}}{\mu}$ asymptotically. Hence, the region

$\mathcal{D}$ attracts all solutions in

${\mathbb{R}}_{+}^{10}$. □

Since the region

$\mathcal{D}$ is positively invariant, it is sufficient to consider the dynamics of the flow generated by the model (

1) in

$\mathcal{D}$, where the usual existence, uniqueness, continuation results hold for the system [

21].

#### Next-Generation Method

Suppose that the population is divided into

n compartments, with

$m<n$ infected compartments. At time

$t,$ let

${x}_{i}\left(t\right)$ be the number of infected individuals in the

${i}^{th}$ infected class such that

where

${F}_{i}\left(x\right)$ represents the rate of appearance of new infections in class

i,

${V}_{i}+\left(x\right)$ represents the rate of transfer of individuals into class

i by all other means, and

${V}_{i}^{-}\left(x\right)$ represents the rate of transfer of individuals out of class

i. System can be rewritten as follows

with,

$F\left(X\right)={({F}_{1},{F}_{2},\dots ,{F}_{m})}^{T}$ and

$V\left(X\right)={({V}_{1},{V}_{2},\dots ,{V}_{m})}^{T}.$**Lemma** **2.** (van den Driessche and Watmough [

22]).

If $\overline{x}$ is a DFE of (5), then the derivatives $D\mathrm{F}\left(\overline{x}\right)$ and $D\mathrm{V}\left(\overline{x}\right)$ are partitioned aswhere F and V are the $m\times m$ matrices defined by,Furthermore, F is non-negative, V is a non-singular $M-$matrix and ${J}_{3},{J}_{4}$ are matrices associated with the transition terms of the model, and all eigenvalues of ${J}_{4}$ have positive real parts.Now, the next-generation matrix is given by

$F{V}^{-1}$ and the spectral radius (the largest eigenvalue) of

$F{V}^{-1}$ is the basic reproduction number of the model (

5) [

22].