Transition from growth to decay of an epidemic due to lockdown

We study the transition of an epidemic from growth phase to decay of the active infections in a population when lockdown health measures are introduced to reduce the probability of disease transmission. Although in the case of uniform lockdown, a simple compartmental model would indicate instantaneous transition to decay of the epidemic, this is not the case when partially isolated active clusters remain with the potential to create a series of small outbreaks. We model this using the Gillespie stochastic simulation algorithm based on a connected set of stochastic susceptible-infected-removed/recovered networks representing the locked-down majority population (in which the reproduction number is less than 1) weakly coupled to a large set of small clusters in which the infection may propagate. We find that the presence of such active clusters can lead to slower than expected decay of the epidemic and significantly delayed onset of the decay phase. We study the relative contributions of these changes, caused by the active clusters within the population, to the additional total infected population. We also demonstrate that limiting the size of the inevitable active clusters can be efficient in reducing their impact on the overall size of the epidemic outbreak. The deceleration of the decay phase becomes apparent when the active clusters form at least 5% of the population.     

应加强对单细胞生物分批培养生长曲线中减速期的认识和教学

根据单细胞生物分批培养过程中比生长速率(μ)的变化,其生长曲线分为延滞期、加速期、对数期、减速期、稳定期和衰亡期6个时期.与其他生长时期相比,在减速期生物的生长、基质的利用、产物的合成和基因表达谱等方面有显著的不同,并对发酵生产有着重要作用.然而,长期以来,对减速期的认识和教学相当薄弱,亟需加强对减速期的认识和教学.     

A Model for Growth of a Single Fungal Hypha Based on Well-Mixed Tanks in Series: Simulation of Nutrient and Vesicle Transport in Aerial Reproductive Hyphae

Current models that describe the extension of fungal hyphae and development of a mycelium either do not describe the role of vesicles in hyphal extension or do not correctly describe the experimentally observed profile for distribution of vesicles along the hypha. The present work uses the n-tanks-in-series approach to develop a model for hyphal extension that describes the intracellular transport of nutrient to a sub-apical zone where vesicles are formed and then transported to the tip, where tip extension occurs. The model was calibrated using experimental data from the literature for the extension of reproductive aerial hyphae of three different fungi, and was able to describe different profiles involving acceleration and deceleration of the extension rate. A sensitivity analysis showed that the supply of nutrient to the sub-apical vesicle-producing zone is a key factor influencing the rate of extension of the hypha. Although this model was used to describe the extension of a single reproductive aerial hypha, the use of the n-tanks-in-series approach to representing the hypha means that the model has the flexibility to be extended to describe the growth of other types of hyphae and the branching of hyphae to form a complete mycelium.     

Change in hyphal morphology of Aspergillus oryzae during fed-batch cultivation

Industrial enzymes are often produced by filamentous fungi in fed-batch cultivations. During cultivation, the different morphological forms displayed by the fungi have an impact on the overall production. The morphology of a recombinant lipase producing Aspergillus oryzae strain was investigated during fed-batch cultivations. During the exponential batch phase of the fed-batch cultivations, the average hyphal length increased as did the number of tips per hyphal element. Most striking was the finding that the diameter of the hyphal elements increased with an average factor of 1.5 during the batch phase from 2.8–2.9 up to 4.0–4.4 μm. The diameter of the hyphal elements remained constant, around 4 μm, after the feed was started. However, the diameter of the immediate hyphal tip, where the enzyme secretion is thought to take place, increased dramatically with up to a factor 2.5 during the feeding period. The expression of the recombinant lipase was induced by the feeding with maltose, and an increase in lipase activity was seen in parallel to a swelling of the tips. The results indicate that the two events are linked as a return to normal growth was observed upon cessation of production due to oxygen limitations.     

