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By V. Gelford. Georgetown University. 2019.

The same models were used to estimate num- culate regional estimates for total goiter rates generic nasonex nasal spray 18 gm on-line allergy treatment 4 addiction. Most The Burden of Disease and Mortality by Condition: Data generic nasonex nasal spray 18gm without a prescription allergy testing qmc, Methods buy nasonex nasal spray 18gm amex allergy testing procedure codes, and Results for 2001 | 81 studies of diabetes prevalence did not indicate the type of were excluded from prevalence estimates. Point prevalence estimates for dependence and harmful use, excluding cases with comor- episodes of unipolar major depression were derived from a bid depressive episode. All available population-based surveys population studies on depressive disorders, which identified using diagnostic criteria that could be mapped to this case 56 studies from all World Bank regions (Ustun and others definition were identified. Variations in the prevalence of unipolar depressive prevalence of alcohol use disorders were obtained from disorders in some European countries, Australia, Japan, and 55 studies (Mathers and Ayuso-Mateos 2003). New Zealand were estimated directly from relevant popula- Published data on alcohol production, trade, and sales, tion studies (Ayuso-Mateos and others 2001). For other high- adjusted for estimates of illegally produced alcohol, were income European countries, country-specific prevalences used to estimate country averages of the volume of alcohol were estimated using a regression model of available preva- consumed. These preliminary estimates were then further lence data on suicide rates (for ages 15 to 59, both sexes com- adjusted on the basis of survey data on alcohol consumption bined). For other regions,prevalence estimates were based on to estimate the prevalence of alcohol use disorders for coun- regional prevalence rates applied to country-specific popula- tries where recent population-based survey data were not tion estimates for 2002. This resulted in an overall disability weight for as does the quality of data collected. This compares reasonably well with a more recent dependence and harmful use or cocaine dependence and analysis of the distribution of depression by severity and dis- harmful use, excluding cases with comorbid depressive ability weights for a Dutch community, which resulted in an episodes. Data on the prevalence of problematic illicit drug overall disability weight of 0. A literature search was conducted of all studies episodes were estimated separately using the disability weight that estimated the prevalence of problematic drug use and for mild depressive disorders. Other data sources Subregional prevalence rates for bipolar disorder were included the United Nations Drug Control Program and the derived from a systematic review of all available published European Monitoring Centre for Drugs and Drug Addiction. Persons with comorbid lence rates for panic disorder, obsessive-compulsive disor- depressive disorder or alcohol or drug use disorders were der, and post-traumatic stress disorder were also derived excluded from the prevalence estimates. Those with comor- ondary to other diseases or injury, were derived from sys- bid depressive disorder or alcohol or drug use disorders tematic reviews of available published and unpublished 82 | Global Burden of Disease and Risk Factors | Colin D. For countries for which no data recent epidemiological studies (Warren and Warren 2001). The prevalence rates, incidence rates, and durations for DisMod software was then used to obtain internally consis- Alzheimer’s disease and other dementias were estimated tent age- and sex-specific estimates of incidence, prevalence, based on 110 available population studies and assumed to remission, and relative risks of mortality. Ratios of blindness apply to countries within each subregion (Mathers and to low vision for each region were used to estimate the preva- Leonardi 2003). Regional incidence to mortality rates for Parkinson’s disease estimated by Murray and Lopez Hearing Loss. Despite the number of published studies on (1996d) were used to derive country-specific estimates for hearing loss, many of them use different criteria and relate incidence from the estimated country-specific mortality rates. Migraine has been ing threshold level in the better ear is 41 decibels or greater treated as a chronic disease lasting from 15 years to around averaged over 0. The case definition was or greater hearing loss (hearing threshold level in the better taken from the International Headache Society’s definition ear is 61 decibels or greater averaged over 0. Regional tion provided prevalence estimates that were quite similar estimates of the prevalence of hearing aid use were used in across most regions. For details of methods and data sources see Fewtrell and others (2004) and Pruss- Angina Pectoris. Both regional and subregional the prevalence and case fatality rates for angina pectoris prevalences for blindness and low vision were