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• Fellow in Cardiovascular Disease
• Department of Cardiovascular Medicine
• Cleveland Clinic Foundation
• Cleveland, Ohio

This quantity is called the number fraction treatment 12th rib syndrome discount 400 mg albenza otc, and it is expressed as a fraction of 1 7 medications that can cause incontinence purchase albenza pills in toronto. The 6 M anual of basic techniques for a health labora to ry num ber fraction of each type m ight be 0 treatment zamrud buy albenza 400mg mastercard. This is defined as the volum e of a specified com ponent of a m ixture divided by the to tal volum e of the m ixture medications journal order generic albenza canada. For exam ple medicine upset stomach albenza 400 mg amex, if the to tal volum e occupied by all the erythrocytes in 1 litre (1000m l) of blood is 450m l medicine search purchase albenza with paypal, the erythrocyte volum e fraction is 450/1000 = 0. The erythrocyte volum e fraction is im portant for the diagnosis of m any diseases and you will often m easure it in the labora to ry. In the traditional system volum e fraction had no special nam e: instead, each different volum e fraction had a different nam. In this case, each of the conversion fac to rs listed m ust be m ultiplied or divided by 100, as in the follow ing exam ples: 0. Setting up a peripheral health labora to ry 9 Part I 10 M anual of basic techniques for a health labora to ry 2. It shows a labora to ry suitable for carrying out som e or all of the techniques described in the m anual. The plan is lim ited to one room, since often this is all the space that is available for the labora to ry. D irty and/or contam inated m aterial should be rem oved from the labora to ry working area as quickly as possible, both for the safety of the workers and to avoid errors and cross-contam ination. Rem ote labora to ries often have problem s in ensuring a continuous supply of elec trical power and m ay need to generate electricity by using a local genera to r or a solar energy supply system. It is possible to use the com bustion engine of a m o to r car or a purpose-built genera to r. A purpose-built genera to r produces an alternating current of 110 volts (V) or 220V and can usually generate m ore energy than a car engine. A car engine provides a direct current of 12V or 24V, which can be fed in to rechargeable batteries (see below). The installation of a direct current network is sim ple and it is safe to operate. H owever, for instrum ents that require a low-voltage (6V, 12V or 24V) direct cur rent, the high voltage produced from the direct current network m ust be converted by m eans of a transform er. Alternatively, for instrum ents that require alternating current (110V, 220V or 240V), the direct current m ust be converted in to alternat ing current by m eans of an inverter. Inverters are heavy and expensive and significant energy losses occur in the conversion process. It is therefore preferable to use either direct current or alternating current appliances, depending on your supply, and avoid the need for conversion. If no genera to r is available or if a m ains power supply is accessible, but the electri cal current fiuctuates or is prone to frequent breakdowns, a solar energy supply m ay be preferable (see below). Solar energy supply system s (pho to voltaic system s) A labora to ry with a few instrum ents with low energy requirem ents can work with a sm all energy supply. For labora to ries located in rem ote areas, a solar energy supply system m ay be m ore suitable than a genera to r since there are no problem s of fuel supplies and it can be easily m aintained. Am orphous silicon panels are less expensive, but produce solar energy less eficiently than crystalline silicon panels. The m inim um distance of the underside of the panel from the surface of the supporting construction m ust be m ore than 5cm to avoid heating of the panel, which would reduce the eficiency of energy production. Electronic charge regula to rs A charge regula to r controls the charging and discharging of the batteries au to m ati cally. When the battery voltage falls below a threshold value during discharge, the labora to ry instrum ent will be disconnected from the battery. A good charge regula to r adapts the m axim al voltage of the battery to the change in the tem perature of the am bient environm ent. It is advisable to choose a charge regula to r with an integrated digital display that allows the bat tery charge to be m oni to red easily. Lead batteries are preferred and m any types are available com m ercially (see Table 2. H igh-eficiency batteries have practical advantages, although they are m ore expensive than norm al batteries. When purchasing batteries choose 12V batteries with the highest capacity (1000 am pere-hours (Ah)). Several types of m aintenance-free lead batteries are com m ercially available, but they are expensive and less eficient than those that require m aintenance. The de velopm ent of this type of battery is still in progress; it has not been thoroughly tested in tropical clim ates. Transport of lead batteries Lead batteries should be em ptied before being transported. It is im portant to re m em ber that if lead batteries are to be transported by air they must be em pty of electrolyte solution, which should be replaced on arrival at the destination. If the batteries are repeatedly discharged to 40% of their capacity, they will last for only about 600 cycles. H igh-eficiency batteries cannot be replaced by norm al car batteries in case of a breakdown. When only car batteries are available to replace a defective high-eficiency battery, all the