Bulletin of Mathematical Biology

Liebscher, Volkmar

doi: 10.1007/s11538-017-0385-zpmid: 29297144

We present a new class of metrics for unrooted phylogenetic X-trees inspired by the Gromov–Hausdorff distance for (compact) metric spaces. These metrics can be efficiently computed by linear or quadratic programming. They are robust under NNI operations, too. The local behaviour of the metrics shows that they are different from any previously introduced metrics. The performance of the metrics is briefly analysed on random weighted and unweighted trees as well as random caterpillars.

Hanin, Leonid; Rose, Jason

doi: 10.1007/s11538-017-0388-9pmid: 29302774

We study metastatic cancer progression through an extremely general individual-patient mathematical model that is rooted in the contemporary understanding of the underlying biomedical processes yet is essentially free of specific biological assumptions of mechanistic nature. The model accounts for primary tumor growth and resection, shedding of metastases off the primary tumor and their selection, dormancy and growth in a given secondary site. However, functional parameters descriptive of these processes are assumed to be essentially arbitrary. In spite of such generality, the model allows for computing the distribution of site-specific sizes of detectable metastases in closed form. Under the assumption of exponential growth of metastases before and after primary tumor resection, we showed that, regardless of other model parameters and for every set of site-specific volumes of detected metastases, the model-based likelihood-maximizing scenario is always the same: complete suppression of metastatic growth before primary tumor resection followed by an abrupt growth acceleration after surgery. This scenario is commonly observed in clinical practice and is supported by a wealth of experimental and clinical studies conducted over the last 110 years. Furthermore, several biological mechanisms have been identified that could bring about suppression of metastasis by the primary tumor and accelerated vascularization and growth of metastases after primary tumor resection. To the best of our knowledge, the methodology for uncovering general biomedical principles developed in this work is new.

Wang, Wei; Zhang, Tongqian

doi: 10.1007/s11538-017-0389-8pmid: 29349609

Caspase-1-mediated pyroptosis is the predominance for driving CD4 $$^{+}$$ + T cells death. Dying infected CD4 $$^{+}$$ + T cells can release inflammatory signals which attract more uninfected CD4 $$^{+}$$ + T cells to die. This paper is devoted to developing a diffusive mathematical model which can make useful contributions to understanding caspase-1-mediated pyroptosis by inflammatory cytokines IL-1 $$\beta $$ β released from infected cells in the within-host environment. The well-posedness of solutions, basic reproduction number, threshold dynamics are investigated for spatially heterogeneous infection. Travelling wave solutions for spatially homogeneous infection are studied. Numerical computations reveal that the spatially heterogeneous infection can make $$\mathscr {R}_0>1$$ R 0 > 1 , that is, it can induce the persistence of virus compared to the spatially homogeneous infection. We also find that the random movements of virus have no effect on basic reproduction number for the spatially homogeneous model, while it may result in less infection risk for the spatially heterogeneous model, under some suitable parameters. Further, the death of infected CD4 $$^{+}$$ + cells which are caused by pyroptosis can make $$\mathscr {R}_0<1$$ R 0 < 1 , that is, it can induce the extinction of virus, regardless of whether or not the parameters are spatially dependent.

Lee, Seunggyu

doi: 10.1007/s11538-018-0390-xpmid: 29344759

In this paper, a mathematical model of contractile ring-driven cytokinesis is presented by using both phase-field and immersed-boundary methods in a three-dimensional domain. It is one of the powerful hypotheses that cytokinesis happens driven by the contractile ring; however, there are only few mathematical models following the hypothesis, to the author’s knowledge. I consider a hybrid method to model the phenomenon. First, a cell membrane is represented by a zero-contour of a phase-field implicitly because of its topological change. Otherwise, immersed-boundary particles represent a contractile ring explicitly based on the author’s previous work. Here, the multi-component (or vector-valued) phase-field equation is considered to avoid the emerging of each cell membrane right after their divisions. Using a convex splitting scheme, the governing equation of the phase-field method has unique solvability. The numerical convergence of contractile ring to cell membrane is proved. Several numerical simulations are performed to validate the proposed model.

