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Mathematical Biosciences & Engineering

December 2018 , Volume 15 , Issue 6

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Modeling crowd dynamics through coarse-grained data analysis
Sebastien Motsch, Mehdi Moussaïd, Elsa G. Guillot, Mathieu Moreau, Julien Pettré, Guy Theraulaz, Cécile Appert-Rolland and Pierre Degond
2018, 15(6): 1271-1290 doi: 10.3934/mbe.2018059 +[Abstract](8263) +[HTML](1946) +[PDF](2444.92KB)

Understanding and predicting the collective behaviour of crowds is essential to improve the efficiency of pedestrian flows in urban areas and minimize the risks of accidents at mass events. We advocate for the development of crowd traffic management systems, whereby observations of crowds can be coupled to fast and reliable models to produce rapid predictions of the crowd movement and eventually help crowd managers choose between tailored optimization strategies. Here, we propose a Bi-directional Macroscopic (BM) model as the core of such a system. Its key input is the fundamental diagram for bi-directional flows, i.e. the relation between the pedestrian fluxes and densities. We design and run a laboratory experiments involving a total of 119 participants walking in opposite directions in a circular corridor and show that the model is able to accurately capture the experimental data in a typical crowd forecasting situation. Finally, we propose a simple segregation strategy for enhancing the traffic efficiency, and use the BM model to determine the conditions under which this strategy would be beneficial. The BM model, therefore, could serve as a building block to develop on the fly prediction of crowd movements and help deploying real-time crowd optimization strategies.

Dynamical analysis for a hepatitis B transmission model with immigration and infection age
Suxia Zhang, Hongbin Guo and Robert Smith?
2018, 15(6): 1291-1313 doi: 10.3934/mbe.2018060 +[Abstract](6437) +[HTML](406) +[PDF](436.46KB)

Hepatitis B virus (HBV) is responsible for an estimated 378 million infections worldwide and 620, 000 deaths annually. Safe and effective vaccination programs have been available for decades, but coverage is limited due to economic and social factors. We investigate the effect of immigration and infection age on HBV transmission dynamics, incorporating age-dependent immigration flow and vertical transmission. The mathematical model can be used to describe HBV transmission in highly endemic regions with vertical transmission and migration of infected HBV individuals. Due to the effects of immigration, there is no disease-free equilibrium or reproduction number. We show that the unique endemic equilibrium exists only when immigration into the infective class is measurable. The smoothness and attractiveness of the solution semiflow are analyzed, and boundedness and uniform persistence are determined. Global stability of the unique endemic equilibrium is shown by a Lyapunov functional for a special case.

Modeling the control of infectious diseases: Effects of TV and social media advertisements
Arvind Kumar Misra, Rajanish Kumar Rai and Yasuhiro Takeuchi
2018, 15(6): 1315-1343 doi: 10.3934/mbe.2018061 +[Abstract](7272) +[HTML](589) +[PDF](1627.5KB)

Public health information through media plays an important role to curb the spread of various infectious diseases as most of the populations rely on what media projects to them. Social media and TV advertisements are important mediums to communicate people regarding the spread of any infectious disease and methods to prevent its spread. Therefore, in this paper, we propose a mathematical model to see how TV and social media advertisements impact the dynamics of an infectious disease. The susceptible population is assumed vulnerable to infection as well as information (through TV and social media ads). It is also assumed that the growth rate of TV and social media ads is proportional to the number of infected individuals with decreasing function of aware individuals. The feasibility of possible equilibria and their stability properties are discussed. It is shown that the increment in growth rate of TV and social media ads destabilizes the system and periodic oscillations arise through Hopf-bifurcation. It is also found that the increase in dissemination rate of awareness among susceptible population also gives rise interesting dynamics about the stability of endemic equilibrium and causes stability switch. It is observed that TV and social media advertisements regarding the spread of infectious diseases have the potential to bring behavioral changes among the people and control the spread of diseases. Numerical simulations also support analytical findings.