updated using (Mathers, Truelson, and others 2004). Observed correlations all available data gathered since 1980 (Resnikoff and others between the prevalence of acute myocardial infarction sur- 2004; Thylefors and others 1995). Subregional prevalences vivors and the prevalence of angina pectoris (whether inci- were estimated from more than 50 cross-sectional, dent before or after acute myocardial infarction) were used The Burden of Disease and Mortality by Condition: Data, Methods, and Results for 2001 | 83 to estimate the prevalence of angina pectoris from the mod- populations based on spirometry were available, both direct eled prevalences of acute myocardial infarction survivors. Asthma prevalence estimates were based on a case rates for acute myocardial infarction. Because accurate prevalence A total of 149 population-based studies were used to data based on spirometry are not available in many regions, derive estimates of asthma prevalence for a wide range of an alternative approach was used to infer disease occurrence countries for children, teenagers, and adults. The relative risk of mortality due to chron- European Community Respiratory Health Survey of adults ic obstructive pulmonary disease across subregions was esti- ages 20 to 44 using self-reported symptoms and bronchial mated as a function of its two leading risk factors—tobacco hyper-responsiveness (Chinn and others 1997; Pearce and smoking and indoor air pollution from solid fuel used for others 2000). Estimates from the population-based studies cooking—along with regional fixed effects (Lopez and oth- were then used to derive subregional average prevalence ers forthcoming). Data on risk factors were derived from the rates, which were assumed to apply in countries without comparative risk assessment carried out for the World specific population studies. Subregional prevalence rates for estimated regional prevalence with data from available pop- rheumatoid arthritis were derived from available published ulation studies. For regions where surveys of representative population studies using case definitions for definite or 84 | Global Burden of Disease and Risk Factors | Colin D. Subregional prevalence rates for in determining the overall health status of populations in all osteoarthritis were derived from available published popu- regions of the world. Prevalence numbers were based on regional causes dominates the overall burden of nonfatal disabling prevalence rates for edentulism estimated by Murray and conditions. The disabling burden of neuropsychiatric condi- tions is almost the same for males and females, but the major contributing causes are different. While depression is Injuries the leading cause for both males and females, the burden of An incident episode of a nonfatal injury is defined as an depression is 50 percent higher for females than for males, episode that is severe enough for the person to be hospital- and females also have a higher burden from anxiety disor- ized or that requires emergency room care (if such care is ders, migraine, and senile dementias. In higher than that for females and accounts for one-quarter of brief, the incidence of nonfatal injuries by external cause the male neuropsychiatric burden. Adult-onset hearing loss is extremely prevalent, with of health facility data provided by 18 countries in five World more than 27 percent of men and 24 percent of women aged Bank regions. For most cause categories, extrapolations 45 and over experiencing mild hearing loss or greater. The total attributable burden of disability due to alcohol use is much larger (see chapter 4). Although healthy life lost through time spent in states of less than full the prevalences of disabling conditions such as dementia health. From 1991 to 1994, average, poor health resulted in a loss of nearly eight years of the risk of premature death increased by 50 percent for healthy life globally. This once again illustrates the importance of Latin America and the Caribbean taking nonfatal conditions into account, as well as deaths, Middle East and North Africa when assessing the causes of loss of health in populations. East Asia and Pacific In 2001, the leading causes of the burden of disease in low- and middle-income countries were broadly similar to South Asia those for the world as a whole (table 3. Between ed for 36 percent of the world’s total burden of disease and 1994 and 1998, life expectancy for males improved, but injury in 2001 and adults ages 15 to 59 accounted for almost declined again significantly between 1998 and 2001 (Men 50 percent. While the proportion of the total burden of disease stantially higher burden of noncommunicable disease than borne by adults ages 15 to 59 was the same in both groups of high-income countries (figure 3. Other uninten- top four causes of the burden of disease, four nonfatal condi- tional injuries and violence were the third and fourth Table 3. Low- and middle-income countries High-income countries around 85 percent in adults ages 15 and older,the proportion 0–4 in middle-income countries has already exceeded 70 percent.