Morphology and clavulanic acid production of Streptomyces clavuligerus: Effect of stirrer speed in batch fermentations

Streptomyces clavuligerus ATCC 20764 was grown from spore-inocuia on a glycerol, malt extract, bacteriological peptone medium in 5-L batch fermentations at 490, 990, and 1300 rpm. Dry cell weights, clavulanic acid production, and the morphological parameters main hyphal length, total hyphal length, number of tips, and hyphal growth unit were measured. Growth and productivity were hardly dependent on stirrer speed. After early growth fragmentation of long, highly branched mycelia to shorter, less branched fragments occurred. This was followed by regrowth and, at 1300 rpm, a second fragmentation phase. The effect of increasing stirrer speed was to accelerate the initial fragmentation phase. It was clearly possible to obtain the same biomass concentration and clavulanic acid liter, with different morphologies depending on stirrer speed. This shows that for this fermentation at least there is no direct link between morphology and productivity and, hence, that it might be possible to manipulate them independently to improve fermentor performance.     

A model for hyphal growth and branching.

A mathematical model for hyphal growth and branching is described which relates cytological events within hyphae to mycelial growth kinetics. Essentially the model quantifies qualitative theories of hyphal growth in which it is proposed that vesicles containing wall precursors and/or enzymes required for wall synthesis are generated at a constant rate throughout a mycelium and travel to the tips of hyphae where they fuse with the plasma membrane, liberating their contents into the wall and increasing the surface area of the hypha to give elongation. The hypothesis that there is a duplication cycle in hyphae which is equivalent to the cell cycle observed in unicellular micro-organisms is also included in the model. Predictions from the model are compared with experimentally observed growth kinetics of mycelia of Geotrichum candidum and Aspergillus nidulans. The finite difference model which was constructed is capable of predicting changes in hyphal length and in the number and positions of branches and septa on the basis of changes in vesicle and nuclear concentration. Predictions were obtained using the model which were in good agreement with experimentally observed data.     

High cell density culture of Anabaena variabilis with controlled light intensity and nutrient supply

Controlling the light energy and major nutrients is important for high cell density culture of cyanobacterial cells. The growth phase of Anabaena variabilis can be divided into an exponential growth phase and a deceleration phase. In this study, the cell growth in the deceleration phase showed a linear growth pattern. Both the period of the exponential growth phase and the average cell growth rate in the deceleration phase increased by controlling the light intensity. To control the light intensity, the specific irradiation rate was maintained above 10 micromol/s/g dry cell by increasing the incident light intensity stepwise. The final cell density increased by controlling the nutrient supply. For the control of the nutrient supply, nitrate, phosphate, and sulfate were intermittently added based on the growth yield, along with the combined control of light intensity and nutrient concentration. Under these control conditions, both final cell concentration and cell productivity increased, to 8.2 g/l and 1.9 g/l/day, respectively.     

Morphology and physiology of an alpha-amylase producing strain of Aspergillus oryzae during batch cultivations

The microscopic morphology, that is, total hyphal length and total number of tips, has been characterized during batch cultivations of Aspergillus oryzae. The specific growth rate estimated by measuring the total hyphal length (mu(h)) corresponds well with the specific growth rate estimated from dry weight measurements during cultures grown as free hyphal elements. The average tip extension rate can be described with a saturation type kinetics with respect to the average total hyphal length, and the branching frequency is closely related to the total hyphal length. For the applied strain of A. oryzae, pellet formation occurs by coagulation of spores. The agglomeration process is pH dependent and pellets are formed at pH values higher than 5, whereas low pH (<3.5) results in growth as freely dispersed hyphal elements. The maximum specific growth rate has a broad pH optimum between 3 and 7, whereas the alpha-amylase production has a sharper maximum at about pH 6. During batch cultivation with pellets the growth is described well by the cube-root law when pellet fragmentation can be neglected. The kinetic parameter k in the cube-root law is derived from the growth kinetics with no mass transfer limitation, k = mu(h)/3. Based on an oxygen balance, the active growth layer in the pellet is estimated to be 200 to 325 mum and, consequently, up to 50% of the biomass is limited by oxygen for large pellets. Ethanol production (up to 1 g L(-1)) was observed during batch cultivations with pellets, suggesting that ethanol is produced in the oxygen limited part of the biomass. A constitutive, low alpha-amylase production was observed at high glucose concentration. The specific alpha-amylase production was significantly higher for filamentous growth than for pellets and oxygen appears to be necessary for production of alpha-amylase. (c) 1996 John Wiley & Sons, Inc.     

