Maroussia Slavtchova-Bojkova

Time to extinction of infectious diseases through age-dependent branching models

This paper is concerned with a Sevast’yanov age-dependent branching process, describing outbreaks of an infectious disease with incubation period. The main goal was to define the optimal proportion of susceptible individuals that has to be vaccinated in order to eliminate the disease. To this end we study the properties of the time to extinction of an infection according to the proportion of immune individuals in the population. The results lead us to suggest a vaccination policy based on the mean of the infection survival time. Finally, we provide a simulation-based method to determine the optimal vaccination level, and as an illustration analyze the data of outbreaks of avian influenza spreading in Vietnam at the end of 2006.

Age-Dependent Branching Processes with State-Dependent Immigration

We consider a model of Bellman-Harris branching processes with immigration only in the state zero (BHIO) which permits in addition an immigration component of i.i.d. BHIO processes entering the population at i.i.d. times of an independent renewal process (BHIOR). Asymptotic properties and limit theorems are proved in non-critical cases.

Stochastic monotonicity and continuity properties of functions defined on Crump–Mode–Jagers branching processes, with application to vaccination in epidemic modelling

This paper is concerned with Crump-Mode-Jagers branching processes, describing spread of an epidemic depending on the proportion of the population that is vaccinated. Births in the branching process are aborted independently with a time-dependent probability given by the fraction of the population vaccinated. Stochastic monotonicity and continuity results for a wide class of functions (e.g., extinction time and total number of births over all time) defined on such a branching process are proved using coupling arguments, leading to optimal vaccination schemes to control corresponding functions (e.g., duration and final size) of epidemic outbreaks. The theory is illustrated by applications to the control of the duration of mumps outbreaks in Bulgaria.

Computation of waiting time to successful experiment using age-dependent branching model

This paper considers the properties of the waiting time to survive forever of the supercritical age-dependent branching process modified with an immigration component. Conditioning on ultimate extinction of the process we analyze the conditional distribution of the cycle length and its expectation. We then derive and give estimates of the conditional expected total progeny of a cycle and its higher moments. In general, the model originates from the problem of estimating the waiting time to a successful experiment in industrial wastewater treatment by bacterial culture systems. Computer runs were made for two reproduction laws and different reproduction means.

Branching processes in continuous time as models of mutations: Computational approaches and algorithms

The appearance of mutations in cancer development plays a crucial role in the disease control and its medical treatment. Motivated by the practical significance, it is of interest to model the event of occurrence of a mutant cell that will possibly lead to a path of indefinite survival. A two-type branching process model in continuous time is proposed for describing the relationship between the waiting time till the first escaping extinction mutant cell is born and the lifespan distribution of cells, which due to the applied treatment have small reproductive ratio. A numerical method and related algorithm for solving the integral equations are developed, in order to estimate the distribution of the waiting time to the escaping extinction mutant cell is born. Numerical studies demonstrate that the proposed approximation algorithm reveals the substantial difference of the results in discrete-time setting. In addition, to study the time needed for the mutant cell population to reach high levels a simulation algorithm for continuous two-type decomposable branching process is proposed. Two different computational approaches together with the theoretical studies might be applied to different kinds of cancer and their proper treatment.

On Two-Type Decomposable Branching Processes in Continuous Time and Time to Escape Extinction

The main goal of this paper is to consider branching processes with two types and in continuous time to model the dynamics of the number of different types of cells, which due to a small reproductive ratio are fated to become extinct. However, mutations occurring during the reproduction process may lead to the appearance of a new type of cells that may escape extinction. This is a typical real world situation with the emergence of scatters after local eradication of a certain type of cancer during the chemotherapy. Mathematically, we are deriving the numbers of mutations of the escape type and their moments. A cell of the “mutation” type, which leads possibly to the beginning of a lineage, that will allow indefinite survival is called “successful mutant”. Using the results about the probability generating function of the single-type branching processes, an answer about the distribution of the waiting time to produce a “successful mutant” in continuous-time setting is obtained. In general, our results aim to prove the limits of expanding the methods used by Serra and Haccou (Theor. Popul. Biol. 72:167–178, 2007) for different schemes leading to mutation.

LIMIT THEOREMS FOR SUBCRITICAL AGE-DEPENDENT BRANCHING PROCESSES WITH TWO TYPES OF IMMIGRATION

ABSTRACT For the classical subcritical age-dependent branching process the effect of the following two-type immigration pattern is studied. At a sequence of renewal epochs a random number of immigrants enters the population. Each subpopulation stemming from one of these immigrants or one of the ancestors is revived by new immigrants and their offspring whenever it dies out, possibly after an additional delay period. All individuals have the same lifetime distribution and produce offspring according to the same reproduction law. This is the Bellman-Harris process with immigration at zero and immigration of renewal type (BHPIOR). We prove a strong law of large numbers and a central limit theorem for such processes. Similar conclusions are obtained for their discrete-time counterparts (lifetime per individual equals one), called Galton-Watson processes with immigration at zero and immigration of renewal type (GWPIOR). Our approach is based on the theory of regenerative processes, renewal theory and occupation measures and is quite different from those in earlier related work using analytic tools.

Total Progeny of Crump-Mode-Jagers Branching Processes: An Application to Vaccination in Epidemic Modelling

This paper is concerned with the use of vaccination schemes to control an epidemic in terms of the total number of individuals infected. In particular, monotonicity and continuity properties of total progeny of Crump-Mode-Jagers branching processes are derived depending on vaccination level. Furthermore, optimal vaccination policies based on the mean and quantiles of the total number of infected individuals are proposed. Finally, how to apply the proposed methodology in real situations is shown through a simulated example motivated by an outbreak of influenza virus in humans, in Indonesia.

Multi-Type Age-Dependent Branching Processes with State-Dependent Immigration

This work continues the study of the age-dependent branching processes allowing two types of immigration, i.e. one in the state zero and another one according to the i.i.d. times of an independent ergodic renewal process. The multidimensional case is considered and asymptotic properties and limit theorems are established. These results generalise both the results of the discrete theory and those for the one-dimensional continuous-time model.