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Mathematical analysis of complex SIR model with coinfection and density dependence
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-8422-6140
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Biology. Linköping University, Faculty of Science & Engineering.
2019 (English)In: Computational and Mathematical Methods, ISSN 2577-7408, Vol. 1, no 4Article in journal (Refereed) Published
Abstract [en]

An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed, and several threshold conditions are obtained, which describes the establishment of diseases in the population. We prove that, for small carrying capacity K, there exists a globally stable disease-free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, that is, we prove that, (1) for small values of K, there exists a unique globally stable equilibrium point, and (b) it moves continuously as K is growing (while its face type may change). This indicates that the carrying capacity is the crucial parameter and an increase in resources in terms of carrying capacity promotes the risk of infection.

Place, publisher, year, edition, pages
Wiley-Blackwell Publishing Inc., 2019. Vol. 1, no 4
Keywords [en]
carrying capacity, coinfection, global stability, SIR model
National Category
Mathematics Immunology
Identifiers
URN: urn:nbn:se:liu:diva-160284DOI: 10.1002/cmm4.1042OAI: oai:DiVA.org:liu-160284DiVA, id: diva2:1351901
Conference
2019/09/17
Available from: 2019-09-17 Created: 2019-09-17 Last updated: 2019-10-16Bibliographically approved
In thesis
1. Dynamics of Coinfection: Complexity and Implications
Open this publication in new window or tab >>Dynamics of Coinfection: Complexity and Implications
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Living beings are always on risk from multiple infectious agents in individual or in groups. Though multiple pathogens' interactions have widely been studied in epidemiology. Despite being well known, the co-existence of these pathogens and their coinfection remained a mystery to be uncovered. Coinfection is one of the important and interesting phenomenon in multiple interactions when two infectious agents coexist at a time in a host. The aim of this thesis is to understand the complete dynamics of coinfection and the role of different factors affecting these interactions.

Mathematical modelling is one of the tools to study the coinfection dynamics. Each model has its own limitations and choice of the model depends on the questions to be addressed. There is always a crosstalk between the choice of model and limitation of their solvability. The complexity of the problem defines the restriction in analytical possibilities.

In this thesis we formulate and analyse the mathematical models of coinfection with different level of complexities. Since viral infections are a major class of infectious diseases, in the first three papers we formulated a susceptible, infected, recovered (SIR) model for coinfection of the two viral strains in a single host population introducing carrying capacity as limited growth factor in susceptible class. In the first study, we made some assumptions for the transmission of coinfection in the model. In the following papers, the analysis is expanded by relaxing these assumptions which has generated the complexity in dynamics. We showed that the dynamics of stable equilibrium points depends on the fundamental parameters including carrying capacity K. A parameter dependent transition dynamics exists starting from disease free state to a level where coinfection can persists only with susceptible class. A disease-free equilibrium point is stable only when K is small. With increase in carrying capacity to a level where only single infection can invade and persists. Further increase in carrying capacity becomes large enough for the existence and persistence of coinfection due to the high density of susceptible class. In paper I, we proved the existence of a globally stable equilibrium point for any set of parameter values, revealing persistence of disease in a population. This shows a close relationship between the intensity of infection and carrying capacity as a crucial parameter of the population. So there is always a positive correlation between risk of infection and carrying capacity which leads to destabilization of the population.

In paper IV, we formulated mathematical models using different assumptions and multiple level of complexities to capture the effect of additional phenomena such as partial cross immunity, density dependence in each class and a role of recovered population in the dynamics. We found the basic reproduction number for each model which is the threshold that describes the invasion of disease in population. The basic reproduction number in each model shows that the persistence of disease or strains depends on the carrying capacity K. In the first model of this paper, we have also shown the local stability analysis of the boundary equilibrium points and showed that the recovered population is not uniformly bounded with respect to K.

Paper V uses simulations to analyse the dynamics and specifically studies how temporal variation in the carrying capacity of the population affects its dynamics. The degree of autocorrelation in variability of carrying capacity influences whether the different classes exhibit temporal variation or not. The fact that the different classes respond differently to the variation depends in itself on whether their equilibrium densities show a dependence on the carrying capacity or not. An important result is that at high autocorrelation, the healthy part of the population is not affected by the external variation and at the same time the infected part of the population exhibits high variation. A transition to lower autocorrelation, more randomness, means that the healthy population varies over time and the size of the infected population decreases in variation.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2019. p. 30
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 2011
National Category
Mathematical Analysis
Identifiers
urn:nbn:se:liu:diva-160932 (URN)10.3384/diss.diva-160932 (DOI)9789175190167 (ISBN)
Public defence
2019-11-07, Nobel BL32, B Building, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2019-10-15 Created: 2019-10-15 Last updated: 2019-10-15Bibliographically approved

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