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An SEIR type of compartmental model with nonlinear incidence and recovery rates was formulated to study the combined impacts of psychological effect and available resources of public health system especially the number of hospital beds on the transmission and control of A(H7N9) virus. Global stability of the disease-free and endemic equilibria is determined by the basic reproduction number as a threshold parameter and is obtained by constructing Lyapunov function and second additive compound matrix. The results obtained reveal that psychological effect and available resources do not change the stability of the steady states but can indeed diminish the peak and the final sizes of the infected. Our studies have practical implications for the transmission and control of A(H7N9) virus.

Avian influenza A(H7N9) is a subtype of influenza viruses that have been detected in birds and confirmed to be low pathogenic among poultry in the past [

There are different types of models to analyze the dynamical behavior of avian influenza virus and assess useful control measures. Iwami et al. [

When a disease breaks out, people’s awareness of its severity can generate a profound psychological impact on the individuals’ behaviors to reduce unnecessary contact with infections [

In previous dynamic models of avian influenza A(H7N9), one usually assumed the recovery rate as a constant, which means that the treatments were always sufficient. But in fact, hospital resources (such as doctors, drugs, hospital bed, and isolation places) are limited to public, especially when a disease breaks out [

This paper is organized as follows. In Section

Based on information reported, there is no evidence of sustained human-to-human transmission, although there have been two family clusters reported. Thus, we always assume the transmission of A(H7N9) virus is not from person to person. In our model, we divide the poultry into two subclasses: susceptible

(i) Taking into account the factors such as poultry market mobility, environment capacity, and the existing populations, the susceptible poultry is subject to the logistic growth [

(ii) Due to psychological effect, the infection force may decrease when the number of infectious individuals increases. Hence, we modify a nonlinear incidence rate proposed by Liu et al. [

(iii) We assume that latent humans

Due to the above assumptions, we can formulate the system as follows:

Detailed descriptions of system parameters and their estimated values are listed in Table

Description of parameters.

Parameter | Description | Value | Reference |
---|---|---|---|

| Intrinsic growth rate of poultry | | [ |

| Maximal carrying capacity of the poultry | 50000 | [ |

| Transmission rate from infectious poultry to susceptible poultry | - | - |

| Natural death rate of poultry (chicken) | 1/5–1/10 | [ |

| Disease induced death rate of poultry | | [ |

| New recruitment and newborn of human | 30 | [ |

| Transmission rate from infectious poultry to susceptible human | | Assumed |

| Natural death rate of human | 1/70 | Assumed |

| Disease induced death rate of human | 0.077 | [ |

| Progression to latent rate of human | 1/7 | CDC |

| Minimum recovery rate of human | (0.067–0.100) | [ |

| Maximum recovery rate of human | | [ |

| Hospital bed-population ratio | | [ |

| Psychological effect parameter | - | - |

For system (

The set

For system (

Let

Therefore, each solution of system (

In this section, we study the existence of equilibria of system (

Therefore, the coordinates of equilibria are determined by nonnegative solutions of equations (

Next, we discuss the endemic equilibrium denoted by

And

(i) Assuming there are three real roots, Vieta Theorem indicates that

(ii) Otherwise, suppose that there are a pair of complex roots and a positive real root, denoted by

In summary, we can conclude that (

In system (

In order to better discuss the full system, we first learn the poultry-only subsystem in

Linearizing the subsystem (

The disease-free equilibrium

The following theorem shows the global stability of the equilibria.

If

If

If

In this section, we will discuss the dynamical behavior of system (

The disease-free equilibrium

(i) The Jacobian matrix at

(ii) The Jacobian matrix at

(iii) The Jacobian matrix at

If

If

If

We consider the norm

We calculate

In this section, we carry out numerical simulations for system (

Our theoretical results show that the basic reproduction number

(a) All solutions of

We then use Latin hypercube sampling (LHS) [

(a) PRCCs for the endemic equilibrium prevalence. All the parameters came from Latin hypercube sampling. (b) Plot of the endemic equilibrium prevalence with respect to the psychological effect parameter

To further examine the impact of psychological effect and hospital resources on infections, respectively, we take

Fix

In this work, in order to evaluate the combined impact of psychological effect and available hospital resources on the transmission of A(H7N9) virus from poultry to humans, we formulated and analyzed a dynamical model with a nonlinear incidence rate and a nonlinear recovery rate. From the mathematical point of view, we obtained the basic reproduction number

Both the psychological effect and available hospital resources cannot neither change the stability of endemic equilibrium nor alter the basic reproduction number, but they indeed play a significant role in affecting the number of infectious humans, seen from PRCC results (Figure

Different from the previous avian influenza dynamics models, which usually use bilinear and standard incidence rates and constant recovery rate, in this work, incorporating the combined impact of psychological effect and available hospital resources, we formulate A(H7N9) dynamic model with nonlinear incidence rate and nonlinear recovery rate. We introduce the recovery function

Note that, from current data for A(H7N9) infection, there is an incubation period between infection and symptom onset in both avian and human populations [

The authors declare that they have no conflicts of interest.

This work is supported by the National Natural Science Foundation of China (NSFC 11401349) and the Foundation for Outstanding Young Scientist in Shandong Province (BS2014SF008).