Modeling Earth Systems and Environment

, Volume 5, Issue 1, pp 21–32

# Modelling and analysis of the effects of density dependent contact rates on the spread of carrier dependent infectious diseases with environmental discharges

• Shikha Singh
• Jitendra Singh
• J. B. Shukla
Original Article

## Abstract

In this paper, an SIS mathematical model is proposed and analyzed to study the effects of density dependent contact rates on the spread of carrier dependent infectious diseases by considering the role of environmental discharges on the growth of carriers. The model consists of four dependent variables, namely, the density of susceptible population, the density of infective population, the density of carrier population and the cumulative density of environmental discharges in the environment of the habitat. In the modelling process, it is assumed that the density of carrier population follows the logistic model, the intrinsic growth rate of which is linear function of the density of environmental discharges. The cumulative density of environmental discharges is assumed to be directly proportional to the number of people who remain in the habitat. The nonlinear model is analyzed by using the stability theory of differential equations and numerical simulations. The analysis shows that as the cumulative density of environmental discharges increases then not only the density of carrier population increases but the spread of carrier dependent infectious diseases also increases. It is shown further that the effect of density dependent contact rates which is linear function of emigration is to decrease the spread of the infectious diseases. The numerical simulation confirms these analytical results. The analysis shows that it is very useful to keep the household environment clean to prevent the spread of carrier dependent infectious diseases.

## Keywords

Mathematical modelling Stability Environmental discharges Carrier population Contact rates

## Notes

### Acknowledgements

The first Author (Dr. Shikha Singh) is thankful to Innovative Internet University for Research (A think tank), Kanpur India for the help and support.

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© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Shikha Singh
• 1
• Jitendra Singh
• 1
• J. B. Shukla
• 2
• 3
1. 1.Department of Mathematics, PPN PG CollegeCSJM UniversityKanpurIndia
2. 2.Innovative Internet University for Research (A Think Tank)KanpurIndia
3. 3.IIT KanpurKanpurIndia