Bulletin of Mathematical Biology

Phillips, Tricia; Lenhart, Suzanne; Strickland, W. Christopher

doi: 10.1007/s11538-021-00925-0pmid: 34402967

Opioid addiction represents a major national health issue spanning decades. In recent years, prescription opioid use disorder has increasingly led to heroin and fentanyl use, with subsequent increases in mortality rates due to overdose. In this paper, we present a mechanistic, epidemic model for prescription opioid addiction and illicit heroin or fentanyl addiction which aims to better understand and predict the dynamics between these two stages of opioid use disorder. Our model aims to be both parsimonious and robust: as a system of five differential equations it is appropriate for use in theory advancement and yet it remains powerful enough to capture state-level data from Tennessee for the period 2013–2018. A key finding from our data-driven analysis is that, in the face of changing policy around prescription opioids, heroin and fentanyl are now the driving force behind the Tennessee opioid epidemic. Model projections suggest that both addictions and overdoses related to heroin and fentanyl will continue to increase in the next few years (2020–2022), even as addiction to prescription drugs continues to fall. Finally, management strategy analysis suggests that in the changing face of the epidemic, the most successful approach will target availability of treatment with subsequent monitoring of stably recovered individuals to see that they do not relapse, coincident with direct efforts to decrease opioid overdose fatalities (e.g., further availability of Naloxone).

Fowler, A. C.

doi: 10.1007/s11538-021-00936-xpmid: 34463830

This paper addresses the problem of extinction in continuous models of population dynamics associated with small numbers of individuals. We begin with an extended discussion of extinction in the particular case of a stochastic logistic model, and how it relates to the corresponding continuous model. Two examples of ‘small number dynamics’ are then considered. The first is what Mollison calls the ‘atto-fox’ problem (in a model of fox rabies), referring to the problematic theoretical occurrence of a predicted rabid fox density of 10-18\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$10^{-18}$$\end{document} (atto-) per square kilometre. The second is how the production of large numbers of eggs by an individual can reliably lead to the eventual survival of a handful of adults, as it would seem that extinction then becomes a likely possibility. We describe the occurrence of the atto-fox problem in other contexts, such as the microbial ‘yocto-cell’ problem, and we suggest that the modelling resolution is to allow for the existence of a reservoir for the extinctively challenged individuals. This is functionally similar to the concept of a ‘refuge’ in predator–prey systems and represents a state for the individuals in which they are immune from destruction. For what I call the ‘frogspawn’ problem, where only a few individuals survive to adulthood from a large number of eggs, we provide a simple explanation based on a Holling type 3 response and elaborate it by means of a suitable nonlinear age-structured model.

Hamis, Sara; Yates, James; Chaplain, Mark A. J.; Powathil, Gibin G.

doi: 10.1007/s11538-021-00935-ypmid: 34459993

We combine a systems pharmacology approach with an agent-based modelling approach to simulate LoVo cells subjected to AZD6738, an ATR (ataxia–telangiectasia-mutated and rad3-related kinase) inhibiting anti-cancer drug that can hinder tumour proliferation by targeting cellular DNA damage responses. The agent-based model used in this study is governed by a set of empirically observable rules. By adjusting only the rules when moving between monolayer and multi-cellular tumour spheroid simulations, whilst keeping the fundamental mathematical model and parameters intact, the agent-based model is first parameterised by monolayer in vitro data and is thereafter used to simulate treatment responses in in vitro tumour spheroids subjected to dynamic drug delivery. Spheroid simulations are subsequently compared to in vivo data from xenografts in mice. The spheroid simulations are able to capture the dynamics of in vivo tumour growth and regression for approximately 8 days post-tumour injection. Translating quantitative information between in vitro and in vivo research remains a scientifically and financially challenging step in preclinical drug development processes. However, well-developed in silico tools can be used to facilitate this in vitro to in vivo translation, and in this article, we exemplify how data-driven, agent-based models can be used to bridge the gap between in vitro and in vivo research. We further highlight how agent-based models, that are currently underutilised in pharmaceutical contexts, can be used in preclinical drug development.

Zheng, Collin Y.; Kim, Peter S.

doi: 10.1007/s11538-021-00933-0pmid: 34477976

We introduce a set of ordinary differential equations (ODEs) that qualitatively reproduce delayed responses observed in immune checkpoint blockade therapy (e.g. anti-CTLA-4 ipilimumab). This type of immunotherapy has been at the forefront of novel and promising cancer treatments over the past decade and was recognised by the 2018 Nobel Prize in Medicine. Our model describes the competition between effector T cells and non-effector T cells in a tumour. By calibrating a small subset of parameters that control immune checkpoint expression along with the patient’s immune-system cancer readiness, our model is able to simulate either a complete absence of patient response to treatment, a quick anti-tumour T cell response (within days) or a delayed response (within months). Notably, the parameter space that generates a delayed response is thin and must be carefully calibrated, reflecting the observation that a small subset of patients experience such reactions to checkpoint blockade therapies. Finally, simulations predict that the anti-tumour T cell storm that breaks the delay is very short-lived compared to the length of time the cancer is able to stay suppressed. This suggests the tumour may subsist off an environment hostile to effector T cells; however, these cells are—at rare times—able to break through the tumour immunosuppressive defences to neutralise the tumour for a prolonged period. Our simulations aim to qualitatively describe the delayed response phenomenon without making precise fits to particular datasets, which are limited. It is our hope that our foundational model will stimulate further interest within the immunology modelling field.