Valega-Mackenzie, Wencel; Ríos-Soto, Karen

doi: 10.1007/s11538-018-0393-7pmid: 29359251

Zika virus (ZIKV) is a vector-borne disease that has rapidly spread during the year 2016 in more than 50 countries around the world. If a woman is infected during pregnancy, the virus can cause severe birth defects and brain damage in their babies. The virus can be transmitted through the bites of infected mosquitoes as well as through direct contact from human to human (e.g., sexual contact and blood transfusions). As an intervention for controlling the spread of the disease, we study a vaccination model for preventing Zika infections. Although there is no formal vaccine for ZIKV, The National Institute of Allergy and Infectious Diseases (part of the National Institutes of Health) has launched a vaccine trial at the beginning of August 2016 to control ZIKV transmission, patients who received the vaccine are expected to return within 44 weeks to determine if the vaccine is safe. Since it is important to understand ZIKV dynamics under vaccination, we formulate a vaccination model for ZIKV spread that includes mosquito as well as sexual transmission. We calculate the basic reproduction number of the model to analyze the impact of relatively, perfect and imperfect vaccination rates. We illustrate several numerical examples of the vaccination model proposed as well as the impact of the basic reproduction numbers of vector and sexual transmission and the effect of vaccination effort on ZIKV spread. Results show that high levels of sexual transmission create larger cases of infection associated with the peak of infected humans arising in a shorter period of time, even when a vaccine is available in the population. However, a high level of transmission of Zika from vectors to humans compared with sexual transmission represents that ZIKV will take longer to invade the population providing a window of opportunities to control its spread, for instance, through vaccination.

Verma, Maitri; Misra, A.

doi: 10.1007/s11538-018-0394-6pmid: 29368079

The extinction of species is a major threat to the biodiversity. The species exhibiting a strong Allee effect are vulnerable to extinction due to predation. The refuge used by species having a strong Allee effect may affect their predation and hence extinction risk. A mathematical study of such behavioral phenomenon may aid in management of many endangered species. However, a little attention has been paid in this direction. In this paper, we have studied the impact of a constant prey refuge on the dynamics of a ratio-dependent predator–prey system with strong Allee effect in prey growth. The stability analysis of the model has been carried out, and a comprehensive bifurcation analysis is presented. It is found that if prey refuge is less than the Allee threshold, the incorporation of prey refuge increases the threshold values of the predation rate and conversion efficiency at which unconditional extinction occurs. Moreover, if the prey refuge is greater than the Allee threshold, situation of unconditional extinction may not occur. It is found that at a critical value of prey refuge, which is greater than the Allee threshold but less than the carrying capacity of prey population, system undergoes cusp bifurcation and the rich spectrum of dynamics exhibited by the system disappears if the prey refuge is increased further.

Asaduzzaman, Sarder Mohammed; Ma, Junling; van den Driessche, P.

doi: 10.1007/s11538-018-0395-5pmid: 29372495

To determine the cross-immunity between influenza strains, we design a novel statistical method, which uses a theoretical model and clinical data on attack rates and vaccine efficacy among school children for two seasons after the 1968 A/H3N2 influenza pandemic. This model incorporates the distribution of susceptibility and the dependence of cross-immunity on the antigenic distance of drifted strains. We find that the cross-immunity between an influenza strain and the mutant that causes the next epidemic is 88%. Our method also gives estimates of the vaccine protection against the vaccinating strain, and the basic reproduction number of the 1968 pandemic influenza.

Hiller, Josh; Keesling, James

doi: 10.1007/s11538-018-0397-3pmid: 29383584

We examine basic asymptotic properties of relative risk for two families of generalized Erlang processes (where each one is based off of a simplified Armitage and Doll multistage model) in order to predict relative risk data from cancer. The main theorems that we are able to prove are all corroborated by large clinical studies involving relative risk for former smokers and transplant recipients. We then show that at least some of these theorems do not extend to other Armitage and Doll multistage models. We conclude with suggestions for lifelong increased cancer screening for both former smoker and transplant recipient subpopulations of individuals and possible future directions of research.