batteries in the energy s to rage system m ust be replaced with car batteries. This apparent unreliability is caused by an increased rate of discharge rather than ineficient re charging of the battery at high am bient tem peratures (see below). You m ay have to m ake sim ple connections or repairs to this equipm ent in the labo ra to ry. The explanations given below are intended to help the labora to ry technician to do this and are lim ited to the steps to follow in each case. Inexperienced persons should start by carrying out the procedures in the presence of an instruc to r. Setting up new electrical equipm ent Voltage Check that the voltage m arked on the instrum ent is the sam e as that of your elec tricity supply. The instrum ent has a label on it stating the voltage with which it m ust be used. Dual-voltage equipm ent D ual-voltage instrum ents can be used with two different voltage supplies. There is a device on the instrum ent that enables you to select the appropriate voltage, i. Setting up a peripheral health labora to ry 17 the electrical pow er of the instrum ent the electrical power is m easured in watts (W) and is m arked on the plate that shows the correct voltage for the instrum ent. Each piece of electrical equipm ent in the labora to ry uses a certain am ount of power. The to tal power used at any one tim e m ust not exceed the power of your electricity supply. You can work out how m uch power is available from the figures shown on the m eter: m ultiply the voltage (V) by the current (A). For exam ple, if the voltage is 220V and the current is 30A, the electrical power supplied will be 220 fi 30 = 6600 watts or 6. Using a transform er If an instrum ent is intended for use with a voltage different from that of the labora to ry electricity supply, it can be used with a transform er. Plug the centrifuge in to the 110V connection of the transform er supplied, then plug the 220V lead from the transform er in to the labora to ry electricity supply (wall socket). Sw itching off electrical equipm ent After an instrum ent has been switched off, it m ust be unplugged from the wall socket. If the wire is broken or m elted, the current no longer passes: the fuse has blown. Replace it with new fuse wire of the sam e gauge (thickness), or with thinner wire if the sam e size is not available. Once the fuse has been repaired, check the whole circuit before switching on the electricity supply. Checking the plug If a fault is suspected in a plug, it m ust be repaired or replaced. There are m any different types of plug; som e have a screw on the outside that can be unscrewed so that the cover can be rem oved. Fitting a new plug To fit a new plug, rem ove the insulating m aterial along a length of 1. This can be done by scraping with a knife but take care not to dam age the wire inside. Twist the exposed ends of both wires to allow them to fit neatly in to the term inal once the screw has been Fig. It is m ost im portant to connect each of the three wires in the cable to the correct pin, and the plug usually contains instructions that should be strictly followed. It provides an escape for the electric current in case of poor insulation, thus avoid ing passage of the current through the hum an body. They have to be unscrewed and opened if you want to check that they are working properly. M ake sure that the two incom ing wires and the two outgoing wires are firm ly fixed in their respective term inals (Fig. The fe m ale plug is fixed to the cable by two term inals inside the plug, just as in the norm al m ale plug. Checking the w all socket To check a wall socket, plug in a lam p that you know to be working. If this is not the case, it is usually wise to call in an electrician to repair a wall socket. Som e sim ple rem edies are described below, in case a plum ber is not readily available. Important: Before starting any plum bing operation, cut off the water at the m ains. Setting up a peripheral health labora to ry 21 W hat to do if w ater fiow s w hen the tap is turned off If water continues to fiow when the tap is turned off, the washer needs to be replaced. U nscrew the head of the tap using an adjustable wrench (turn in an anticlockwise direction) (Fig. If the tap continues to leak after the washer has been re placed, the seating (S) that receives the washer (Fig. W hat to do if w ater leaks out of the head of the tap If water leaks out of the head of the tap, the joint needs to be replaced. W ind new to w around the screw thread, starting at the to p and winding in a clockwise direction (Fig. Replacing the w hole tap U nscrew the faulty tap, using a pipe wrench (turn in an anticlockwise direction). W ind to w from around the around the thread and sm ear with jointing com pound as described above. The wastewater fiows in to the trap, which is perm anently filled with water (the seal). Setting up a peripheral health labora to ry 23 Unblocking w ith a plunger Place the plunger over the waste pipe. Leave for 5 m inutes, then rinse the sink thoroughly with cold water from the tap. W arning: Sodium hydroxide solution is highly corrosive and should be used with extrem e care. If it is splashed on the skin or in the eyes, wash the affected areas im m ediately with large quantities of water. W hat to do if the sink trap is leaking If foul sm ells com e up through the waste pipe of the sink, the perm anent reservoir of water (the seal) at the bot to m of the trap m ust have leaked because of a fault in joint J2. Filtering Using a porous unglazed porcelain or sintered glass filter this type of filter can be attached to a tap. Alternatively, it can be kept im m ersed in a container of the water to be filtered (Fig.

