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The first year of the COVID-19 pandemic put considerable strain on healthcare systems worldwide. In order to predict the effect of the local epidemic on hospital capacity in England, we used a variety of data streams to inform the construction and parameterisation of a hospital progression model, EpiBeds, which was coupled to a model of the generalised epidemic. In this model, individuals progress through different pathways (e.g. may recover, die, or progress to intensive care and recover or die) and data from a partially complete patient-pathway line-list was used to provide initial estimates of the mean duration that individuals spend in the different hospital compartments. We then fitted EpiBeds using complete data on hospital occupancy and hospital deaths, enabling estimation of the proportion of individuals that follow the different clinical pathways, the reproduction number of the generalised epidemic, and to make short-term predictions of hospital bed demand. The construction of EpiBeds makes it straightforward to adapt to different patient pathways and settings beyond England. As part of the UK response to the pandemic, EpiBeds provided weekly forecasts to the NHS for hospital bed occupancy and admissions in England, Wales, Scotland, and Northern Ireland at national and regional scales.

In this work, we have investigated the evolutionary dynamics of a generalist pathogen, e.g., a virus population, that evolves toward specialization in an environment with multiple host types. We have particularly explored under which conditions generalist viral strains may rise in frequency and coexist with specialist strains or even dominate the population. By means of a nonlinear mathematical model and bifurcation analysis, we have determined the theoretical conditions for stability of nine identified equilibria and provided biological interpretation in terms of the infection rates for the viral specialist and generalist strains. By means of a stability diagram, we identified stable fixed points and stable periodic orbits, as well as regions of bistability. For arbitrary biologically feasible initial population sizes, the probability of evolving toward stable solutions is obtained for each point of the analyzed parameter space. This probability map shows combinations of infection rates of the generalist and specialist strains that might lead to equal chances for each type becoming the dominant strategy. Furthermore, we have identified infection rates for which the model predicts the onset of chaotic dynamics. Several degenerate Bogdanov–Takens and zero-Hopf bifurcations are detected along with generalized Hopf and zero-Hopf bifurcations. This manuscript provides additional insights into the dynamical complexity of host–pathogen evolution toward different infection strategies.

Altres ajuts: CERCA Programme/Generalitat de Catalunya, We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov-Takens and zero- Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

