The effect of using different types of periodic contact rate on the behaviour of infectious diseases: A simulation study
Introduction
Many epidemiological models have been studied using computer simulations to examine the effect of a seasonally varying contact rate on the behaviour of the disease. Most of these models performed computer simulations using the sinusoidal function of period one year for the seasonally varying contact rate. Examples of such studies include [1], [2], [3], [4], [5], [6]. Also most of these simulations have been done for parameter values which correspond to measles, as measles is usually a notifiable disease and is therefore one of the diseases for which the data is best.
Dietz [1] showed numerically that there is a biennial periodic solution for an SIRS model with vital dynamics when the contact rate is a sinusoidally varying periodic function of period one year. Aron and Schwartz [3] studied an SEIR deterministic model for measles with the same contact rate and they found that coexisting periodic solutions of period one, two and three years are possible and increasing , the amplitude of the seasonal variation in the contact rate generates a sequence of period doubling bifurcations. They showed that the appearance of period doubling also depends on the value of the basic reproduction number. Schwartz [6] studied an SEIRS model and showed that periodic outbreaks are demonstrated for single population models, but chaotic cases can also occur for these models. Grossman et al. [2] studied an SEIR model and showed that there exists a biennial periodic solution for this model with the same sinusoidal contact rate.
This paper aims to broaden the scope of simulation from previous studies to cover diseases other than measles, and other types of periodic functional forms for the transmission rate than a sinusoidal transmission rate. We look at parameter values corresponding to the childhood diseases of measles, mumps, chickenpox and rubella. The parameter values which we use in our simulation are taken from [3], [7], [8], [9]. We concentrate on the following three particular forms for the disease transmission function:
- 1.
A sinusoidal function of period one year.
- 2.
A step function of period one year, where and is the largest integer less than or equal to
- 3.
A linear combination of the above sinusoidal and step functions .
The basic idea is that for a given set of parameter values we obtain the endemic equilibrium solution by running the system for a long time to eliminate transient solutions. We then section the equilibrium solutions by looking at Poincaré sections of them taken every year (recall that the underlying seasonal variation in the contact rate has period one year). By plotting the sections of the long term endemic equilibrium solutions against the amplitude we obtain a number of points in a vertical line corresponding to each value chosen for the amplitude parameter . These points represent the period of the stable long term periodic solution of our model. So this simulation study plans to solve our models numerically using a variety of particular periodic functional forms of the transmission rate. We also plan to simulate the model using different values of the disease parameters.
Section snippets
The model
We have considered a more complicated and realistic SEIR model and investigated the effect of the seasonal variation in the contact rate using this type of model which takes account of the disease latency. It has been shown that with a non-constant transmission rate our model has one and only one equilibrium point which represents the disease free equilibrium (DFE) of our SEIR model [10]. In this paper, simulation results have been conducted for parameter values which insure that the basic
Simulation results
In this section we shall study the global bifurcation diagrams of the SEIR model plotted from our simulations. These diagrams are presented for our three different suggested periodic functions, the sinusoidal contact rate, a suggested reparameterised step periodic contact rate (reparameterisation of the binary step function studied in [16]) and the linear combination of this new step function and the sinusoidal one. Comparing the results for each one of the three different contact rate with the
Summary and conclusion
The computer simulations are conducted in this paper for different infectious diseases including measles. Measles is the disease investigated most often as the data available for measles are very good. However our simulation results have been extended to other forms of seasonal variation in the contact rate. For the other diseases the simulation results are completely original even with the sinusoidal contact rate. Our simulations show numerically the existence of endemic periodic solutions
Acknowledgements
I would like to thank Dr. David Greenhalgh for his support and acknowledge his help in writing these results.
Islam A. Abdel-Moneim was born in Kalub, Egypt, in 1968. He received the M.Sc. degree from the University of Benha, Egypt, in 1995 and a Ph.D. in 2001 from University of Strathclyed, UK. He is Assistant Professor in the Department of Basic and Applied Science, Community College Al-majma’a, King Saud University, Al-majma’h 11952, Saudi Arabia temporarily. Permanently he is Assistant Professor at University of Benha, Faculty of Science, Department of Mathematics. His research interests focus on
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Islam A. Abdel-Moneim was born in Kalub, Egypt, in 1968. He received the M.Sc. degree from the University of Benha, Egypt, in 1995 and a Ph.D. in 2001 from University of Strathclyed, UK. He is Assistant Professor in the Department of Basic and Applied Science, Community College Al-majma’a, King Saud University, Al-majma’h 11952, Saudi Arabia temporarily. Permanently he is Assistant Professor at University of Benha, Faculty of Science, Department of Mathematics. His research interests focus on Modelling and Simulation in Biology and Medicine specially in infectious diseases.