If the store room is too hot or poorly ventilated cheap 18 gm nasonex nasal spray with amex allergy symptoms vs flu symptoms, improve air circulation by installing a fan or perforated airbrick cheap nasonex nasal spray 18 gm amex allergy testing redding ca. Many items order 18gm nasonex nasal spray with amex allergy medicine 003, such as laboratory reagents, are made in countries with cool climates so, for example, instructions to protect from excessive heat mean store below 25-30°C. Organise the store in a simple and logical way so that items can be found quickly and easily, using the following guidelines and the diagrams. Putting shelves all around the walls takes up a lot of space and wastes the space in the middle of the store. Section 2 Procurement and management of supplies and equipment 23 • Organise the stock into different sections for different categories of supplies, for example, drugs, dressings, instruments, medical stationery, equipment and spare parts, laboratory supplies, disinfectants. Keep a reserve stock of instruments that are used frequently, such as scissors, suture needles and forceps. Put items that have the latest expiry date at quickly and easily the back and items with the earliest expiry date at the front. Stock control Stock control is about the management of supplies in a health facility. Stock control includes keeping accurate and reliable records of stock received and issued, stock taking (checking stocks on a regular basis), and carrying out an inventory of stock at least once a year. A stock control system uses tools such as stock record cards and a stock control ledger. Effective stock Desk control is important to help you order the right quantities of supplies and equipment and, ideally, you should carry out a Open shelves and stock take before placing an storage for bulky items order for more supplies. Floor plan of a small medical storeroom 24 Section 2 Procurement and management of supplies and equipment Store records Every health facility needs a system for recording stock movement. Reliable record keeping is very important, because errors caused by poor record keeping will affect service delivery. For example, if stock records used to estimate requirements are not accurate, you may order too much or too little. Stock cards (stock record cards) and stock control ledgers are the simplest, and examples are provided below. They provide information about quantities received, issued and in stock at any time, can be used to calculate orders, and are a useful tool for preventing shortages and over-stocking (see Figure 2. Stock cards help supervisors to monitor overall consumption and use by different services, and to check stock levels, assess wastage and identify theft. Maximum level: Reserve stock level: Lead time: Minimum level: Order quantity: Date Received from/ No. Lead time, Order quantity, Minimum level, Maximum level, Reserve stock level: see Section 2. Calculate and record the new stock balance (old balance + quantity received or – quantity issued). Once a month the information on the stock cards is transferred to the stock control ledger. It is simpler to make an order using the summary in the stock control ledger than using all the individual stock cards. The stock control ledger is also a useful tool for analysing stock management and reviewing the accuracy of stock levels (see Figure 2. You can either obtain a stock control ledger from the district health team or make one yourself, using a separate page to keep records for each type of item. Unit/pack size: Order quantity: Date Quantity received Quantity used Balance Quantity to order Signature Example 2 Item: Code no. Unit/pack size: Order quantity: Order quantity: Date Previous count Amount Amount Present count Quantity to Signature (physical) received used (physical) order Practical tips for stock storage and record keeping ● Assign responsibility for stock control and the store room and develop written procedures. An organised store saves time when ordering or locating items and prevents stock from getting lost. This enables the person responsible for stock control to update the cards after every transaction (ordering, receiving and issuing stock). Write each transaction on a separate line, even if there is more than one transaction on the same day. Otherwise you may only discover that an item is incorrect, damaged or poor quality when it needs to be used and when it is too late to ask the supplier for a replacement. The following simple checks can avoid these problems and save time in the long term: • Check the delivery note, packing list and contents against a copy of the order. If the colour has changed or the temperature is incorrect, there may be a problem with the cold storage used during transport and you should report this to the supplier. Only accept items nearing the end of their shelf life if you are sure you can use them before the expiry date. The expiry date is the time up to which the manufacturer guarantees the quality of the product and many products, e. Report any problems to the supplier and the carrier immediately, explaining the nature of the problem, for example under-supply or damaged goods. When you unpack supplies, enter the details on the stock card and