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Browsing by study line "Biomathematics"

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  • Rantala, Frans (2023)
    Cancer consists of heterogeneous cell populations that repeatedly undergo natural selection. These cell populations contest with each other for space and nutrients and try to generate phenotypes that maximize their ecological fitness. For achieving this, they evolve evolutionarily stable strategies. When an oncologist starts to treat cancer, another game emerges. While affected by the cellular evolution processes, modeling of this game owes to the results of the classical game theory. This thesis investigates the theoretical foundations of adaptive cancer treatment. It draws from two game theoretical approaches, evolutionary game theory and Stackelberg leader-follower game. The underlying hypothesis of adaptive regimen is that the patient's cancer burden can be administered by leveraging the resource competition between treatment-sensitive and treatment-resistant cells. The intercellular competition is mathematically modelled as an evolutionary game using the G function approach. The properties of the evolutionary stability, such as ESS, the ESS maximum principle, and convergence stability, that are relevant to tumorigenesis and intra-tumoral dynamics, are elaborated. To mitigate the patient's cancer burden, it is necessary to find an optimal modulation and frequency of treatment doses. The Stackelberg leader-follower game, adopted from the economic studies of duopoly, provides a promising framework to model the interplay between a rationally playing oncologist as a leader and the evolutionary evolving tumor as a follower. The two game types applied simultaneously to cancer therapy strategisizing can nourish each other and improve the planning of adaptive regimen. Hence, the characteristics of the Stackelberg game are mathematically studied and a preliminary dose-optimization function is presented. The applicability of the combination of the two games in the planning of cancer therapy strategies is tested with a theoretical case. The results are critically discussed from three perspectives: the biological veracity of the eco-evolutionary model, the applicability of the Stackelberg game, and the clinical relevance of the combination. The current limitations of the model are considered to invite further research on the subject.
  • Siurua, Joel (2023)
    Contacts between individuals play a central part in infectious disease modelling. Social or physical contacts are often determined through surveys. These types of contacts may not accurately represent the truly infectious contacts due to demographic differences in susceptibility and infectivity. In addition, surveyed data is prone to statistical biases and errors. For these reasons, a transmission model based on surveyed contact data may make predictions that are in conflict with real-life observations. The surveyed contact structure must be adjusted to improve the model and produce reliable predictions. The adjustment can be done in multiple different ways. We present five adjustment methods and study how the choice of method impacts a model’s predictions about vaccine effectiveness. The population is stratified into n groups. All five adjustment methods transform the surveyed contact matrix such that its normalised leading eigenvector (the model-predicted stable distribution of infections) matches the observed distribution of infections. The eigenvector method directly adjusts the leading eigenvector. It changes contacts antisymmetrically: if contacts from group i to group j increase, then contacts from j to i decrease, and vice versa. The susceptibility method adjusts the group-specific susceptibility of individuals. The changes in the contact matrix occur row-wise. Analogously, the infectivity method adjusts the group-specific infectivity; changes occur column-wise. The symmetric method adjusts susceptibility and infectivity in equal measure. It changes contacts symmetrically with respect to the main diagonal of the contact matrix. The parametrised weighting method uses a parameter 0 ≤ p ≤ 1 to weight the adjustment between susceptibility and infectivity. It is a generalisation of the susceptibility, infectivity and symmetric methods, which correspond to p = 0, p = 1 and p = 0.5, respectively. For demonstrative purposes, the adjustment methods were applied to a surveyed contact matrix and infection data from the COVID-19 epidemic in Finland. To measure the impact of the method on vaccination effectiveness predictions, the relative reduction of the basic reproduction number was computed for each method using Finnish COVID-19 vaccination data. We found that the eigenvector method has no impact on the relative reduction (compared to the unadjusted baseline case). As for the other methods, the predicted effectiveness of vaccination increased the more infectivity was weighted in the adjustment (that is, the larger the value of the parameter p). In conclusion, our study shows that the choice of adjustment method has an impact on model predictions, namely those about vaccination effectiveness. Thus, the choice should be considered when building infectious disease models. The susceptibility and symmetric methods seem the most natural choices in terms of contact structure. Choosing the ”optimal” method is a potential topic to explore in future research.