Time course of auxin stimulations of growth

Measurements of the time course of growth responses of corn coleoptile sections to pulses of auxin (10⁻⁵m indoleacetic acid) establish that the growth rate changes in a regular pattern around the auxin pulse: a latent phase of 12 to 15 minutes is followed by an acceleration of growth rate lasting 15 to 20 minutes, after which a fairly steady rate is maintained. When the auxin source is withdrawn, there is an after-effect of about 15 minutes followed by a decay of growth rate, which reaches 50% decay after a further 15 to 40 minutes. The decay phase appears to be a function of the transport of auxin out of the sections. The 50% decay of growth for single cells is estimated at 30 minutes from the time of withdrawal of an exogenous supply of auxin. The regulation of growth by auxin is rapidly imposed or dissipated as auxin enters and exits, respectively, suggesting a facile association and disassociation of auxin with a growth-limiting site in the cell. It is proposed that the growth-stimulated state is dissipated at once when the transportable auxin has passed out of the cell.     

Kinetics of human lymphocyte responses in vitro: the phase of exponential growth

The kinetics of thymidine uptake in human peripheral lymphocytes stimulated by allogenic cells, antigen E (ragweed allergen) and a variety of mitogens can generally be divided into four consecutive phases. First, a lag period with no increase in thymidine uptake, then a short period of rapid change in uptake, followed by a log-linear growth period and finally a decay phase. In this report we examine in detail the characteristics of the third, log-linear growth phase. Since, as discussed in the preceding paper, thymidine uptake is proportional to the number of cells acumulating thymidine, we can calculate from the log-linear growth period an apparent doubling time. We show that for five different stimulating agents the cells reach a log-linear growth phase of varying length and that the doubling times show little variation. This invariance indicates that, despite possible variation in cell death and recruitment rates, the rate of proliferation is in all cases dominated by the generation time of human lymphocytes.     

Dynamics of the long jump

A mechanical model is proposed which quantitatively describes the dynamics of the centre of gravity (c.g.) during the take-off phase of the long jump. The model entails a minimal but necessary number of components: a linear leg spring with the ability of lengthening to describe the active peak of the force time curve and a distal mass coupled with nonlinear visco-elastic elements to describe the passive peak. The influence of the positions and velocities of the supported body and the jumper's leg as well as of systemic parameters such as leg stiffness and mass distribution on the jumping distance were investigated. Techniques for optimum operation are identified: (1) There is a minimum stiffness for optimum performance. Further increase of the stiffness does not lead to longer jumps. (2) For any given stiffness there is always an optimum angle of attack. (3) The same distance can be achieved by different techniques. (4) The losses due to deceleration of the supporting leg do not result in reduced jumping distance as this deceleration results in a higher vertical momentum. (5) Thus, increasing the touch-down velocity of the jumper's supporting leg increases jumping distance.     

KINETICS OF HUMAN LYMPHOCYTE RESPONSES IN VITRO: THE PHASE OF EXPONENTIAL GROWTH

The kinetics of thymidine uptake in human peripheral lymphocytes stimulated by allogenic cells, antigen E (ragweed allergen) and a variety of mitogens can generally be divided into four consecutive phases. First, a lag period with no increase in thymidine uptake, then a short period of rapid change in uptake, followed by a log-linear growth period and finally a decay phase. In this report we examine in detail the characteristics of the third, log-linear growth phase. Since, as discussed in the preceding paper, thymidine uptake is proportional to the number of cells acumulating thymidine, we can calculate from the log-linear growth period an apparent doubling time. We show that for five different stimulating agents the cells reach a log-linear growth phase of varying length and that the doubling times show little variation. This invariance indicates that, despite possible variation in cell death and recruitment rates, the rate of proliferation is in all cases dominated by the generation time of human lymphocytes.     