L'aparició de nous virus causants de malalties està estretament lligada a l'especialització de subpoblacions virals cap a nous tipus d'amfitrions. La modelització matemàtica proporciona un marc quantitatiu que pot ajudar amb la predicció de processos a llarg termini com pot ser l'especialització. A causa de la naturalesa complexa que presenten les interaccions intra i interespecífiques en els processos evolutius, cal aplicar eines matemàtiques complexes, com ara l'anàlisi de bifurcacions, al estudiar dinàmiques de població. Aquesta tesi desenvolupa una jerarquia de models de població per poder comprendre l'aparició i les dinàmiques d'especialització, i la seva dependència dels paràmetres del sistema. Utilitzant un model per a un virus de tipus salvatge i un virus mutat que competeixen pel mateix amfitrió, es determinen les condicions per a la supervivència únicament de la subpoblació mutant, juntament amb la seva coexistència amb el cep de tipus salvatge. Els diagrames d'estabilitat que representen regions de dinàmiques diferenciades es construeixen en termes de taxa d'infecció, virulència i taxa de mutació; els diagrames s'expliquen en base a les característiques biològiques de les subpoblacions. Per a paràmetres variables, s'observa i es descriu el fenomen d'intersecció i intercanvi d'estabilitat entre diferents solucions sistemàtiques i periòdiques en l'àmbit dels ceps de tipus salvatge i els ceps mutants en competència directa. En el cas de que diversos tipus d'amfitrions estiguin disponibles per a ser disputats per ceps especialitzats i generalistes existeixen regions de biestabilitat, i les probabilitats d'observar cada estat es calculen com funcions de les taxes d'infecció. S'ha trobat un rar atractor caòtic i s'ha analitzat amb l'ús d'exponents de Lyapunov. Això, combinat amb els diagrames d'estabilitat, mostra que la supervivència del cep generalista en un entorn estable és un fet improbable. A més, s'estudia el cas dels diversos ceps N»1 que competeixen pe, La aparición de nuevos virus causantes de enfermedades está estrechamente ligada a la especialización de las subpoblaciones virales hacia nuevos tipos de anfitriones. La modelizaci ón matemática proporciona un marco cuantitativo que puede ayudar a la predicción de procesos a largo plazo como la especialización. Debido a la naturaleza compleja que presentan las interacciones intra e interespecíficas en los procesos evolutivos, aplicar herramientas matemáticas complejas, tales como el análisis de bifurcación, al estudiar dinámicas de población. Esta tesis desarrolla una jerarquía de modelos de población para poder comprender la aparición y las dinámicas de especialización, y su dependencia de los parámetros del sistema. Utilizando un modelo para un virus de tipo salvaje y un virus mutado que compiten por el mismo anfitrión, se determinan las condiciones para la supervivencia únicamente de la subpoblación mutante, junto con su coexistencia con la cepa de tipo salvaje. Los diagramas de estabilidad que representan regiones de dinámicas diferenciadas se construyen en términos de tasa de infección, virulencia y tasa de mutación; los diagramas se explican en base a las características biológicas de las subpoblaciones. Para parámetros variables, se observa y se describe el fenómeno de intersección e intercambio de estabilidad entre diferentes soluciones sistemáticas y periódicas en el ámbito de las cepas de tipo salvaje y las cepas mutantes en competencia directa. En el caso de que varios tipos de anfitriones estén disponibles para ser disputados por cepas especializadas y generalistas existen regiones de biestabilidad, y las probabilidades de observar cada estado se calculan como funciones de las tasas de infección. Se ha encontrado un raro atractor caótico y se ha analizado con el uso de exponentes de Lyapunov. Esto, combinado con los diagramas de estabilidad, muestra que la supervivencia de la cepa generalista en un entorno estable es un hecho improbable. Además, se estud, The occurrence of new disease-causing viruses is tightly linked to the specialisation of viral sub-populations towards new host types. Mathematical modelling provides a quantitative framework that can aid with the prediction of long-term processes such as specialisation. Due to the complex nature of intra- and interspecific interactions present in evolutionary processes, elaborate mathematical tools such as bifurcation analysis must be employed while studying population dynamics. In this thesis, a hierarchy of population models is developed to understand the onset and dynamics of specialisation and their dependence on the parameters of the system. Using a model for a wild-type and mutant virus that compete for the same host, conditions for the survival of only the mutant subpopulation, along with its coexistence with the wild-type strain, are determined. Stability diagrams that depict regions of distinct dynamics are constructed in terms of infection rates, virulence and the mutation rate; the diagrams are explained in terms of the biological characteristics of the sub-populations. For varying parameters, the phenomenon of intersection and exchange of stability between different periodic solutions of the system is observed and described in the scope of the competing wild-type and mutant strains. In the case of several types of hosts being available for competing specialist and generalist strains, regions of bistability exist, and the probabilities of observing each state are calculated as functions of the infection rates. A strange chaotic attractor is discovered and analysed with the use of Lyapunov exponents. This, combined with the stability diagrams, shows that the survival of the generalist in a stable environment is an unlikely event. Furthermore, the case of N=1 different strains competing for different types of host cells is studied. For this case, a counterintuitive and non-monotonic dependence of the specialisation time on the burst size and mutation rate

We investigate the dynamics of a wild-type viral strain which generates mutant strains differing in phenotypic properties for infectivity, virulence and mutation rates. We study, by means of a mathematical model and bifurcation analysis, conditions under which the wild-type and mutant viruses, which compete for the same host cells, can coexist. The coexistence conditions are formulated in terms of the basic reproductive numbers of the strains, a maximum value of the mutation rate and the virulence of the pathogens. The analysis reveals that parameter space can be divided into five regions, each with distinct dynamics, that are organized around degenerate Bogdanov–Takens and zero-Hopf bifurcations, the latter of which gives rise to a curve of transcritical bifurcations of periodic orbits. These results provide new insights into the conditions by which viral populations may contain multiple coexisting strains in a stable manner.