enter new items in the inventory. It is also important to keep a goods received record for equipment items, listing the supplier, date, invoice number and the serial number or other unique identification. Keep all equipment packaging materials in case you need to transport it again in future. Make sure at least two staff members receive and check supplies Section 2 Procurement and management of supplies and equipment 27 Issuing supplies Every health facility also needs a system for recording issue of supplies. The following information should be recorded every time an item is issued: date of issue, item and quantities issued, name of receiving service or individual, and the signature of the recipient. After issue, the receiving service or individual should be responsible for care of the item and accountable for loss or breakage. For example, microscope care should be the responsibility of the laboratory or the laboratory technician in charge. A stock take involves physically counting what is in stock and comparing the counted figures with the balance figures on the stock cards, checking expiry dates and the condition of stock. If there is a difference between the counted figures and the balance figures on the stock cards, you need to find out why. For example, stock may have been received or issued without being recorded or may have been stolen. If this is not possible because you order stocks very frequently then carry out a stock take at least three times a year. Inventory of stock An inventory is a list of non-expendable supplies and equipment that are kept at the health facility (see Figure 2. The person in charge of the health facility should keep a master copy of all items and update this list each time an item is received and issued. The person in charge of each service should keep an updated list of all the equipment and supplies they receive and include items damaged, broken or sent for repair. An inventory should be carried out at regular intervals (at least once a year) to check the condition and location of supplies and equipment in use and in stock.

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As a result purchase 18gm nasonex nasal spray visa allergy forecast brooklyn ny, there are an increasing number of studies that have utilized culture-independent methods to demonstrate that stressor exposure can affect more than just a few gut microbes; community-wide alterations of the gut microbiota have been demonstrated to occur in response to multiple types of stressors generic nasonex nasal spray 18gm mastercard allergy migraine. This was first realized in rats that were separated from their mothers for 3 h per day early in life (i discount 18gm nasonex nasal spray with amex allergy forecast arlington tx. Studies in this laboratory have also used culture-independent methods to assess the effects of stressor exposure on the intestinal microbiota [34, 35]. Studies Involving Prolonged Restraint Prolonged restraint is a widely used murine stressor that has been extensively characterized in the literature and is the most commonly used murine stressor in biomedical and biobehavioral research [36]. Exposure to the prolonged restraint stressor induces a physiological stress response that results in the elevation of endogenous corticosterone, epinephrine, and norepinephrine [36–39]. Thus, mice were exposed to the prolonged restraint stressor to determine the effects of the stress response on the stability of the intestinal microbiota. In this initial experiment, approximately 100,000 sequences from the cecal contents of 32 mice (approximately 3,000 sequences per mouse) were analyzed to characterize microbial diversity within the cecum. In microbial ecology, there are two primary measures of diversity, with α-diversity assessing diversity of 12 Influence of Stressor-Induced Nervous System Activation on the Intestinal. Hierarchical clustering ana- lyses indicated that the profile of the top ten most abundant bacterial types was significantly different in the mice exposed to 3, 5, or 7 days of restraint compared to profiles found in control animals [34]. Mice will not eat or drink while in restraining tubes, even if food and water is provided. Because changes in diet can have a profound impact on the microbiota [5, 40], a food and water deprived control group was included in the study. Mice that were restrained for one night had microbial profiles that were similar to food and water deprived control mice. This indicates that at least some of the effects of the stressor on the microbiota are due to food and water deprivation, but that repeated cycles of the stressor had additional effects on the microbiota that were not accounted for by food and water deprivation. In addition to changes in microbial community β-diversity, exposure to prolonged restraint also results in changes to α-diversity. Rarefaction analysis indicated that species diversity decreased with repeated cycles of restraint. This is important, because it is generally believed that loss of α-diversity leads to increased susceptibility to enteric infection [41]. Thus, it was hypothesized that mice exposed to the prolonged restraint stressor would have an increased susceptibility to enteric infection [34]. To