Effect of irrigation systems on temporal distribution of malaria vectors in semi-arid regions

Previous research models have used climate data to explain habitat conditions of Anopheles mosquitoes transmitting malaria parasites. Although they can estimate mosquito populations with sufficient accuracy in many areas, observational data show that there is a tendency to underestimate the active growth and reproduction period of mosquitoes in semi-arid agricultural regions. In this study, a new, modified model that includes irrigation as a factor was developed to predict the active growing period of mosquitoes more precisely than the base model for ecophysiological and climatological distribution of mosquito generations (ECD-mg). Five sites with complete sets of observational data were selected in semi-arid regions of India for the comparison. The active growing period of mosquitoes determined from the modified ECD-mg model that incorporated the irrigation factor was in agreement with the observational data, whereas the active growing period was underestimated by the previous ECD-mg model that did not incorporate irrigation. This suggests that anthropogenic changes in the water supply due to extensive irrigation can encourage the growth of Anopheles mosquitoes through the alteration of the natural water balance in their habitat. In addition, it was found that the irrigation systems not only enable the active growth of mosquitoes in dry seasons but also play an important role in stabilizing the growth in rainy seasons. Consequently, the irrigation systems could lengthen the annual growing period of Anopheles mosquitoes and increase the maximum generation number of mosquitoes in semi-arid subtropical regions.     

Modeling surface growth of Escherichia coli on agar plates

Surface growth of Escherichia coli cells on a membrane filter placed on a nutrient agar plate under various conditions was studied with a mathematical model. The surface growth of bacterial cells showed a sigmoidal curve with time on a semilogarithmic plot. To describe it, a new logistic model that we presented earlier (H. Fujikawa et al., Food Microbiol. 21:501-509, 2004) was modified. Growth curves at various constant temperatures (10 to 34 degrees C) were successfully described with the modified model (model III). Model III gave better predictions of the rate constant of growth and the lag period than a modified Gompertz model and the Baranyi model. Using the parameter values of model III at the constant temperatures, surface growth at various temperatures was successfully predicted. Surface growth curves at various initial cell numbers were also sigmoidal and converged to the same maximum cell numbers at the stationary phase. Surface growth curves at various nutrient levels were also sigmoidal. The maximum cell number and the rate of growth were lower as the nutrient level decreased. The surface growth curve was the same as that in a liquid, except for the large curvature at the deceleration period. These curves were also well described with model III. The pattern of increase in the ATP content of cells grown on a surface was sigmoidal, similar to that for cell growth. We discovered several characteristics of the surface growth of bacterial cells under various growth conditions and examined the applicability of our model to describe these growth curves.     

Model for Branch Initiation in Aspergillus nidulans Based on Measurements of Growth Parameters

The rate of hyphal elongation and the number of branches per hypha were measured on short sporelings of Aspergillus nidulans growing at different rates. The rate of elongation was proportional to total length in unbranched and branched hyphae. At each growth rate, the number of branches per hypha increased with increasing length and gave approximately straight-line graphs when plotted against length. The average number of branches per unit of hyphal length was quite different for the various growth rates and increased in direct proportion to the growth rate. The results are interpreted to mean that (i) growing tips have a maximum rate at which they can elongate and which is reached at hyphal lengths characteristic of the particular growth rate and (ii) a new branch is formed when the capacity of the hypha to elongate exceeds that of the existing tips.     

Determination of proliferative parameters from growth curves

Abstract. Data on total cell numbers and proportions of non-proliferative cells during the development of a tissue can be obtained by experiment. This paper shows, via a simple mathematical model, that data collected over a period of time determine the time profile of two proliferative parameters. These are the rate per proliferative cell per day at which cells divide and the average number of proliferative daughters which result from the division of a proliferative cell. One dataset alone does not provide sufficient information to determine either time profile. The findings are applicable where there is no significant cell death during the growth period. The mathematical model is fitted to data from chick neural retina.