Early and late stage profiles for a chemotaxis model with density-dependent jump probability
Tianyuan Xu, Shanming Ji, Chunhua Jin, Ming Mei and Jingxue Yin
2018, 15(6): 1345-1385 doi: 10.3934/mbe.2018062 +[Abstract](6205) +[HTML](380) +[PDF](772.08KB)

In this paper, we derive a chemotaxis model with degenerate diffusion and density-dependent chemotactic sensitivity, and we provide a more realistic description of cell migration process for its early and late stages. Different from the existing studies focusing on the case of non-degenerate diffusion, this model with degenerate diffusion causes us some essential difficulty on the boundedness estimates and the propagation behavior of its compact support. In the presence of logistic damping, for the early stage before tumour cells spread to the whole domain, we first estimate the expanding speed of tumour region as \begin{document}$O(t^{β})$ \end{document} for \begin{document}$ 0 < β < \frac{1}{2}$ \end{document}. Then, for the late stage of cell migration, we further prove that the asymptotic profile of the original system is just its corresponding steady state. The global convergence of the original weak solution to the steady state with exponential rate \begin{document}$O(e^{-ct})$ \end{document} for some \begin{document}$c>0$ \end{document} is also obtained.

Ebola: Impact of hospital's admission policy in an overwhelmed scenario
Mondal Hasan Zahid and Christopher M. Kribs
2018, 15(6): 1387-1399 doi: 10.3934/mbe.2018063 +[Abstract](7587) +[HTML](381) +[PDF](565.3KB)

Infectious disease outbreaks sometimes overwhelm healthcare facilities. A recent case occurred in West Africa in 2014 when an Ebola virus outbreak overwhelmed facilities in Sierra Leone, Guinea and Liberia. In such scenarios, how many patients can hospitals admit to minimize disease burden? This study considers what type of hospital admission policy during a hypothetical Ebola outbreak can better serve the community, if overcrowding degrades the hospital setting. Our result shows that which policy minimizes loss to the community depends on the initial estimation of the control reproduction number, \begin{document}$R_0$\end{document}. When the outbreak grows extremely fast (\begin{document}$R_0$\end{document}\begin{document}$ \gg $\end{document}1) it is better (in terms of total disease burden) to stop admitting patients after reaching the carrying capacity because overcrowding in the hospital makes the hospital setting ineffective at containing infection, but when the outbreak grows only a little faster than the system's ability to contain it (\begin{document}$R_0 \gtrsim 1$\end{document}), it is better to admit patients beyond the carrying capacity because limited overcrowding still reduces infection more in the community. However, when \begin{document}$R_0$\end{document} is no more than a little greater than 1 (for our parameter values, 1.012), both policies result the same because the number of patients never exceeds the maximum capacity.

Dynamics of a stochastic delayed Harrison-type predation model: Effects of delay and stochastic components
Feng Rao, Carlos Castillo-Chavez and Yun Kang
2018, 15(6): 1401-1423 doi: 10.3934/mbe.2018064 +[Abstract](5996) +[HTML](1618) +[PDF](1231.06KB)

This paper investigates the complex dynamics of a Harrison-type predator-prey model that incorporating: (1) A constant time delay in the functional response term of the predator growth equation; and (2) environmental noise in both prey and predator equations. We provide the rigorous results of our model including the dynamical behaviors of a positive solution and Hopf bifurcation. We also perform numerical simulations on the effects of delay or/and noise when the corresponding ODE model has an interior solution. Our theoretical and numerical results show that delay can either remain stability or destabilize the model; large noise could destabilize the model; and the combination of delay and noise could intensify the periodic instability of the model. Our results may provide us useful biological insights into population managements for prey-predator interaction models.

Impact of behavioral change on the epidemic characteristics of an epidemic model without vital dynamics
Jianquan Li, Xiaoqin Wang and Xiaolin Lin
2018, 15(6): 1425-1434 doi: 10.3934/mbe.2018065 +[Abstract](6016) +[HTML](282) +[PDF](348.47KB)

The epidemic characteristics of an epidemic model with behavioral change in [V. Capasso, G. Serio, A generalizaition of the Kermack-McKendrick deterministic epidemic model, Math. Bios., 42 (1978), 43-61] are investigated, including the epidemic size, peak and turning point. The conditions on the appearance of the peak state and turning point are represented clearly, and the expressions determining the corresponding time for the peak state and turning point are described explicitly. Moreover, the impact of behavioral change on the characteristics is discussed.