Exceptions may be made for specific activities such as field trips outside the child care facility medicine 877 order albenza on line. Measures to assure proper s to rage and refrigeration of sack lunches are required of the child care facility symptoms glaucoma purchase generic albenza. For programs which operate for three or more days a week treatment quality assurance unit cheap 400 mg albenza mastercard, but which operate only one full day during the week symptoms 6 days past ovulation order discount albenza online, sack lunches provided by parents shall be permitted on that day medicine woman cheap 400 mg albenza free shipping. Facilities operating six hours or less per day are allowed to have children bring sack lunches provided all of the following requirements are met: a treatment stye albenza 400 mg amex. The facility shall have a written policy about sack lunches and a copy shall be given to parents. The notice shall contain instructions as to what foods are proper for a sack lunch. In child care facilities where all the children are present for four hours or less per day during normal hours of operation, snacks may be provided by parents. If any child is present for more than four hours per day on a routine basis the facility shall meet the standard of having snacks prepared by the facility or a permitted catering establishment. All kitchens and/or food/snack preparation areas in a child care facility must be inspected as part of the child care inspection process. Each child care facility must have a Certified Food Manager meeting the standards set forth in Rule 1. In the absence of the Certified Food Manager, an individual trained in food service must be present. Any form of emotional abuse, including rejecting, terrorizing, ignoring, isolating (out of view of a caregiver), or corrupting a child. Children shall not be given medicines or drugs that will affect their behavior except as prescribed by a licensed physician and with specific written instructions from the licensed physician for use of the medicines or drugs. Children board or leave the vehicle from the curb-side of the street and/or are safely accompanied to their destinations. All children will be properly restrained whenever they are being transported in a mo to r vehicle. Every person transporting a child under the age of four (4) years in a passenger mo to r vehicle, and operated on a public roadway, street or highway, shall provide for the protection of the child by properly using a child passenger restraint device or system meeting applicable federal mo to r vehicle safety standards. No children shall be transported in the front seat of vehicles equipped with passenger-side air bags. The driver of the vehicle shall not be counted as a caregiver while transporting the children. A diapering area shall contain a hand washing lava to ry with hot and cold running water, a smooth and easily cleanable surface, a plastic-lined, covered garbage receptacle, and sanitizing solution. Hand washing sinks at diaper changing stations shall not be used for any other purpose. Example: the diaper changing sink may not be used for washing cups, baby bottles, food, dishes, utensils, etc. All rest period equipment shall be cleaned twice a week with a germicidal solution. The bottle shall be removed at once when empty or when the child has fallen asleep. Breast-feeding mothers, including employees, shall be provided a sanitary place that is not a to ilet stall to breast-feed their child or to express milk. Milk must be s to red in accordance with the American Academy of Pediatrics and Centers for Disease Control guidelines. Child care staff shall be trained in the safe and proper s to rage and handling of human milk. A person having an American Red Cross lifeguard certificate, or the equivalent as recognized by the licensing agency, shall be present at all swimming and water activities. