Author Topic: A Study of Effects of Disease Caused Death in A Simple Epidemic Model  (Read 1944 times)

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Author : Dr. Sunil Kumar Singh, Dr. Shekh Aqeel
International Journal of Scientific & Engineering Research, Volume 2, Issue 5, May-2011
ISSN 2229-5518
Download Full Paper : PDF

Abstract- This paper is concerned with various effects of disease caused death on the host population in an epidemic model of SIR type. Various effects of disease caused death on the host population are studied in this epidemic model. The basic problem discussed in this paper is to be describing the spread of an infection caused death within a population. It is further assumed that there is no substantial development of immunity and that removed infectious are in effect cured of disease. The rate of natural birth and death is assumed to be balanced.

Key Word - Mathematical modeling, Population size , Birth- rate , Death- rate & Infection rate.
 
INTORDUCTION
Anderson and May [7] studied the effects of disease caused death on the population size in model for a disease which spreads through direct infection within a population whose size is allowed to very in time. Two important new phenomena a was revealed by their study.

 A threshold for the population size exists that determines whether the population can sustain an epidemic; fatal diseases are found to have a regulating effect on the growth of the population. Many subsequent works have fallowed this line of research [5]. Another characteristic of this body of research is that the emphasis is on the interplay between the net intrinsic growth rate r and the rate of disease caused death  : it r >  then the disease it likely to become endemic. To explain this phenomenon, potential mechanisms   to endemicity other then a large intrinsic growth rate r need to be studied. In recent study, we discovered that a long incubation period incorporated into a SIR model may provide an explanation for concurrence of high pathogenicity and long life span of infectiousness. We took different approach to study of epidemic models by assuming that the population a small intrinsic growth rate r so that disease caused mortality rate
 is relatively large. This approach has following advantage.
1.   If greatly reduces the technicality in mathematical analysis, One may start with the case r = 0 and then consider the case of small positive r.
2.   If enables us to isolate those effects on the population that directly   related to the disease e.g. we discovered our essential difference between a model that incorporates an incubation period and one that does not. Even in a simple model that does not contain an incubation period, this new approach leads  to the discovery of several interesting details not found in literature.

3.   By keeping the mathematical technicalities at its minimum, this approach may allow our models more accessible to field  epidemiologist and hence encourage of application of  mathematics in epidemiological studies.

In the present work we demonstrate our approach through a very simple model. We assume that the disease spreads through direct contact among the hosts, the disease has no incubation period as considered in most of previous works [2,4] and that the intrinsic growth rate of host population is zero i.e r = 0 so that in absence of disease, the population size remains constant. The mathematical analysis of model is very elementary, and it provides epidemiologically interesting details about an epidemic. Also, we demonstrate that the kind of phenomenon one many observe in the case of a small positive intrinsic growth is essentially the some as we obtained here.  In particular, this seems to suggest that, an SIR model is essentially a model for an epidemic; it does not provide an epidemiologically relevant mechanism for disease endimicity.
For other studies on epidemological models with varying population size closely related to the one we consider her, see Greenhalgh [1] and Mena- Lorca and Hethcote [6] and references there on. Other models with varying population and disease- caused death have been studies by Brauer [2], Bremerman and Thieme [3], Gao and Hethcote [4], and Hethcote [6], and Pugliese [7].

FORMULATION OF MATHEMATICAL MODEL :

The population is partitioned into classes of susceptible, infectious and immune individuals, with population N(t) is
           n(t)  = x(t) + y(t) +z (t)                  (1)
Let us consider the per capital birth rate is a constant b and all newborns are susceptible. The per capital natural death rate assumed to be b so that total population remains constant in absence of disease. Suppose the disease spreads through direct contact between susceptible and infections individuals. We assume that the   transmission coefficient per unit time by  x(t)y(t). This is equivalent to assuming that the contact rate between individual is  n(t), proportional to n (t).
   The disease is assumed to causes death to infected individuals, with a death rate constant  . Let the average infectious period for an infectious individual be   so that transfer from infectious class to immune class is at a constant rate  . It is also assumed that the disease confers permanent immunity so that no transfer from infectious class to immune class exists. Since vaccination is one of the major means of control and prevention of many viral infections, the effect of a vaccination strategy is also considered. All susceptible individuals are vaccinated at a constant per capita rate p. Based on these modeling hypotheses, the following set of differential equations is derived.

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