Enhancement of chemotherapy using oncolytic virotherapy: Mathematical and optimal control analysis
Joseph Malinzi, Rachid Ouifki, Amina Eladdadi, Delfim F. M. Torres and K. A. Jane White
2018, 15(6): 1435-1463 doi: 10.3934/mbe.2018066 +[Abstract](6564) +[HTML](324) +[PDF](704.6KB)

Oncolytic virotherapy has been emerging as a promising novel cancer treatment which may be further combined with the existing therapeutic modalities to enhance their effects. To investigate how virotherapy could enhance chemotherapy, we propose an ODE based mathematical model describing the interactions between tumour cells, the immune response, and a treatment combination with chemotherapy and oncolytic viruses. Stability analysis of the model with constant chemotherapy treatment rates shows that without any form of treatment, a tumour would grow to its maximum size. It also demonstrates that chemotherapy alone is capable of clearing tumour cells provided that the drug efficacy is greater than the intrinsic tumour growth rate. Furthermore, virotherapy alone may not be able to clear tumour cells from body tissue but would rather enhance chemotherapy if viruses with high viral potency are used. To assess the combined effect of virotherapy and chemotherapy we use the forward sensitivity index to perform a sensitivity analysis, with respect to chemotherapy key parameters, of the virus basic reproductive number and the tumour endemic equilibrium. The results from this sensitivity analysis indicate the existence of a critical dose of chemotherapy above which no further significant reduction in the tumour population can be observed. Numerical simulations show that a successful combinational therapy of the chemotherapeutic drugs and viruses depends mostly on the virus burst size, infection rate, and the amount of drugs supplied. Optimal control analysis was performed, by means of the Pontryagin's maximum principle, to further refine predictions of the model with constant treatment rates by accounting for the treatment costs and sides effects. Results from this analysis suggest that the optimal drug and virus combination correspond to half their maximum tolerated doses. This is in agreement with the results from stability and sensitivity analyses.

State feedback impulsive control of computer worm and virus with saturated incidence
Meng Zhang, Kaiyuan Liu, Lansun Chen and Zeyu Li
2018, 15(6): 1465-1478 doi: 10.3934/mbe.2018067 +[Abstract](6491) +[HTML](312) +[PDF](564.37KB)

A state feedback impulsive model is set up to discuss the spreading and control of the computer worm and virus. Considering the transmission features, saturated infectious is adopted to describe the spreading in the model, and all the treatment measures, such as patching operating system and updating antivirus software, are assumed to take effect instantly. Then the model is analyzed with a novel method, and the existence and stability of order-1 limit cycle are discussed. Finally, the numerical simulation is listed to verify the result of the paper.

Coexistence of a cross-diffusive West Nile virus model in a heterogenous environment
Abdelrazig K. Tarboush, Jing Ge and Zhigui Lin
2018, 15(6): 1479-1494 doi: 10.3934/mbe.2018068 +[Abstract](5822) +[HTML](373) +[PDF](711.34KB)

This paper is concerned with a strongly-coupled elliptic system, which describes a West Nile virus (WNv) model with cross-diffusion in a heterogeneous environment. The basic reproduction number is introduced through the next generation infection operator and some related eigenvalue problems. The existence of coexistence states is presented by using a method of upper and lower solutions. The true positive solutions are obtained by monotone iterative schemes. Our results show that a cross-diffusive WNv model possesses at least one coexistence solution if the basic reproduction number is greater than one and the cross-diffusion rates are small enough, while if the basic reproduction number is less than or equal to one, the model has no positive solution. To illustrate the impact of cross-diffusion and environmental heterogeneity on the transmission of WNv, some numerical simulations are given.

Review of stability and stabilization for impulsive delayed systems
Xueyan Yang, Xiaodi Li, Qiang Xi and Peiyong Duan
2018, 15(6): 1495-1515 doi: 10.3934/mbe.2018069 +[Abstract](10140) +[HTML](549) +[PDF](2010.21KB)

This paper reviews some recent works on impulsive delayed systems (IDSs). The prime focus is the fundamental results and recent progress in theory and applications. After reviewing the relative literatures, this paper provides a comprehensive and intuitive overview of IDSs. Five aspects of IDSs are surveyed including basic theory, stability analysis, impulsive control, impulsive perturbation, and delayed impulses. Then the research prospect is given, which provides a reference for further study of IDSs theory.

2018 Impact Factor: 1.313




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