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MATHEMATICAL BIOSCIENCES doi:10.3934/mbe.2013.10.445

AND ENGINEERING

Volume 10, Number 2, April 2013 pp. 445–461

DYNAMICS OF AN INFECTIOUS DISEASES WITH

MEDIA/PSYCHOLOGY INDUCED NON-SMOOTH INCIDENCE

Yanni Xiao and Tingting Zhao

Department of Applied Mathematics

Xi’an Jiaotong University, Xi’an, 710049, China

Sanyi Tang

College of Mathematics and Information Science

Shaanxi Normal University, Xi’an, 710062, China.

(Communicated by Haiyan Wang)

Abstract. This paper proposes and analyzes a mathematical model on an

infectious disease system with a piecewise smooth incidence rate concerning

media/psychological eﬀect. The proposed models extend the classic models

with media coverage by including a piecewise smooth incidence rate to repre-

sent that the reduction factor because of media coverage depends on both the

number of cases and the rate of changes in case number. On the basis of prop-

erties of Lambert W function the implicitly deﬁned model has been converted

into a piecewise smooth system with explicit deﬁnition, and the global dynamic

behavior is theoretically examined. The disease-free is globally asymptotically

stable when a certain threshold is less than unity, while the endemic equilib-

rium is globally asymptotically stable for otherwise. The media/psychological

impact although does not aﬀect the epidemic threshold, delays the epidemic

peak and results in a lower size of outbreak (or equilibrium level of infected

individuals).

1. Introduction. During the outbreak of inﬂuenza A (H1N1) in 2009, media cov-

erage plays an important role in helping both the government authority make inter-

ventions to contain the disease and people response to the disease [14]. Individuals’

reactions range from avoiding social contact with infected individuals to wearing

protective masks, or vaccination. Human behavior change consequently leads to

reduction in number of real susceptible individuals or eﬀective contact rates. The

mass media have been used as a way of delivering preventive health messages due

to their potential inﬂuence on people’s behavior [7]. To curb the spread of infec-

tious diseases it is then crucial to examine the role of media coverage on disease

outbreaks.

Although, the precise functioning of media coverage of epidemics is not well

understood, a number of mathematical models has been formulated to describe

2010 Mathematics Subject Classiﬁcation. 92C50, 92C60, 54B20.

Key words and phrases. Media coverage, SIR epidemic model, outbreak, epidemic threshold.

The authors are supported by the National Mega-project of Science Research No.

2012ZX10001-001, by the National Natural Science Foundation of China (NSFC, 11171268(YX),

11171199(ST)), by the Fundamental Research Funds for the Central Universities (08143042 (YX),

GK 201003001 (ST)).

445

446 YANNI XIAO, TINGTING ZHAO AND SANYI TANG

the impact of media coverage on the transmission dynamics of infectious diseases

[19, 11, 10, 24, 25]. To model the reduction in contacts due to behavioral change,

as awareness of the presence of the disease, functions of the number of infectives

are formulated to response to the reported information of cases [28]. The negative

exponential function βe

−mI

or βe

−α

E−α

I−α

, where H denotes hospitalized in-

dividuals, has been embedded into the transmission term by Cui et al. [10] and Liu

et al. [19] to examine the impact of media on the spreading of the disease. While

Cui et al. [11], Tchuenche et al. [24] and Sun et al. [23] incorporated non-linear

function of the number of infectives, c(I) = c

− c

f(I), in their transmission term

to investigate the eﬀects of media coverage on the transmission dynamics. Liu et

al.[19] obtained the potential for multiple outbreaks and sustained oscillations of

emerging infectious diseases, and Cui et al. [10] concluded that multiple positive

equilibria are possible when the media eﬀect is suﬃciently strong.

However, these changes of individual avoidance and contact patterns, as aware-

ness of the presence of the disease, are response to not only the reported information

of cases but also the changing rate in case number. During the early stage of out-

break of an emergent infectious disease like A/H1N1 inﬂuenza and SARS [18], the

reported high growth rate of cases puts great pressure on people and cause people

to stay home or wear face masks when going outside. While as disease further

spreads, the massive cases number exhibits great inﬂuence on people’s behaviors.

So, during the whole outbreak people are aware of presence of disease either in the

case number or in their changing rate or both reported by mass media.

The purpose of this paper is to formulate a particular function to describe the re-

duction factor induced either by large number of cases (I), or by signiﬁcant changes

in the number of cases (dI/dt). Awareness through media and education plays a

tremendous role in changing behaviors or contact patterns, and hence in limiting

the spread of infectious disease. Whether this media function induced by both cases

and changes in cases aﬀect the size of an epidemic outbreak or the peak time, or

bring interesting ﬁndings remains unknown, and falls within the scope of this study.

We then examine the system with incident function dependent on both cases num-

ber and their changing rate. Firstly we convert this implicitly deﬁned system into a

explicitly deﬁned system, which is actually a piecewise smooth system [4, 6, 29], by

using the property of the Lambert W function [9]. The piecewise smooth system is

then theoretically and technically analyzed. Finally the epidemiological conclusions

on eﬀect of media coverage are addressed.

2. The SIR model. We consider the dynamics of susceptibles S, infecteds I and

recovereds R. Let Λ be the (constant) recruitment rate and µ be the natural

death rate of the population. Assume that β is basic the transmission coeﬃcient,

γ is the recovery rate from infection, and α is disease-related death rate. Such

media/psychological impact, as awareness of the presence of the disease, is simply

described by an exponential decreasing factor, resulting the transmission coeﬃcient

as βe

−M(I,dI/dt)

, where

M(I, dI/dt) = max



0, p

I(t) + p

dI(t)



, (1)

where p

, p

are non-negative parameters to measure the eﬀect of media/psychologic-

al impact of media reported cases and changing rate. Then we have the model

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