# American Institute of Mathematical Sciences

ISSN:
1551-0018

eISSN:
1547-1063

All Issues

## Mathematical Biosciences & Engineering

April 2018 , Volume 15 , Issue 2

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2018, 15(2): 337-359 doi: 10.3934/mbe.2018015 +[Abstract](3757) +[HTML](230) +[PDF](688.1KB)
Abstract:

The 2014 outbreak of Ebola virus disease (EVD) in West Africa was multinational and of an unprecedented scale primarily affecting the countries of Guinea, Liberia, and Sierra Leone. One of the qualities that makes EVD of high public concern is its potential for extremely high mortality rates (up to 90%). A prophylactic vaccine for ebolavirus (rVSV-ZEBOV) has been developed, and clinical trials show near-perfect efficacy. We have developed an ordinary differential equations model that simulates an EVD epidemic and takes into account (1) transmission through contact with infectious EVD individuals and deceased EVD bodies, (2) the heterogeneity of the risk of becoming infected with EVD, and (3) the increased survival rate of infected EVD patients due to greater access to trained healthcare providers. Using fitted parameter values that closely simulate the dynamics of the 2014 outbreak in Sierra Leone, we utilize our model to predict the potential impact of a prophylactic vaccine for the ebolavirus using various vaccination strategies including ring vaccination. Our results show that an rVSV-ZEBOV vaccination coverage as low as 40% in the general population and 95% in healthcare workers will prevent another catastrophic outbreak like the 2014 outbreak from occurring.

2018, 15(2): 361-392 doi: 10.3934/mbe.2018016 +[Abstract](3965) +[HTML](184) +[PDF](3563.9KB)
Abstract:

In this paper, a novel multiscale method is proposed for the study of heterogeneous tumor spheroid growth in vitro. The entire tumor spheroid is described by an ellipsoid-based model while nutrient and other environmental factors are treated as continua. The ellipsoid-based discrete component is capable of incorporating mechanical effects and deformability, while keeping a minimum set of free variables to describe complex shape variations. Moreover, our purely cell-based description of tumor avoids the complex mutual conversion between a cell-based model and continuum model within a tumor, such as force and mass transformation. This advantage makes it highly suitable for the study of tumor spheroids in vitro whose size are normally less than 800 \begin{document} $μ m$ \end{document} in diameter. In addition, our numerical scheme provides two computational options depending on tumor size. For a small or medium tumor spheroid, a three-dimensional (3D) numerical model can be directly applied. For a large spheroid, we suggest the use of a 3D-adapted 2D cross section configuration, which has not yet been explored in the literature, as an alternative for the theoretical investigation to bridge the gap between the 2D and 3D models. Our model and its implementations have been validated and applied to various studies given in the paper. The simulation results fit corresponding in vitro experimental observations very well.

2018, 15(2): 393-406 doi: 10.3934/mbe.2018017 +[Abstract](2902) +[HTML](149) +[PDF](665.5KB)
Abstract:

It is a common understanding that rotational cattle grazing provides better yields than continuous grazing, but a quantitative analysis is lacking in agricultural literature. In rotational grazing, cattle periodically move among paddocks in contrast to continuous grazing, in which the cattle graze on a single plot for the entire grazing season. We construct a differential equation model of vegetation grazing on a fixed area to show that production yields and stockpiled forage are greater for rotational grazing than continuous grazing. Our results show that both the number of cattle per acre and stockpiled forage increase for many rotational configurations.

2018, 15(2): 407-428 doi: 10.3934/mbe.2018018 +[Abstract](1530) +[HTML](170) +[PDF](705.65KB)
Abstract:

In this work we formulate a model for the population dynamics of Mycobacterium tuberculosis (Mtb), the causative agent of tuberculosis (TB). Our main interest is to assess the impact of the competition among bacteria on the infection prevalence. For this end, we assume that Mtb population has two types of growth. The first one is due to bacteria produced in the interior of each infected macrophage, and it is assumed that is proportional to the number of infected macrophages. The second one is of logistic type due to the competition among free bacteria released by the same infected macrophages. The qualitative analysis and numerical results suggests the existence of forward, backward and S-shaped bifurcations when the associated reproduction number \begin{document}$R_0$\end{document} of the Mtb is less unity. In addition, qualitative analysis of the model shows that there may be up to three bacteria-present equilibria, two locally asymptotically stable, and one unstable.