test this hypothesis, mice were orally challenged with Citrobacter rodentium, which is a natural murine colonic pathogen, with patho- genesis and resulting colonic pathology that are nearly indistinguishable from that produced in humans infected with enteropathogenic E. As the infection progresses, the colonic inflammatory response resembles many aspects of the colitis found in patients with inflammatory bowel disease [44, 45]. Interestingly, exposing mice to six consecutive nights of prolonged restraint prior to oral challenge with C. However, exposing mice to the prolonged restraint stressor for 1 night prior to oral challenge with C. In some cases, neutrophilic inflammation extended from the mucosa to the submucosa and was frequently associated with epithelial erosion and ulceration [46]. However, simply expos- ing mice to the prolonged restraint stressor during oral challenge with C. Stressor exposure is well known to affect tight junctional protein expression and the permeability of intestinal tissue [47– 49]. Our study involving a colonic pathogen suggests that pairing stressor exposure and colonic infection can further degrade colonic epithelial barrier integrity [46]. It is not yet known whether stressor-induced alterations in the intestinal microbiota contribute to the enhancive effects of stressor exposure on C. Exposure to the prolonged restraint stressor reduces both relative and absolute levels of commensal L. It is further hypothesized that this internal environment leads to increased epithelial permeability and the translocation of pathogenic (as well as commensal) microbes from the lumen of the intestines to the interior of the body where they stimulate increases in inflammatory cytokines that alter the behavior of the host (Fig. Further studies are needed to confirm this hypothesis, and to determine whether commensal and probiotic microbes in addition to L. These disrupted homeostatic interactions lead to increases in suscep- tibility to intestinal infection and inflammation, and enhances epithelial barrier permeability and subsequent translocation from the lumen of the intestines to the interior of the body. The disruptions in epithelial barrier integrity lead to increases in circulating cytokines that have the capacity to change animal behavior and further stimulate the endocrine response. The hypothesis that alterations in the intestinal microbiota are responsible for these disrupted homeostatic interactions comes from data indicating that stressor exposure reduces beneficial microbes, such as bacteria in the genus Lactobacillus. Feeding mice lactobacilli to prevent the stressor-induced reduction in Lactobacillus spp. Social stressors often involve aggressive interactions between dominant and subordinate animals and are widely used to study the effects of stress on animal behavior and physiological functioning [51–54]. Social disruption involves aggressive interactions between a dominant intruder mouse (i. The aggressive interactions occur over a 2 h period at the beginning of the active cycle, when the aggressor is placed into the cage of the resident subordinate mice. The aggressor physically interacts with the residents for short periods of time until the residents display an upright defeat posture. Because the mice are housed together, the subordinate mice cannot escape and the aggres- sive intruder mouse will repeatedly attack and defeat the residents. Bailey response marked by elevated corticosterone [57, 58], epinephrine, and norepineph- rine [59]. These differ- ences were evident immediately after the last cycle of stressor exposure, as well as the morning following the last cycle of the stressor [35] indicating that the effects of the stressor occur rapidly in response to stressor exposure and can persist for at least 15 h after termination of stressor exposure. In addition, pathogen-induced colonic histopathology, which was mild in mice left undisturbed during oral challenge with C. Stressor exposed mice had increases in colonic epithelial cell hyperplasia and dysplasia, as well as epithelial defects, generalized edema and leukocyte infiltration. These effects were not evident in the mice that were not exposed to the stressor during pathogen challenge (Galley et al. In addition, colonic histopathol- ogy was not evident in any of the mice fed the L. Much has been learned about the effects of probiotic microbes on host immune responses over the past 10 years, and it is tempting to speculate on the mechanisms by which L. Mice exposed to either the prolonged restraint stressor or the social stressor during oral challenge with C. It is also possible that the production of immunomodulatory neuroendocrine mediators by 268 M. Potential pathways by which stress, the microbiota, and probiotics impact colonic inflammation are illustrated in Fig. Microbiota and Stressor-Induced Immunomodulation in Systemic Compartments Stressor exposure often results in increases in nonspecific inflammatory responses. The mechanisms by which these stressor-induced increases in inflammatory cytokines occur in otherwise healthy individuals are not completely understood. But data from our laboratory, as well as others, suggest that the intestinal microbiota are involved [35, 81–83].

