Multitype Branching Processes Research Articles

Critical Multitype Branching Processes with Random Migration

Abstract The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in [N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 1980, 4, 473–475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

Dynamic analysis of HIV infection model with CTL immune response and cell-to-cell transmission

HIV infection is still a serious worldwide public health problem. During the early stage of HIV infection, the number of infected cells and viruses are extremely low and the process of infection is stochastic. In this paper, we use a stochastic model consisting of continuous-time Markov chain to investigate CTL immune response and two modes of infection (viral transmission and cellular transmission). A multitype branching process is used to approximate the probability of clearance and outbreak of HIV infection near the infection-free equilibrium. In addition, we compare the dynamics of the deterministic model with the results of the stochastic model, and the differences are as follows: (1) when R0>1, the deterministic model shows HIV infection outbreak, while the stochastic model indicates a positive probability of HIV infection elimination; (2) the initial number of infected cells and viruses have no effect on the dynamics of the deterministic model, whereas the dynamics of the stochastic model are highly dependent on the initial sizes and cannot be ignored.

Diffusion approximation of critical controlled multi-type branching processes

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs.

A Simple Finite Delayed Multi-type Branching Process for Infectious Disease Modeling

We study a model for the spread of an infectious disease which incorporates spatial and temporal effects. The model is a delayed multi-type branching process in which types represent geographic regions while infected individuals reproduce offspring during a finite time interval and have convales- cence times and random death/recovery outcomes. We give simple expressions for the limit of the geometrically weighted mean evolution of the process.

A stochastic multi-host model for West Nile virus transmission

When initially introduced into a susceptible population, a disease may die out or result in a major outbreak. We present a Continuous-Time Markov Chain model for enzootic WNV transmission between two avian host species and a single vector, and use multitype branching process theory to determine the probability of disease extinction based upon the type of infected individual initially introducing the disease into the population – an exposed vector, infectious vector, or infectious host of either species. We explore how the likelihood of disease extinction depends on the ability of each host species to transmit WNV, vector biting rates on host species, and the relative abundance of host species, as well as vector abundance. Theoretical predictions are compared to the outcome of stochastic simulations. We find the community composition of hosts and vectors, as well as the means of disease introduction, can greatly affect the probability of disease extinction.

The impacts of spatial–temporal heterogeneity of human-to-human contacts on the extinction probability of infectious disease from branching process model

In this study, we focus on the impacts of spatial–temporal heterogeneity of human-to-human contacts on the spread of infectious diseases and develop a multi-type branching process model by introducing random human-to-human contact mode into a structured population. We provide the general formulas of the generation size, extinction probability, and basic reproduction number of the proposed branching process model. The result shows that the natural temporal heterogeneity (i.e. random contacts over time) can lead to a higher extinction probability while remains the same basic reproduction number and generation size. This is also numerically verified by choosing the real contact distributions from different circumstances of four countries. In addition, we observe a non-monotonic pattern of the differences, against the transmission probability and the mean contact rate, between the extinction probabilities under the constant and random contact patterns. Given the spatial heterogeneity, we show that it can contribute to the increase of basic reproduction number, but also increase the extinction probability of the infectious disease. This study adds novel insights to the course of the impact of heterogeneity on the transmission dynamics and also provides additional evidence for the limited role of reproduction numbers.

The coalescent structure of uniform and Poisson samples from multitype branching processes

The coalescent structure of uniform and Poisson samples from multitype branching processes

A generating-function approach to modelling complex contagion on clustered networks with multi-type branching processes

Abstract Understanding cascading processes on complex network topologies is paramount for modelling how diseases, information, fake news and other media spread. In this article, we extend the multi-type branching process method developed in Keating et al., (2022), which relies on networks having homogenous node properties, to a more general class of clustered networks. Using a model of socially inspired complex contagion we obtain results, not just for the average behaviour of the cascades but for full distributions of the cascade properties. We introduce a new method for the inversion of probability generating functions to recover their underlying probability distributions; this derivation naturally extends to higher dimensions. This inversion technique is used along with the multi-type branching process to obtain univariate and bivariate distributions of cascade properties. Finally, using clique-cover methods, we apply the methodology to synthetic and real-world networks and compare the theoretical distribution of cascade sizes with the results of extensive numerical simulations.