2018, 15(2): 429-440 doi: 10.3934/mbe.2018019 +[Abstract](3072) +[HTML](173) +[PDF](605.9KB)
Abstract:

The precise regulation of cell life division is indispensable to the reliable inheritance of genetic material, i.e. DNA, in successive generations of cells. This is governed by dedicated biochemical networks which ensure that all requirements are met before transition from one phase to the next. The Spindle Assembly Checkpoint (SAC) is an evolutionarily mechanism that delays mitotic progression until all chromosomes are properly linked to the mitotic spindle. During some asymmetric cell divisions, such as those observed in budding yeast, an additional mechanism, the Spindle Position Checkpoint (SPOC), is required to delay exit from mitosis until the mitotic spindle is correctly aligned. These checkpoints are complex and their elaborate spatiotemporal dynamics are challenging to understand intuitively. In this study, bistable mathematical models for both activation and silencing of mitotic checkpoints were constructed and analyzed. A one-parameter bifurcation was computed to show the realistic biochemical switches considering all signals. Numerical simulations involving systems of ODEs and PDEs were performed over various parameters, to investigate the effect of the diffusion coefficient. The results provide systems-level insights into mitotic transition and demonstrate that mathematical analysis constitutes a powerful tool for investigation of the dynamic properties of complex biomedical systems.

2018, 15(2): 441-460 doi: 10.3934/mbe.2018020 +[Abstract](2353) +[HTML](203) +[PDF](860.2KB)
Abstract:

In this paper, a network model has been proposed to control dengue disease transmission considering host-vector dynamics in \begin{document} $n$ \end{document} patches. The control of mosquitoes is performed by SIT. In SIT, the male insects are sterilized in the laboratory and released into the environment to control the number of offsprings. The basic reproduction number has been computed. The existence and stability of various states have been discussed. The bifurcation diagram has been plotted to show the existence and stability regions of disease-free and endemic states for an isolated patch. The critical level of sterile male mosquitoes has been obtained for the control of disease. The basic reproduction number for \begin{document} $n$ \end{document} patch network model has been computed. It is evident from numerical simulations that SIT control in one patch may control the disease in the network having two/three patches with suitable coupling among them.

2018, 15(2): 461-483 doi: 10.3934/mbe.2018021 +[Abstract](2773) +[HTML](141) +[PDF](2267.1KB)
Abstract:

Pair approximation models have been used to study the spread of infectious diseases in spatially distributed host populations, and to explore disease control strategies such as vaccination and case isolation. Here we introduce a pair approximation model of individual uptake of non-pharmaceutical interventions (NPIs) for an acute self-limiting infection, where susceptible individuals can learn the NPIs either from other susceptible individuals who are already practicing NPIs ("social learning"), or their uptake of NPIs can be stimulated by being neighbours of an infectious person ("exposure learning"). NPIs include individual measures such as hand-washing and respiratory etiquette. Individuals can also drop the habit of using NPIs at a certain rate. We derive a spatially defined expression of the basic reproduction number \begin{document}$R_0$\end{document} and we also numerically simulate the model equations. We find that exposure learning is generally more efficient than social learning, since exposure learning generates NPI uptake in the individuals at immediate risk of infection. However, if social learning is pre-emptive, beginning a sufficient amount of time before the epidemic, then it can be more effective than exposure learning. Interestingly, varying the initial number of individuals practicing NPIs does not significantly impact the epidemic final size. Also, if initial source infections are surrounded by protective individuals, there are parameter regimes where increasing the initial number of source infections actually decreases the infection peak (instead of increasing it) and makes it occur sooner. The peak prevalence increases with the rate at which individuals drop the habit of using NPIs, but the response of peak prevalence to changes in the forgetting rate are qualitatively different for the two forms of learning. The pair approximation methodology developed here illustrates how analytical approaches for studying interactions between social processes and disease dynamics in a spatially structured population should be further pursued.