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We also assume that the population has reached an equilibrium age distribution with exponential growth in the form U(a order 18gm nasonex nasal spray overnight delivery allergy symptoms back pain, t)=eqtA(a) given by (4 buy discount nasonex nasal spray 18 gm line allergy levels austin. Iterative use of this equation leads to the following equation for Pi in terms of P1: ci−1 ···c1P1 (4 buy nasonex nasal spray 18 gm with mastercard allergy testing columbus ohio. If the population reproduction number Rpop is less than, equal to, or greater than 1, then the q solution of (4. As in the continuous demographic model, it is assumed that the population starts at a steady state age distribution with total size 1 at time 0, so that the group sizes Pi remain fixed and add up to 1. See [105] for more details on the derivation of this demographic model for age groups. For many endemic mod- els the basic reproduction number can be determined analytically by either of two methods. One method is to find the threshold condition above which a positive (en- demic) equilibrium exists for the model and to interpret this threshold condition as R0 > 1. The second method is to do a local stability analysis of the disease-free equi- librium and to interpret the threshold condition at which this equilibrium switches from asymptotic stability to instability as R0 > 1. Here we use the appearance of an endemic steady state age distribution to identify expressions for the basic reproduction number R0, and then show that the disease-free steady state is globally asymptotically stable if and only if R0 ≤ 1. The age distributions of the numbers in the classes are denoted by M(a, t), S(a, t), E(a, t), I(a, t), and R(a, t), where a is age and t is time, so that, for example, the number of susceptible individuals at time t in the age interval [a1,a2] is the integral of S(a, t) from a1 to a2. Because informa- tion on age-related fertilities and death rates is available for most countries and because mixing is generally heterogeneous, epidemiology models with age groups are now used frequently when analyzing specific diseases. However, special cases with homogeneous mixing and asymptotic age distributions that are a negative ex- ponential or a step function are considered in sections 5. For example, the negative exponential age distribution is used for measles in Niger in section 7. Here it is assumed that the contact rate be- tween people of age a and age a˜ is separable in the form b(a)˜b(˜a), so that the force of infection λ is the integral over all ages of the contact rate times the infectious ∞ fraction I(˜a, t)/ 0 U(˜a, t)da˜ at time t. The division by the total population size ∞ 0 U(a, t)da makes the contact rate λ(a, t) independent of the population size, so the contact number is independent of the population size [57, 97, 102, 159]. One example of separable mixing is proportionate mixing, in which the contacts of a person of age a are distributed over those of other ages in proportion to the ac- tivity levels of the other ages [103, 174]. If l(a) is the average number of people contacted by a person of age a per unit time, u(a) is the steady state age distribu- ∞ tion for the population, and D = 0 l(a)u(a)da is the total number of contacts per unit time of all people, then b(a)=l(a)/D1/2 and b(˜a)=l(˜a)/D1/2. An- other example of separable mixing is age-independent mixing given by b(a)=1and ˜b(˜a)=β. Thus the boundary conditions at age 0 are ∞ M(0,t)= f(a)[M + E + I + R]da, 0 ∞ S(0,t)= f(a)Sda, 0 while the other distributions at age 0 are zero. For each age a the fractional age distributions of the population in the epidemi- ological classes at time t are m(a, t)=M(a, t)/U(a, t), s(a, t)=S(a, t)/U(a, t), etc. Because the numerators and denominator contain the asymptotic growth factor eqt, these fractional distributions do not grow exponentially. Determining the local stability of the disease-free