A comparison of mutation and amplification-driven resistance mechanisms and their impacts on tumor recurrence.

Tumor recurrence, driven by the evolution of drug resistance is a major barrier to therapeutic success in cancer. Tumor drug resistance is often caused by genetic alterations such as point mutation, which refers to the modification of a single genomic base pair, or gene amplification, which refers to the duplication of a region of DNA that contains a gene. These mechanisms typically confer varying degrees of resistance, and they tend to occur at vastly different frequencies. Here we investigate the dependence of tumor recurrence dynamics on these mechanisms of resistance, using stochastic multi-type branching process models. We derive tumor extinction probabilities and deterministic estimates for the tumor recurrence time, defined as the time when an initially drug sensitive tumor surpasses its original size after developing resistance. For models of amplification-driven and mutation-driven resistance, we prove law of large numbers results regarding the convergence of the stochastic recurrence times to their mean. Additionally, we prove sufficient and necessary conditions for a tumor to escape extinction under the gene amplification model, discuss behavior under biologically relevant parameters, and compare the recurrence time and tumor composition in the mutation and amplification models both analytically and using simulations. In comparing these mechanisms, we find that the ratio between recurrence times driven by amplification versus mutation depends linearly on the number of amplification events required to acquire the same degree of resistance as a mutation event, and we find that the relative frequency of amplification and mutation events plays a key role in determining the mechanism under which recurrence is more rapid for any specific system. In the amplification-driven resistance model, we also observe that increasing drug concentration leads to a stronger initial reduction in tumor burden, but that the eventual recurrent tumor population is less heterogeneous, more aggressive and harbors higher levels of drug-resistance.

A stochastic model of antibiotic misuse, economy, and drug resistance: relating mutant extinction probability to socioeconomic and epidemiological factors

ABSTRACT Controlling antibiotic drug resistance requires understanding extinction and emergence mechanisms of emerging bacteria. Selective pressure from prolonged antibiotic misuse may cause high-level antimicrobial resistance. Self-medication and other socioeconomic factors reduce antibiotic use, accelerating the emergence and extinction of resistant pathogens through stochastic fluctuation. This continuous antibiotic self-medication exposes individuals and communities to antibiotic resistance, especially in low- and lower-middle-income countries, according to current literature. We developed a stochastic drug-resistance model that integrates socio-economic growth and antibiotic use to study extinction and strain establishment in this paper. We analytically derived the extinction threshold using the multi-type branching process and obtained pathogen extinction conditions that match numerical experiments. The model's sensitivity analysis identifies extinction dynamics' key parameters. Our results show that higher income, awareness, and lower antibiotic use may increase the chance of extinction by reducing antibiotic misuse, along with strain transmission potential. These analyses may help public health policymakers combat drug resistance.

INVASION AND SUPERINFECTION IN DETERMINISTIC AND STOCHASTIC TWO-STRAIN DENGUE MODELS WITH DEMOGRAPHIC AND SEASONAL VARIATION

The effects of seasonality on disease invasion and superinfection are investigated in dengue epidemic models with two viral strains. A two-strain host–vector nonautonomous ODE is formulated with seasonal periodicity in either vector recruitment or transmission. The basic reproduction number is derived and the dynamics of the single-strain periodic subsystem are summarized. Conditions for uniform persistence and existence of a single-strain periodic solution are verified and the invasion reproduction number is derived. A time-nonhomogeneous, continuous-time Markov chain (CTMC) is formulated from the ODE framework. The basic and invasion reproduction numbers apply to the ODE and CTMC. In the case of invasion, after one strain is established and the invasion reproduction number exceeds one, the ODE predicts a successful invasion, but in the CTMC, there is a periodic probability of invasion that depends on time and size of the invasion. A multitype branching process approximation of the CTMC provides an analytical method to predict the periodic probability of invasion. The numerical results show that the peak time of invasion is related to the current endemic strain and the dominant seasonal driver but it often precedes the peak time of the seasonal driver.

