TURING PATTERN FORMATIONS ARISING FROM A SPATIAL EPIDEMIC MODEL HAVING CROSS-DIFFUSION TERMS IN THE SUSCEPTIBLE AND INFECTED POPULATIONS

• Main Author: Triska, Anita
• Format: Dissertations
• Language: Indonesia
• Online Access: https://digilib.itb.ac.id/gdl/view/64993
• Institution: Institut Teknologi Bandung

One form of the spatial epidemic model is through a system of reaction-diffusion. There are two types of diffusion terms, namely self-diffusion and cross-diffusion. The self-diffusion term represents the natural movement of each individual as the basis of the spatial aspect, while cross-diffusion states the movement of individuals caused by other group. Spatial dependence becomes very important to consider since every individual moves to carry out their daily activities so that it expands the interaction space. Besides self-diffusion term, the model in this study also involves cross-diffusion, not only cross-diffusion of the susceptible but also of the infected. The cross-diffusion of the susceptible expresses their movement to areas with a lower density of the infected to represents the tendency of the susceptible to stay away from the infected. However, cross-diffusion of the infected in this study implies a tendency of the infected to move to areas with a higher density of the susceptible, such as for work, school, or urbanization. Therefore, the movement of the infected to more densely populated areas of the susceptible is interesting to be studied scientifically in discussing of the spread of a disease, especially in postpandemic situations. The spatial epidemic model in this study are reviewed through two aspects, namely analytically and numerically. Analytic studies are carried out by Turing bifurcation analysis which leads to pattern formations in a spatial domain. The patterns which are also known as Turing pattern are formed as a result of the instability which is known as Turing instability. The patterns can provide an overview of the dynamics of the spread of an infectious disease spatially. Through the analysis of the Turing bifurcation, the conditions for the occurrence of Turing instability are obtained. Next, the amplitude equation is determined with the help of multiple-scale analysis to predict the patterns that appear near to the Turing bifurcation point. Moreover, the stability analysis of the amplitude equation is carried out to determine the stability properties of the predicted patterns. In addition to this analytic study, numerical simulations are also carried out to validate the predicted patterns by the amplitude equation near to the Turing bifurcation point. Simulations are also carried out to obtain an overview of the patterns formed when the bifurcation parameter are far from its bifurcation point. The results of an intensive numerical simulation show that there are five types of patterns of the model, such as the spots, spots-stripes, stripes, stripes-holes, and holes. From an epidemiological point of view, the holes indicate the situation of a disease outbreak occurs in a region, while the spots indicate that the outbreak only occurs in certain areas. Furthermore, numerical simulations are carried out by varying the cross-diffusion coefficient of the susceptible and infected. The simulation results show that if the cross-diffusion coefficient of the infected is much bigger than the cross-diffusion coefficient of the susceptible then the patterns formed lead to the holes. It means that if the infected is free to move to an area with high density of the susceptible then it may trigger an outbreak in a region. The results of this study indicate that the movement of the infected has an important role in the spread of an infectious disease that may cause the next wave of pandemic. This study tries to fill the gaps in the discussion about the movement of infected people to areas with high density of the susceptible so it can be the basis for decision-making in dealing with the situations during and after the pandemic.