[EN]: The occurrence of new disease-causing viruses is tightly linked to the specialisation of viral sub-populations towards new host types. Mathematical modelling provides a quantitative framework that can aid with the prediction of long-term processes such as specialisation. Due to the complex nature of intra- and interspecific interactions present in evolutionary processes, elaborate mathematical tools such as bifurcation analysis must be employed while studying population dynamics. In this thesis, a hierarchy of population models is developed to understand the onset and dynamics of specialisation and their dependence on the parameters of the system. Using a model for a wild-type and mutant virus that compete for the same host, conditions for the survival of only the mutant subpopulation, along with its coexistence with the wild-type strain, are determined. Stability diagrams that depict regions of distinct dynamics are constructed in terms of infection rates, virulence and the mutation rate; the diagrams are explained in terms of the biological characteristics of the sub-populations. For varying parameters, the phenomenon of intersection and exchange of stability between different periodic solutions of the system is observed and described in the scope of the competing wild-type and mutant strains. In the case of several types of hosts being available for competing specialist and generalist strains, regions of bistability exist, and the probabilities of observing each state are calculated as functions of the infection rates. A strange chaotic attractor is discovered and analysed with the use of Lyapunov exponents. This, combined with the stability diagrams, shows that the survival of the generalist in a stable environment is an unlikely event. Furthermore, the case of N >> 1 different strains competing for different types of host cells is studied. For this case, a counterintuitive and non-monotonic dependence of the specialisation time on the burst size and mutat, [CAT]: L'aparició de nous virus causants de malalties està estretament lligada a l'especialització de subpoblacions virals cap a nous tipus d'amfitrions. La modelització matemàtica proporciona un marc quantitatiu que pot ajudar amb la predicció de processos a llarg termini com pot ser l'especialització. A causa de la naturalesa complexa que presenten les interaccions intra i interespecífiques en els processos evolutius, cal aplicar eines matemàtiques complexes, com ara l'anàlisi de bifurcacions, al estudiar dinàmiques de població. Aquesta tesi desenvolupa una jerarquia de models de població per poder comprendre l'aparició i les dinàmiques d'especialització, i la seva dependència dels paràmetres del sistema. Utilitzant un model per a un virus de tipus salvatge i un virus mutat que competeixen pel mateix amfitrió, es determinen les condicions per a la supervivència únicament de la subpoblació mutant, juntament amb la seva coexistència amb el cep de tipus salvatge. Els diagrames d'estabilitat que representen regions de dinàmiques diferenciades es construeixen en termes de taxa d'infecció, virulència i taxa de mutació; els diagrames s'expliquen en base a les característiques biològiques de les subpoblacions. Per a paràmetres variables, s'observa i es descriu el fenomen d'intersecció i intercanvi d'estabilitat entre diferents solucions sistemàtiques i periòdiques en l'àmbit dels ceps de tipus salvatge i els ceps mutants en competència directa. En el cas de que diversos tipus d'amfitrions estiguin disponibles per a ser disputats per ceps especialitzats i generalistes existeixen regions de biestabilitat, i les probabilitats d'observar cada estat es calculen com funcions de les taxes d'infecció. S'ha trobat un rar atractor caòtic i s'ha analitzat amb l'ús d'exponents de Lyapunov. Això, combinat amb els diagrames d'estabilitat, mostra que la supervivència del cep generalista en un entorn estable és un fet improbable. A més, s'estudia el cas dels diversos ceps N >> 1 que comp, [ES]: La aparición de nuevos virus causantes de enfermedades está estrechamente ligada a la especialización de las subpoblaciones virales hacia nuevos tipos de anfitriones. La modelización matemática proporciona un marco cuantitativo que puede ayudar a la predicción de procesos a largo plazo como la especialización. Debido a la naturaleza compleja que presentan las interacciones intra e interespecíficas en los procesos evolutivos, aplicar herramientas matemáticas complejas, tales como el análisis de bifurcación, al estudiar dinámicas de población. Esta tesis desarrolla una jerarquía de modelos de población para poder comprender la aparición y las dinámicas de especialización, y su dependencia de los parámetros del sistema. Utilizando un modelo para un virus de tipo salvaje y un virus mutado que compiten por el mismo anfitrión, se determinan las condiciones para la supervivencia únicamente de la subpoblación mutante, junto con su coexistencia con la cepa de tipo salvaje. Los diagramas de estabilidad que representan regiones de dinámicas diferenciadas se construyen en términos de tasa de infección, virulencia y tasa de mutación; los diagramas se explican en base a las características biológicas de las subpoblaciones. Para parámetros variables, se observa y se describe el fenómeno de intersección e intercambio de estabilidad entre diferentes soluciones sistemáticas y periódicas en el ámbito de las cepas de tipo salvaje y las cepas mutantes en competencia directa. En el caso de que varios tipos de anfitriones estén disponibles para ser disputados por cepas especializadas y generalistas existen regiones de biestabilidad, y las probabilidades de observar cada estado se calculan como funciones de las tasas de infección. Se ha encontrado un raro atractor caótico y se ha analizado con el uso de exponentes de Lyapunov. Esto, combinado con los diagramas de estabilidad, muestra que la supervivencia de la cepa generalista en un entorno estable es un hecho improbable. Además, se