Sequential mutations in exponentially growing populations.

Stochastic models of sequential mutation acquisition are widely used to quantify cancer and bacterial evolution. Across manifold scenarios, recurrent research questions are: how many cells are there with n alterations, and how long will it take for these cells to appear. For exponentially growing populations, these questions have been tackled only in special cases so far. Here, within a multitype branching process framework, we consider a general mutational path where mutations may be advantageous, neutral or deleterious. In the biologically relevant limiting regimes of large times and small mutation rates, we derive probability distributions for the number, and arrival time, of cells with n mutations. Surprisingly, the two quantities respectively follow Mittag-Leffler and logistic distributions regardless of n or the mutations' selective effects. Our results provide a rapid method to assess how altering the fundamental division, death, and mutation rates impacts the arrival time, and number, of mutant cells. We highlight consequences for mutation rate inference in fluctuation assays.

Cystic echinococcosis dynamics in dogs, humans and cattle: Deterministic and stochastic modeling

This work investigates the dynamics of cystic echinococcosis in humans, dogs and cattle. The next generation matrix method is employed to derive the basic reproduction number R0. The normalized forward sensitivity index is used to identify sensitive parameters to disease transmission. The multitype branching process is adopted to compute the likelihood of disease extinction or spread in humans, dogs, and cattle. Numerical results show that both the disease free equilibrium and the endemic equilibria exist. Sensitivity analysis shows that the most positive sensitive parameters to disease transmission are rate at which human and dog infection from contaminated environments and infectious meat respectively, the rate at which exposed cattle progress to an infectious stage, the rate at which infected dogs defecate, and the rates at which dogs and cattle are recruited are most positive sensitive parameters to disease transmission. The most negative sensitive parameters are the natural mortality rates of dogs and cattle. According to the results of the simulation, there is a high likelihood that cystic echinococcosis will become extinct if the disease arises from echinococcus tapeworms, but a low likelihood if it arises from either infectious meat or an infected dog population. The disease dies, nevertheless, if it emerges from infected people. Based on results, we suggest that measures such as dog indoor keeping, meat inspection and treatment of infected dog population be implemented to control cystic echinococcosis in humans, dogs and cattle.

Limiting Behaviors of Stochastic Spread Models Using Branching Processes

In this paper, we introduce a spread model using multi-type branching processes to investigate the evolution of the population during a pandemic in which individuals are classified into different types. We study some limiting behaviors of the population including the growth rate of the population and the spread rate of each type. In particular, the work in this paper focuses on the cases where the offspring mean matrices are non-primitive but can be decomposed into two primitive components, A and B, with maximal eigenvalues ρA and ρB, respectively. It is shown that the growth rate and the spread rate heavily depend on the conditions of these two maximal eigenvalues and are related to the corresponding eigenvectors. In particular, we find the spread rates for the case with ρB>ρA>1 and the case with ρA>ρB>1. In addition, some numerical examples and simulations are also provided to support the theoretical results.

Subcritical multitype Markov branching processes with immigration generated by Poisson random measures

We investigate multitype subcritical Markov branching processes with immigration driven by Poisson random measures. Limiting distributions are established for various rates of the Poisson measures when they are asymptotically equivalent to exponential or regularly varying functions. Results analogous to a strong LLN are proved, and limiting normal distributions are obtained when the local intensity of the Poisson measure increases with time. When it decreases, conditional limiting distributions are established. When the intensity converges to a constant, a stationary limiting distribution is obtained. The asymptotic behavior of the first and second moments of the processes is also investigated.