  • Backlund, Sofia Maria (2022)
    Coral reefs form important marine ecosystems and simultaneously are at risk of deterioration due to rapidly changing environments as a consequence of human actions. Understanding their dynamics is thus important in order to be able to protect them from being destroyed. In this thesis we construct a lattice model for two life-history strategies of corals, brooders and spawners. These two strategies differ mainly in their modes of sexual reproduction, but also differences in growth and death rates as well as competitive ability are considered. We use pair approximation to help analyse the model while keeping its spatial structure. Numerical analysis is used to find the equilibria of the system as well as their stabilities, first for a single strategy and then for the two-strategy system. We find that the two strategies are able to coexist if the spawners have a higher growth rate and higher death rate and are competitively superior to brooders. This requires some reproduction over distance and a trade-off between growth and death rates. Thus we find that brooders are focusing a bigger part of their energy on long-distance reproduction, while spawners are dominating over short distances and having a higher turnover. We also find that both mutual invasibility and coexistence in the broader sense are only possible for low rates of sexual reproduction for both strategies. For higher rates of sexual reproduction we find that whichever strategy invades the lattice first will stay and the other cannot invade. Lastly we look at the effect of a change in environmental conditions, namely the acidification and temperature increase of oceans, on the two strategies and find that it affects the two strategies differently. The spawners are quickly driven to extinction by the change in environmental conditions, while brooders initially benefit from the changing conditions and only start to suffer themselves after the spawners have gone extinct.
  • Lintula, Johannes (2023)
    This work examines how neural networks can be used to qualitatively analyze systems of differential equations depicting population dynamics. We present a novel numerical method derived from physics informed learning, capable of extracting equilibria and bifurcations from population dynamics models. The potential of the framework is showcased three different example problems, a logistic model with outside inference, the Rosenzweig-MacArthur model and one model from a recent population dynamics paper. The key idea behind the method is having a neural network learn the dynamics of a free parameter ODE system, and then using the derivatives of the neural network to find equilibria and bifurcations. We, a bit clunkily, refer to these networks as physics informed neural networks with free parameters and variable initial conditions. In addition to these examples, we also survey how and where these neural networks could be further utilized in the context of population dynamics. To answer the how, we document our experiences choosing good hyperparameters for these networks, even venturing into previously unexplored territory. For the where, we suggest potentially useful neural network frameworks to answer questions from an external survey concerning contemporary open questions in population dynamics. The research of the work is preceded by a short dive on qualitative population dynamics, where we ponder what are the problems we want to solve and what are the tools we have available for that. Special attention is paid to parameter sensitivity analysis of ordinary differential equation systems through bifurcation theory. We also provide a beginner friendly introduction to deep learning, so that the research can be understood even by someone not previously familiar with the field. The work was written, and all included contents were selected, with the goal of establishing a basis for future research.
  • Patieva, Fatima (2023)
    In this thesis, we study epidemic models such as SIR and superinfection to demonstrate the coexistence as well as the competitive exclusion of all but one strain. We show that the strain that can keep its position under the worst environmental conditions cannot be invaded by any other strain when it comes to some models with a constant death rate. Otherwise, the optimization principle does not necessarily work. Nevertheless, Ackleh and Allen proved that in the SIR model with a density-dependent mortality rate and total cross-immunity the strain with the largest basic reproduction number is the winner in competitive exclusion. However, it must be taken into account that the conditions on the parameters used for the proof are sufficient but not necessary to exclude the coexistence of different pathogen strains. We show that the method can be applied to both density-dependent and frequency-dependent transmission incidence. In the latter half, we link the between and within-host models and expand the nested model to allow for superinfection. The introduction of the basic notions of adaptive dynamics contributes to simplifying our task of demonstrating the evolutionary branching leading to diverging dimorphism. The precise conclusions about the outcome of evolution will depend on the host demography as well as on the class of superinfection and the shape of transmission functions.
  • Viitikko, Tanja (2023)
    Pathogens are everywhere in nature, so organisms have developed various defense mechanisms in order to defend themselves against the pathogens. Two of the defense mechanisms are known as resistance and tolerance. Resistance describes the host's ability to avoid being infected by the pathogen, while tolerance describes the host's ability to reduce the fitness loss caused by the infection. We assume that investing into resistance reduces the transmission rate of the pathogens and investing into tolerance reduces the host's virulence. Developing the defense mechanisms is costly to the host. In this thesis, we assume that the resources invested into resistance and tolerance are taken away from the host's fecundity. The independent but simultaneous evolution of resistance and tolerance is modeled with an SIS model. The model is studied with the methods of adaptive dynamics. We concentrate on finding continuously stable strategies, which serve as the evolutionary end points for the population. We study the varying ecological parameters to determine which strategies are optimal for the host in different environments. We find that for low values of transmission rate, the hosts favor resistance over tolerance. When the transmission rate increases, resistance is traded for tolerance and the host benefits more from high tolerance. Low values of virulence result in tolerance being favored over resistance. Increasing virulence leads to a change in the defense mechanism as for high values of virulence investing into resistance is more beneficial to the host. The same holds for recovery rate, as tolerance is favored for low values of recovery rate and changed for resistance when the recovery rate increases. Patterns and associations between resistance and tolerance are also studied. Positive correlation between resistance and tolerance is found with low values of transmission rate, low and high values of virulence and high values of recovery rate. Resistance and tolerance correlate negatively with high values of transmission rate, intermediate values of virulence and low values of recovery rate.