2018, 15(2): 485-505 doi: 10.3934/mbe.2018022 +[Abstract](3662) +[HTML](153) +[PDF](703.3KB)
Abstract:

In this paper an improved SEIR model for an infectious disease is presented which includes logistic growth for the total population. The aim is to develop optimal vaccination strategies against the spread of a generic disease. These vaccination strategies arise from the study of optimal control problems with various kinds of constraints including mixed control-state and state constraints. After presenting the new model and implementing the optimal control problems by means of a first-discretize-then-optimize method, numerical results for six scenarios are discussed and compared to an analytical optimal control law based on Pontrygin's minimum principle that allows to verify these results as approximations of candidate optimal solutions.

2018, 15(2): 507-522 doi: 10.3934/mbe.2018023 +[Abstract](2942) +[HTML](165) +[PDF](1150.6KB)
Abstract:

Translation is a central biological process by which proteins are synthesized from genetic information contained within mRNAs. Here, we investigate the kinetics of translation at the molecular level by a stochastic simulation model. The model explicitly includes RNA sequences, ribosome dynamics, the tRNA pool and biochemical reactions involved in the translation elongation. The results show that the translation efficiency is mainly limited by the available ribosome number, translation initiation and the translation elongation time. The elongation time is a log-normal distribution, with the mean and variance determined by the codon saturation and the process of aa-tRNA selection at each codon binding site. Moreover, our simulations show that the translation accuracy exponentially decreases with the sequence length. These results suggest that aa-tRNA competition is crucial for both translation elongation, translation efficiency and the accuracy, which in turn determined the effective protein production rate of correct proteins. Our results improve the dynamical equation of protein production with a delay differential equation that is dependent on sequence information through both the effective production rate and the distribution of elongation time.

2018, 15(2): 523-541 doi: 10.3934/mbe.2018024 +[Abstract](3194) +[HTML](236) +[PDF](717.9KB)
Abstract:

Dengue, malaria, and Zika are dangerous diseases primarily transmitted by Aedes aegypti, Aedes albopictus, and Anopheles stephensi. In the last few years, a new disease control method, besides pesticide spraying to kill mosquitoes, has been developed by releasing mosquitoes carrying bacterium Wolbachia into the natural areas to infect the wild population of mosquitoes and block disease transmission. The bacterium is transmitted by infected mothers and the maternal transmission was assumed to be perfect in virtually all previous models. However, recent experiments on Aedes aegypti and Anopheles stephensi showed that the transmission can be imperfect. In this work, we develop a model to describe how the imperfect maternal transmission affects the dynamics of Wolbachia spread. We establish two useful identities and employ them to find sufficient and necessary conditions under which the system exhibits monomorphic, bistable, and polymorphic dynamics. These analytical results may help find a plausible explanation for the recent observation that the Wolbachia strain wMelPop failed to establish in the natural populations in Australia and Vietnam.

2018, 15(2): 543-567 doi: 10.3934/mbe.2018025 +[Abstract](3793) +[HTML](162) +[PDF](1368.6KB)
Abstract:

A multigroup model is developed to characterize brucellosis transmission, to explore potential effects of key factors, and to prioritize control measures. The global threshold dynamics are completely characterized by theory of asymptotic autonomous systems and Lyapunov direct method. We then formulate a multi-objective optimization problem and, by the weighted sum method, transform it into a scalar optimization problem on minimizing the total cost for control. The existence of optimal control and its characterization are well established by Pontryagin's Maximum Principle. We further parameterize the model and compute optimal control strategy for Inner Mongolia in China. In particular, we expound the effects of sheep recruitment, vaccination of sheep, culling of infected sheep, and health education of human on the dynamics and control of brucellosis. This study indicates that current control measures in Inner Mongolia are not working well and Brucellosis will continue to increase. The main finding here supports opposing unregulated sheep breeding and suggests vaccination and health education as the preferred necessary emergency intervention control. The policymakers must take a new look at the current control strategy, and, in order to control brucellosis better in Inner Mongolia, the governments have to preemptively press ahead with more effective measures.

2017  Impact Factor: 1.23