Extinction times of multitype continuous-state branching processes

Un processus de branchement multitype, continu (MCSBP) Z=(Zt)t≥0, est un processus de Markov à valeurs dans [0,∞)d qui satisfait à la propriété de branchement. Sa loi est caractérisée par son mécanisme de branchement qui est donné par d exposants de Laplace de processus de Lévy spectralement positifs, à valeurs dans Rd, chacun d’entre eux possédant d−1 coordonnées croissantes. Nous donnons une expression de la probabilité pour un MCSBP de tendre vers 0 à l’infini en terme de son mécanisme de branchement. Nous montrons que cette extinction a lieu en un temps fini si et seulement si une certaine condition portant sur le mécanisme de branchement est satisfaite. Cette condition étend la condition de Grey bien connue en dimension 1. Nos arguments portent sur des éléments de théorie des fluctuations pour les champs de Lévy additifs, spectrallement positifs récemment établis en (Electron. J. Probab. 25 (2020) 26) ainsi qu’une extension de la représentation de Lamperti en dimension supérieure obtenue en (Ann. Inst. Henri Poincaré Probab. Stat. 53 (2017) 1280–1304).

Shift, scale and restart smaller models to estimate larger ones: Agent based simulators in epidemiology

Agent-based simulators (ABS) are a popular epidemiological modelling tool to study the impact of non medical interventions in managing epidemics [1], [2]. They accurately model a heterogeneous population with time and location varying, person specific interactions. Government policies such as partial and location specific lock downs, case isolation, home quarantine, school closures, partially opened workplaces, etc. are easily modelled and ABS allow flexibility to incorporate important pandemic developments over time including variants and vaccines. For accuracy, each person is modelled separately. This however may make computational time prohibitive when the city population and the simulated time are large. We observe that simply considering a smaller representative model and scaling up the output leads to inaccuracies. In this note we focus on the COVID- 19 pandemic and dig deeper into the underlying probabilistic structure of generic ABS to arrive at modifications that allow smaller models to give accurate statistics for larger ones. We exploit the observation that in the initial disease spread phase, the starting infections create a family tree of infected individuals more-or-less independent of the other trees and are modelled well as a multi-type super-critical branching process whose relative proportions across infected population types stabilize soon and thereafter are invariant to shifts in time. Further, for large city population, once enough people have been infected, the future evolution of the pandemic is closely approximated by its mean field limit with a random starting state. We build upon these insights to develop a shifted, scaled and restart based algorithm that accurately evaluates the ABS's performance using a much smaller model while carefully reducing the bias that may arise from just scaling. Our key contributions are: 1) we develop an algorithm by carefully exploiting the closeness of the underlying exposed/infected process (process of number exposed/infected of each type at each time) initially to a branching process, and then the normalised infection process (infection process divided by the city population) to its mean field limit, so that the output from the smaller model accurately matches the output from the larger model in a realistic city setting. 2) we provide theoretical support for the proposed approach through an asymptotic analysis where the population increases to infinity. For brevity, we conduct the analysis in a simpler and yet practically useful setting.

Limit theorems for discrete multitype branching processes counted with a characteristic

For a discrete time multitype supercritical Galton–Watson process (Zn)n∈N and corresponding genealogical tree T, we associate a new discrete time process (ZnΦ)n∈N such that, for each n∈N, the contribution of each individual u∈T to ZnΦ is determined by a (random) characteristic Φ evaluated at the age of u at time n. In other words, ZnΦ is obtained by summing over all u∈T the corresponding contributions Φu, where (Φu)u∈T are i.i.d. copies of Φ. Such processes are known in the literature under the name of Crump–Mode–Jagers (CMJ) processes counted with characteristicΦ. We derive a LLN and a CLT for the process (ZnΦ)n∈N in the discrete time setting, and in particular, we show a dichotomy in its limit behavior. By applying our main result, we also obtain a generalization of the results in Kesten and Stigum (1966).