Mondal, Ritwika; Saha, Suman; Kesh, Dipak; Mukherjee, Debasis

doi: 10.1007/s11538-021-00930-3pmid: 34448068

A simple model on volatile organic compound (VOC)-mediated plant–insect interactions is proposed and examined here, when two different classes of herbivorous insects competing for a common resource (plant) in the presence of a specialist carnivorous enemy, which only predates one of the herbivore species. We, particularly, emphasize the impact of VOCs on plant’s growth fitness. The system experiences several local and global bifurcations with emergent alternative states for variations in recruitment factors and predation rate. Basin transitions and basin of attractions have provided detail descriptions on the selectivity of the alternative states, when only one of the herbivore species can survive depending on the choice of initial population densities of the interacting species and how it provides a steady growth in plant. Additionally, our results support the concept of competitive exclusion principle in an indirect interspecific competition between the two herbivore types for the common resource, plant.

Xue, Shuyang; Li, Meili; Ma, Junling; Li, Jia

doi: 10.1007/s11538-021-00940-1pmid: 34498148

In this paper, we used a generic two-stage population model to derive an adaptive dynamical system for the evolution of reproduction age and studied how this evolution is driven by the harvest of adults. We considered the tradeoffs between maturation rate and fecundity, juvenile mortality, and adult mortality. We analyzed the benefit and cost of faster maturation under each tradeoff that drives the evolution. We found that harvesting adults affects the evolution of maturation by affecting the benefit. For the tradeoff between maturation and juvenile mortality, harvesting adults does not affect the benefit and thus, does not affect optimal maturation strategy. For the other two tradeoffs, harvesting adults affects the benefit through the equilibrium adult/juvenile ratio, which is determined by the density dependence of juveniles. Harvesting adults causes a slower maturation only if it significantly reduces this ratio, which can only happen with very strong adult protection to juveniles. Otherwise, harvesting adults always causes a faster maturation.

Ruiz-Herrera, Alfonso; Torres, Pedro J.

doi: 10.1007/s11538-021-00929-wpmid: 34410514

In this paper, we analyze the influence of the usual movement variables on the spread of an epidemic. Specifically, given two spatial topologies, we can deduce which topology produces less infected individuals. In particular, we determine the topology that minimizes the overall number of infected individuals. It is worth noting that we do not assume any of the common simplifying assumptions in network theory such as all the links have the same diffusion rate or the movement of the individuals is symmetric. Our main conclusion is that the degree of mobility of the population plays a critical role in the spread of a disease. Finally, we derive theoretical insights to management of epidemics.

Jiang, Hongyan; Lam, King-Yeung; Lou, Yuan

doi: 10.1007/s11538-021-00939-8pmid: 34524555

We study the evolution of dispersal in advective three-patch models with distinct network topologies. Organisms can move between connected patches freely and they are also subject to the passive, directed drift. The carrying capacity is assumed to be the same in all patches, while the drift rates could vary. We first show that if all drift rates are the same, the faster dispersal rate is selected for all three models. For general drift rates, we show that the infinite diffusion rate is a local Convergence Stable Strategy (CvSS) for all three models. However, there are notable differences for three models: For Model I, the faster dispersal is always favored, irrespective of the drift rates, and thus the infinity dispersal rate is a global CvSS. In contrast, for Models II and III a singular strategy will exist for some ranges of drift rates and bi-stability phenomenon happens, i.e., both infinity and zero diffusion rates are local CvSSs. Furthermore, for both Models II and III, it is possible for two competing populations to coexist by varying drift and diffusion rates. Some predictions on the dynamics of n-patch models in advective environments are given along with some numerical evidence.

Gross, Kevin; de Roos, André M.

doi: 10.1007/s11538-021-00931-2pmid: 34460011

Sherratt, Jonathan A.; Liu, Quan-Xing; van de Koppel, Johan

doi: 10.1007/s11538-021-00932-1pmid: 34427781

Self-organised regular pattern formation is one of the foremost examples of the development of complexity in ecosystems. Despite the wide array of mechanistic models that have been proposed to understand pattern formation, there is limited general understanding of the feedback processes causing pattern formation in ecosystems, and how these affect ecosystem patterning and functioning. Here we propose a generalised model for pattern formation that integrates two types of within-patch feedback: amplification of growth and reduction of losses. Both of these mechanisms have been proposed as causing pattern formation in mussel beds in intertidal regions, where dense clusters of mussels form, separated by regions of bare sediment. We investigate how a relative change from one feedback to the other affects the stability of uniform steady states and the existence of spatial patterns. We conclude that there are important differences between the patterns generated by the two mechanisms, concerning both biomass distribution in the patterns and the resilience of the ecosystems to disturbances.