In this paper we develop a compartmental model of SIR type (the abbreviation refers to the number of Susceptible, Infected and Recovered people) that models the population dynamics of two diseases that can coinfect. We discuss how the underlying dynamics depends on the carrying capacity K: from a simple dynamics to a more complex. This can also help in understanding the appearance of more complicated dynamics, for example, chaos and periodic oscillations, for large values of K. It is also presented that pathogens can invade in population and their invasion depends on the carrying capacity K which shows that the progression of disease in population depends on carrying capacity. More specifically, we establish all possible scenarios (the so-called transition diagrams) describing an evolution of an (always unique) locally stable equilibrium state (with only non-negative compartments) for fixed fundamental parameters (density independent transmission and vital rates) as a function of the carrying capacity K. An important implication of our results is the following important observation. Note that one can regard the value of K as the natural ‘size’ (the capacity) of a habitat. From this point of view, an isolation of individuals (the strategy which showed its efficiency for COVID-19 in various countries) into smaller resp. larger groups can be modelled by smaller resp. bigger values of K. Then we conclude that the infection dynamics becomes more complex for larger groups, as it fairly maybe expected for values of the reproduction number R0≈1. We show even more, that for the values R0>1 there are several (in fact four different) distinguished scenarios where the infection complexity (the number of nonzero infected classes) arises with growing K. Our approach is based on a bifurcation analysis which allows to generalize considerably the previous Lotka-Volterra model considered previously in Ghersheen et al. (Math Meth Appl Sci 42(8), 2019).

In this paper we continue the stability analysis of the model for coinfection with density dependent susceptible population introduced in Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We consider the remaining parameter values left out from Andersson et al. (Effect of density dependence on coinfection dynamics. arXiv:2008.09987, 2020). We look for coexistence equilibrium points, their stability and dependence on the carrying capacity K. Two sets of parameter value are determined, each giving rise to different scenarios for the equilibrium branch parametrized by K. In both scenarios the branch includes coexistence points implying that both coinfection and single infection of both diseases can exist together in a stable state. There are no simple explicit expression for these equilibrium points and we will require a more delicate analysis of these points with a new bifurcation technique adapted to such epidemic related problems. The first scenario is described by the branch of stable equilibrium points which includes a continuum of coexistence points starting at a bifurcation equilibrium point with zero single infection strain #1 and finishing at another bifurcation point with zero single infection strain #2. In the second scenario the branch also includes a section of coexistence equilibrium points with the same type of starting point but the branch stays inside the positive cone after this. The coexistence equilibrium points are stable at the start of the section. It stays stable as long as the product of K and the rate γ¯γ¯ of coinfection resulting from two single infections is small but, after this it can reach a Hopf bifurcation and periodic orbits will appear.

Multiple viruses are widely studied because of their negative effect on the health of host as well as on whole population. The dynamics of coinfection are important in this case. We formulated an susceptible infected recovered (SIR) model that describes the coinfection of the two viral strains in a single host population with an addition of limited growth of susceptible in terms of carrying capacity. The model describes five classes of a population: susceptible, infected by first virus, infected by second virus, infected by both viruses, and completely immune class. We proved that for any set of parameter values, there exists a globally stable equilibrium point. This guarantees that the disease always persists in the population with a deeper connection between the intensity of infection and carrying capacity of population. Increase in resources in terms of carrying capacity promotes the risk of infection, which may lead to destabilization of the population.

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.