Analysis of an improved epidemic model with stochastic disease transmission

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Abstract

This paper proposes an improved logistic epidemic model and carries out the complete parameters analysis of asymptotic behavior of infectious diseases. Some interesting details such as the threshold value of outbreak of epidemics and critical states of disease spread are derived. The mean and variance of proportion of infected population are given explicitly. The results show that our model is more reasonable and applicable to describe the real situation. Especially, P=12 might be considered as the alarm for relate institutions to make effective policies to prevent and control some epidemics.

Introduction

In the real world there are various kinds of infectious diseases which play important roles in not only threatening people’s lives but also leading to catastrophic economic consequences. Lots of examples abound in our world, for example, plague in the 14th century claimed about 25 million people’s lives, one in four Europeans at the time; HIV, in fact, is causing more death than plague. Recently, the outbreak of foot and mouth disease in Britain brought them substantial loss in economics.

In order to suppress the outbreak of such infectious diseases and reduce the loss brought by those epidemics, many authors payed attention to disease control such as vaccination which would render its recipients immune to the disease (see e.g. [1], [2]). Many deterministic infectious diseases models have been proposed and investigated (see e.g. [3], [4], [5]). However, it is obvious that there are many factors which would produce some uncertain effects on epidemic models. Therefore many authors applied stochastic models to study the behavior of infectious diseases in real world (see e.g. [6], [7], [8], [9], [10]).

In this paper we shall explore our improved model which incorporates several interesting feathers of asymptotic behavior of a disease. One of those concerns is the threshold value of the outbreak of epidemics. As a matter of fact, P=12+α2B might be considered as the threshold value; however, P=12 is much more meaningful when we should make some practical and effective polices to deal with epidemics in real life. The second consists of some critical state such as (1-2p)B+α=0 and (1-2p)B+α<0, which would lead to a catastrophic outbreak of epidemics in finite or infinite time. Moreover, we also compare the improved model with the original one and find that the improved model is more realistic and applicable.

Section snippets

Description of the model

Here we consider a fatal disease without recovery period like HIV. The model we investigated is similar to that of Roberts and Saha [11] but has some significant modifications to make it more realistic and adaptable. According to [11], N is the density of population in per unit area, and its natural density of birth rate and death rate are B(N) and D(N), respectively. In addition, they also assumed that B(N)0,D(N)0, B(0)>D(0) and B(N)=D(N)B(N)D(N). The first two conditions are

Asymptotic behavior analysis

To deal with Eq. (6), it would be better to apply the It oˆ solution of it on the ground that this problem arises from population biology (see [12]). The Fokker–Planck equation of Eq. (6) is (see [13])P(Z,t)t=-Z(F(Z)P(Z,t))+122Z2((G(Z))2P(Z,t)),where P(Z, t) is the probability density function of Z. When the state of the proportion of infected and infectious population Z(t) arrives its steady state, P(Z), we getd2dZ2((G(Z))2P(Z))-2ddZ(F(Z)P(Z))=0.Solving (8) leads toP(Z)=C0Z1-Z2r/ρ2exp-

The mean and variance

Making use of (8) givesddZ[G2(Z)P(Z)]-2[F(Z)P(Z)]=0.According to (6), (13), one can see that01F(Z)P(Z)dZ=0.On the other hand, multiplying (10) by Z-1 and then integrating, we can derive that201Z-1F(Z)P(Z)dZ-01Z-2G2(Z)P(Z)dZ=0.Hence, according to (14), (15) one can observe thatμ=EZ=01ZP(Z)dZ=β0C(2rC2-ρ2C2)ρ2C4r+2β0C(β0C-ρ2C2).At the same time, we can see thatEZ2=C2rβ0Cμ.Using (16), (17) results inδ2=C2rβ0Cμ-μ2,where δ2 stands for the variance of Z(t).

Comparisons and discussions

In this paper, a famous stochastic infectious disease model was improved. The complete parameters analysis of asymptotic behavior of infectious diseases was carries out. The results showed that the threshold value of disease outbreak is P=12+α2B. However, it is more convenient and effective to control and make prevention policies referring to P=12. In addition, infectious diseases would be explosive in finite or infinite time in some special states. By comparing our main results with those in

Acknowledgments

The authors thank the editor and referee for their very important and helpful comments and suggestions. The authors also thank the NSFC of PR China (Nos. 11126219, 11171081 and 11171056), the Postdoctoral Science Foundation of China (Grant No. 20100481339), Shandong Provincial Natural Science Foundation of China (Grant No. ZR2011AM004), the NSFC of Shandong Province (No. ZR2010AQ021), the Key Project of Science and Technology of Weihai (No. 2010-3-96) and the Natural Scientific Research

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