What Influence Could the Acceptance of Visitors Cause on the Epidemic Dynamics of a Reinfectious Disease?: A Mathematical Model.

The globalization in business and tourism becomes crucial more and more for the economical sustainability of local communities. In the presence of an epidemic outbreak, there must be such a decision on the policy by the host community as whether to accept visitors or not, the number of acceptable visitors, or the condition for acceptable visitors. Making use of an SIRI type of mathematical model, we consider the influence of visitors on the spread of a reinfectious disease in a community, especially assuming that a certain proportion of accepted visitors are immune. The reinfectivity of disease here means that the immunity gained by either vaccination or recovery is imperfect. With the mathematical results obtained by our analysis on the model for such an epidemic dynamics of resident and visitor populations, we find that the acceptance of visitors could have a significant influence on the disease's endemicity in the community, either suppressive or supportive.

An epidemic dynamics model with limited isolation capacity.

We consider a modified SIR model with a four-dimensional system of ordinary differential equations to consider the influence of a limited isolation capacity on the final epidemic size defined as the total number of individuals who experienced the disease at the end of an epidemic season. We derive the necessary and sufficient condition that the isolation reaches the capacity in a finite time on the way of the epidemic process, and show that the final epidemic size is monotonically decreasing in terms of the isolation capacity. We find further that the final epidemic size could have a discontinuous change at the critical value of isolation capacity below which the isolation reaches the capacity in a finite time. Our results imply that the breakdown of isolation with a limited capacity would cause a drastic increase of the epidemic size. Insufficient capacity of the isolation could lead to an unexpectedly severe epidemic situation, while such a severity would be avoidable with the sufficient isolation capacity.

A type IV functional response with different shapes in a predator-prey model.

Group defense is a phenomenon that occurs in many predator-prey systems. Different functional responses with substantially different properties representing such a mechanism exist. Here, we develop a functional response using timescale separation. A prey-dependent catch rate represents the group defense. The resulting functional response contains a single parameter that controls whether the group defense functional response is saturating or dome-shaped. Based on that, we show that the catch rate must not increase monotonically with increasing prey density to lead to a dome-shaped functional response. We apply bifurcation analysis to show that non-monotonic group defense is usually more successful. However, we also find parameter regions in which a paradox occurs. In this case, higher group defense can give rise to a stable limit cycle, while for lower values, the predator would go extinct. The study does not only provide valuable insight on how to include functional responses representing group defense in mathematical models, but it also clarifies under which circumstances the usage of different functional responses is appropriate.

An SIS model for the epidemic dynamics with two phases of the human day-to-day activity.

An SIS model is analyzed to consider the contribution of community structure to the risk of the spread of a transmissible disease. We focus on the human day-to-day activity introduced by commuting to a central place for the social activity. We assume that the community is classified into two subpopulations: commuter and non-commuter, of which the commuter has two phases of the day-to-day activity: private and social. Further we take account of the combination of contact patterns in two phases, making use of mass-action and ratio-dependent types for the infection force. We investigate the dependence of the basic reproduction number on the commuter ratio and the daily expected duration at the social phase as essential factors characterizing the community structure, and show that the dependence is significantly affected by the combination of contact patterns, and that the difference in the commuter ratio could make the risk of the spread of a transmissible disease significantly different.

A Population Dynamics Model of Mosquito-Borne Disease Transmission, Focusing on Mosquitoes' Biased Distribution and Mosquito Repellent Use.

We present an improved mathematical model of population dynamics of mosquito-borne disease transmission. Our model considers the effect of mosquito repellent use and the mosquito's behavior or attraction to the infected human, which cause mosquitoes' biased distribution around the human population. Our analysis of the model clearly shows the existence of thresholds for mosquito repellent efficacy and its utilization rate in the human population with respect to the elimination of mosquito-borne diseases. Further, the results imply that the suppression of mosquito-borne diseases becomes more difficult when the mosquitoes' distribution is biased to a greater extent around the human population.

A model for epidemic dynamics in a community with visitor subpopulation.

With a five dimensional system of ordinary differential equations based on the SIR and SIS models, we consider the dynamics of epidemics in a community which consists of residents and short-stay visitors. Taking different viewpoints to consider public health policies to control the disease, we derive different basic reproduction numbers and clarify their common/different mathematical natures so as to understand their meanings in the dynamics of the epidemic. From our analyses, the short-stay visitor subpopulation could become significant in determining the fate of diseases in the community. Furthermore, our arguments demonstrate that it is necessary to choose one variant of basic reproduction number in order to formulate appropriate public health policies.

Growth and cell cycle of Ulva compressa (Ulvophyceae) under LED illumination.

The cell-cycle progression of Ulva compressa is diurnally gated at the G1 phase in accordance with light-dark cycles. The present study was designed to examine the spectral sensitivity of the G1 gating system. When blue, red, and green light-emitting diodes (LEDs) were used for illumination either alone or in combination, the cells divided under all illumination conditions, suggesting that all colors of light were able to open the G1 gate. Although blue light was most effective to open the G1 gate, red light alone or green light alone was also able to open the G1 gate even at irradiance levels lower than the light compensation point of each color. Occurrence of a period of no cell division in the course of a day suggested that the G1 gating system normally functioned as under ordinary illumination by cool-white fluorescent lamps. The rise of the proportion of blue light to green light resulted in increased growth rate. On the other hand, the growth rates did not vary regardless of the proportion of blue light to red light. These results indicate that the difference in growth rate due to light color resulted from the difference in photosynthetic efficiency of the colors of light. However, the growth rates significantly decreased under conditions without blue light. This result suggests that blue light mediates cell elongation and because the spectral sensitivity of the cell elongation regulating system was different from that of the G1 gating system, distinct photoreceptors are likely to mediate the two systems.