Terrorism in all its forms and manifestations and its scope, intensity, cruelty has now become one of the most acute and topical problems of global significance. The article deals with conceptual measures for strengthening the fight against terrorism and extremism financing, effective work with the private sector on providing information necessary for detection of the activity connected with terrorism and extremism financing. This work analyses the increasing level of awareness of representatives of public and private sectors and strengthening of the role of financial intelligence for the use of reconnaissance data as a dominant way of fight against terrorism and extremism. It has been determined that in such circumstances effective fight against terrorism and extremism as the exterritorial phenomenon has to have a complex character, combining efforts at universal, regional and interstate levels. As experts note, the measures taken only at the national level will have a limited effect. For safety insurance, each state needs international cooperation. It has been established that in Israel in order to prevent crises and mass protests, they use in practice the positively recommended Israel’s experiment of information exchange of law enforcement and special bodies with leaders of public opinion (authoritative persons, elders), representatives of official clergy, mass media and NGO by means of WhatsApp mobile application in online mode about the events having a public response. It is established that all interested private structures and government bodies have to reach a compromise in the strategy of ensuring a reliable system of counteraction and development of countermeasures according to the solution of anti-terrorist actions.

Terrorism in all its forms and manifestations and its scope, intensity, cruelty has now become one of the most acute and topical problems of global significance. The article deals with conceptual measures for strengthening the fight against terrorism and extremism financing, effective work with the private sector on providing information necessary for detection of the activity connected with terrorism and extremism financing. This work analyses the increasing level of awareness of representatives of public and private sectors and strengthening of the role of financial intelligence for the use of reconnaissance data as a dominant way of fight against terrorism and extremism. It has been determined that in such circumstances effective fight against terrorism and extremism as the exterritorial phenomenon has to have a complex character, combining efforts at universal, regional and interstate levels. As experts note, the measures taken only at the national level will have a limited effect. For safety insurance, each state needs international cooperation. It has been established that in Israel in order to prevent crises and mass protests, they use in practice the positively recommended Israel’s experiment of information exchange of law enforcement and special bodies with leaders of public opinion (authoritative persons, elders), representatives of official clergy, mass media and NGO by means of WhatsApp mobile application in online mode about the events having a public response. It is established that all interested private structures and government bodies have to reach a compromise in the strategy of ensuring a reliable system of counteraction and development of countermeasures according to the solution of anti-terrorist actions.