steady state (at which λ = kb(a)=0ands = 1) by linearization is possible following the method in [40], but we can construct a Liapunov function to show the global stability of the disease-free steady state when R0 ≤ 1. Consider the Liapunov function ∞ V = [α(a)e(a, t)+β(a)i(a, t)]da, 0 where the positive, bounded functions α(a) and β(a) are to be determined. The formal Liapunov derivative is ∞ V˙ = {α(a)[λs − εe − ∂e/∂a]+β(a)[εe − γi − ∂i/∂a]}da 0 ∞ = {λsα(a)+e[α (a) − εα(a)+εβ(a)]+[β (a) − γβ(a)]i}da. Then ∞ ∞ ∞ ∞ ˙ εa −εz ˜ −D(˜a)−qa˜ V = sb(a)εe e β(z)dzda b(˜a)iρe da˜ + [β − γβ]ida. Then ∞ ∞ ∞ ˙ εa (γ−ε)z ˜ −D(x)−qx−γx V = sb(a)εe e b(x)ρe dxdzda − 1 0 a z ∞ ˜ −D(˜a)−qa˜ × b(˜a)i(˜a, t)ρe da. The set with V˙ = 0 is the boundary of the feasible region with i = 0, but di(a(t),t)/dt = εe on this boundary, so that i moves off this boundary unless e =0. Thus the disease-free steady state is the only positively invariant subset of the set with V˙ = 0. If there is a finite maximum age (so that all forward paths have compact closure), then either Corollary 2. If R > 1, then we have V>˙ 0 for points sufficiently close to the disease-free 0 steady state with s close to 1 and i>0 for some age, so that the disease-free steady state is unstable. Although the endemic steady state would usually be stable, this may not be true in unusual cases. For example, in preferred mixing, certain age groups are more likely to mix with their own age group [103]. For more general mixing, the endemic steady state might not be unique, but some conditions that guarantee existence, uniqueness, and local stability have been given [53, 125]. Although the steady state age distribution of the population is −D(a)−qa −D(a) ∞ −D(a) ρe , the age distribution for a specific birth cohort is e / 0 e da. Thus the rate at which individuals in a birth cohort leave the susceptible class due to −D(a) ∞ −D(a) an infection is λ(a)s(a)e / 0 e da, where s(a) is given in (5. Hence the expected age A for leaving the susceptible class is ∞ a aλ(a)e−D(a)[δF e−Λ(a) + δ(1 − F ) e−δx−Λ(a)+Λ(x)dx]da 0 ∗ λ 0 (5. When the death rate coefficient d(a) is independent of the age a, the age distribution (4. Also, the waiting times in M, E, and I have negative exponential distributions, so that, after adjusting for changes in the popu- lation size, the average period of passive immunity, the average latent period, and the average infectious period are 1/(δ + d + q), 1/(ε + d + q), and 1/(γ + d + q), respectively. Here it is also assumed that the contact rate is independent of the ages of the infectives and susceptibles, so we let b(a)=1and˜b(˜a)=β. Thus the infective replacement number R0s¯ is 1 at the endemic equilibrium for this model. However, this is generally not true, so it is not valid to use R0 =1/s¯ to derive an expression for the basic reproduction number. Thus the rate that individuals in a birth cohort leave the susceptible class due to an infection is λs(a)de−da, where s(a) is given in (5. Here the equation for the expected age A for leaving the susceptible class is δ − λs0 δ(1 − s0) ∞ − λd a[c e−(λ+d)a + c e−(δ+d)a]da 2 2 0 1 2 (λ + d) (δ + d) (5. But the death factor really should be included, since we want to calculate the average age for those who survive long enough to become infected. In the limiting situation every newborn infant has passive immunity, so that m0 → 1 and s0 → 0. Note that the formula for λ is for an endemic steady state for a virulent disease, so it does not imply that R0δ/(δ + d + q) > 1 is the threshold condition for existence of a positive endemic steady state age distribution; compare with [12, p. Thus for a very virulent disease, adding a passively immune class to a model increases the average age of attack by the mean period of passive immunity. Solving for R0 in terms of the average period p of passive immunity and the average lifetime L =1/d, we obtain [q +1/(A − p)](1 + pq) (5.

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