A mathematical model of population dynamics for Batesian mimicry system.

We analyse a mathematical model of the population dynamics among a mimic, a corresponding model, and their common predator populations. Predator changes its search-and-attack probability by forming and losing its search image. It cannot distinguish the mimic from the model. Once a predator eats a model individual, it comes to omit both the model and the mimic species from its diet menu. If a predator eats a mimic individual, it comes to increase the search-and-attack probability for both model and mimic. The predator may lose the repulsive/attractive search image with a probability per day. By analysing our model, we can derive the mathematical condition for the persistence of model and mimic populations, and then get the result that the condition for the persistence of model population does not depend on the mimic population size, while the condition for the persistence of mimic population does depend the predator's memory of search image.

Native intra- and inter-specific reactions may cause the paradox of pest control with harvesting.

We analyse a general time-discrete mathematical model of host-parasite population dynamics with harvesting, in which the host can be regarded as a pest. We harvest a portion of the host population at a moment in each year. Our model involves the density effect on the host population. We investigate the condition in which the harvesting of the host results in a paradoxical increase of its equilibrium population size. Our results imply that for a family of pest-enemy systems, the paradox of pest control could be caused essentially by the interspecific relationship and the intraspecific density effect.

A paradox in discrete single species population dynamics with harvesting/thinning.

We analyze a general time-discrete mathematical model of single species population dynamics with the intraspecific density effect and the harvesting/thinning effect. We harvest a portion of the population at a moment in each year. We investigate the condition under which the harvesting/thinning causes an eventual increase of its population at the equilibrium, and show that such a paradoxical increase could occur for the discrete single species population dynamics with a large family of density effect functions. Some typical models are analyzed in detail according to the possibility of the paradox emergence. Our result implies that the contest competition would never cause the paradox, while the scramble competition would be likely to cause it.

Ecological balance in the native population dynamics may cause the paradox of pest control with harvesting.

We analyze a time-discrete mathematical model of host-parasite population dynamics with harvesting, in which the host can be regarded as a pest. We harvest a portion of the host population at a moment in each parasitism season. The principal target of the harvesting is the host; however, the parasite population may also be affected and reduced by a portion. Our model involves the Beverton-Holt type density effect on the host population. We investigate the condition in which the harvesting of the host results in an eventual increase of its equilibrium population size, analytically proving that the paradoxical increase could occur even when the harvesting does not directly affect the parasite population at all. We show that the paradox of pest control could be caused essentially by the interspecific relationship and the intraspecific density effect.

A mathematical model on the optimal timing of offspring desertion.

We consider the offspring desertion as the optimal strategy for the deserter parent, analyzing a mathematical model for its expected reproductive success. It is shown that the optimality of the offspring desertion significantly depends on the offsprings' birth timing in the mating season, and on the other ecological parameters characterizing the innate nature of considered animals. Especially, the desertion is less likely to occur for the offsprings born in the later period of mating season. It is also implied that the offspring desertion after a partially biparental care would be observable only with a specific condition.

A mathematical modelling for the cheliped regeneration with handedness in fiddler crab.

An enormously developed giant cheliped with the other small one characterizes the adult male fiddler crab. Some experiments with artificial severances of cheliped indicate that such a handedness in the cheliped size is maintained even after the regeneration of severed cheliped. Other experimental researches give some results about an unknown physiological system which controls the emergence and the regeneration of the handedness in the cheliped size. In this paper, with two hypothesized factors relevant to the regeneration of a severed cheliped, we propose a simple mathematical model to describe the experimental result about the cheliped regeneration with a handedness after the cheliped severance for the fiddler crab. Our model gives a suggestion about an underlying system for the cheliped regeneration in the fiddler crab or some other crustacean species.

A mathematical consideration for the optimal shell change of hermit crab.

Shell of the adult hermit crab has some important roles for its fitness. In the same time, the shell size often limits the body growth of its owner. To grow the body size larger, the individual must change the shell to another larger shell. If the individual cannot get another larger one, the individual has to suppress the body size growth as the occupied shell size allows. Growth suppression would result in the lower fitness. With a simple mathematical model, we consider the criterion about whether the individual should try to change the shell or not in order to get the higher fitness. We show that the optimality of a shell change behavior has a relation with the body size and the season length for the shell change. They also affect the optimal timing for the shell change. It is implied that the probability of the success in a shell change and the cost for the shell change behavior do not affect the optimal timing for the shell change at all but significantly do the optimality of the behavioral choice.

A population dynamic model for facultative agamosperms.

Plants that can reproduce both sexually and agamically are called facultative apomicts. Some species, such as Taraxacum, contain both sexual diploids and triploid facultative apomicts. Triploids produce seeds without gamete fusion and recombination, and can also produce pollen and fertilize diploids. We present a population dynamic model that deals with gene flow and competition between diploids and triploids, with differing allocation towards reproductive investment in seeds and pollen. This paper examines whether diploids and triploids of plants with facultative agamospermy can coexist within a single population. We analyse the global behavior of such a dynamic system. Features of the system are significantly affected by the germination rates of diploids and triploids. Either diploids or triploids persist alone when the germination rate of diploids is sufficiently larger or smaller than that of triploids, respectively. Competitive exclusion occurs when both germination rates are sufficiently large. Coexistence is possible under certain specific conditions when: (I) the germination rates of both diploid sexuals and triploids are not sufficiently large, and (II) triploids produce sufficient pollen. When diploid sexuals and triploids coexist, triploids cannot exist alone, implying that the pollen of triploids is necessary to exploit diploid ovules.