Plotting from a life expectancy table on a logarithmic scale versus gives a curve known as a survivorship curve. There are three general classes of survivorship curves, illustrated above. 1. Type I curves are typical of populations in which most mortality occurs among the elderly (e.g., humans in developed countries). 2. Type II curves occur when mortality is not dependent on age (e.g., many species of large birds and fish). For an infinite type II population, , but this cannot hold for a finite population. 3. Type III curves occur when juvenile mortality is extremely high (e.g., plant and animal species producing many offspring of which few survive). In type III populations, it is often true that for small . In other words, life expectancy increases for individuals who survive their risky juvenile period. ..
An SIR model is an epidemiological model that computes the theoretical number of people infected with a contagious illness in a closed population over time. The name of this class of models derives from the fact that they involve coupled equations relating the number of susceptible people , number of people infected , and number of people who have recovered . One of the simplest SIR models is the Kermack-McKendrick model.The Season 1 episode "Vector" (2005) of the television crime drama NUMB3RS features SIR models.
The differential equation describing exponential growth is(1)This can be integrated directly(2)to give(3)where . Exponentiating,(4)This equation is called the law of growth and, in a much more antiquated fashion, the Malthusian equation; the quantity in this equation is sometimes known as the Malthusian parameter.Consider a more complicated growth law(5)where is a constant. This can also be integrated directly(6)(7)(8)Note that this expression blows up at . We are given the initial condition that , so .(9)The in the denominator of (◇) greatly suppresses the growth in the long run compared to the simple growth law.The (continuous) logistic equation, definedby(10)is another growth law which frequently arises in biology. It has solution(11)
The parameter in the exponential equation of population growthwhere is the initial population size (at ) and is the elapsed time.
The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The model is continuous in time, but a modification of the continuous equation to a discrete quadratic recurrence equation known as the logistic map is also widely used.The continuous version of the logistic model is described by the differential equation(1)where is the Malthusian parameter (rate of maximum population growth) and is the so-called carrying capacity (i.e., the maximum sustainable population). Dividing both sides by and defining then gives the differential equation(2)which is known as the logistic equation and has solution(3)The function is sometimes known as the sigmoid function.While is usually constrained to be positive, plots of the above solution are shown for various positive and negative values of and initial conditions ranging from 0.00 to 1.00..
An table is a tabulation of numbers which is used to calculate life expectancies.010002001.000.200.902.702.7018001000.800.120.751.802.2527002000.700.290.601.051.5035003000.500.600.350.450.9042002000.201.000.100.100.505000.00--0.000.00--10002.70: Age category (, 1, ..., ). These values can be in any convenient units, but must be chosen so that no observed lifespan extends past category . : Census size, defined as the number of individuals in the study population who survive to the beginning of age category . Therefore, (the total population size) and . : ; . Crude death rate, which measures the number of individuals who die within age category . : . Survivorship, which measures the proportion of individuals who survive to the beginning of age category . : ; . Proportional death rate, or "risk," which measures the proportion of individuals surviving to the beginning of age category who die within that category...