Taenia solium taeniasis and cysticercosis: extinction or outbreak

Taenia solium taeniasis and cysticercosis are neglected zoonotic diseases that affect human health and economies of developing countries. In this work, we formulate and analyze deterministic and continuous time Markov chain (CTMC) stochastic models to determine parameters that drive Taenia solium taeniasis and cysticercosis and the likelihood of their extinction. The basic reproduction number R_0 is computed by the next generation matrix approach, sensitivity index of each parameter in R_0 is derived by the normalized forward sensitivity index and the likelihood of diseases’ extinction is computed by the multitype branching process. The analysis shows that humans with Taenia solium taeniasis, infectious pork and Taenia solium eggs in the environment play an important role in the transmission of Taenia solium taeniasis and cysticercosis, and the model exhibits forward bifurcation at R_0=1. This implies that R_0<1 is a sufficient condition to eliminate Taenia solium taeniasis and cysticercosis. For CTMC model, analysis shows that the probability of Taenia solium taeniasis and cysticercosis extinction is high if the diseases emerge from humans with Taenia solium cysticercosis and there is an outbreak if the diseases emerge from either humans with Taenia solium taeniasis or infectious pork or Taenia solium eggs in the environment. To control Taenia solium taeniasis and cysticercosis, the intervention strategies should focus on improving hygiene and sanitation for reducing shedding rate of Taenia solium eggs in the environment, inspection of pork for reducing the rate of acquiring Taenia solium taeniasis and spraying of insecticides for killing Taenia solium eggs in the environment.

A Kesten–Stigum type theorem for a supercritical multitype branching process in a random environment

Consider a supercritical d-type branching process $Z_n^{i} = $ $ (Z_n^i(1), \cdots, Z_n^i(d)), \,n\geq 0,$ in an i.i.d. environment $\xi =(\xi_0, \xi_1, \ldots)$, starting with one particle of type $i$, whose offspring distributions of generation $n$ depend on the environment $\xi_n$ at time $n$. In the deterministic environment case, the famous Kesten-Stigum (1966) theorem states essentially that, if the mean matrix of the offspring distribution has spectral radius $\rho >1$, then for all $i,j =1, \ldots, d$, almost surely $ \lim_{n\rightarrow \infty} \frac{Z_{n}^{i}(j)}{\rho^n }$ exists and is finite; moreover, the limit variables are non-degenerate if and only if $\mathbb{E} \left(Z_1^{i}(j) \log^{+} Z_1^{i}(j)\right)<+\infty$ for all $i$ and $j$. The extension to the random environment case with $d=1$ has been done by Athreya and Karlin (1971) and Tanny (1988). Extending the Kesten-Stigum theorem to the random environment case with $d>1$ is a long-standing problem. The main objective of this paper is to resolve this delicate problem. In particular, under simple conditions, we prove that for any $1\le i,j\le d$, as $n \rightarrow +\infty$, $Z_n^{i}(j)/\mathbb{E}_\xi Z_n^{i}(j) \rightarrow W^{i}$ in probability, where $W^i$ is a non-negative random variable, $ \mathbb{E}_\xi Z_n^{i}(j)$ is the conditional expectation of $Z_n^{i}(j)$ given the environment $\xi$, which diverges to $\infty$ with geometric rate in the sense that $ \frac{1}{n} \log \mathbb{E}_\xi Z_n^{i}(j) \rightarrow \gamma >0$ almost surely, $\gamma$ being the Lyapunov exponent of the mean matrices of the offspring distributions; moreover $W^i$ are non-degenerate for all $i $ if and only if $\mathbb{E}\left(\frac{Z_1^{i}(j)}{M_0(i,j)}\log^{+}\frac{Z_1^{i}(j)}{M_0(i,j)}\right)<+\infty$ for all $i$ and $j$, where $M_0(i,j)$ is the conditioned mean of the number of children of type $j$ produced by a particle of type $i$ at time $0$, given the environment $\xi$. The key idea of the proof is the introduction of a non-negative martingale $(W_n^{i})$ which converges a.s. to $W^i$, and which reduces to the well-known fundamental martingale in the deterministic environment case. In addition, we prove that the direction $Z_n^{i}/ \| Z_n^{i} \|$ converges in law conditioned on the explosion event $\{\|Z_n^i\| \rightarrow +\infty\}$. The case of stationary and ergodic environment is also considered. Our results open ways to prove important properties such as central limit theorems with convergence rate and large deviation asymptotics.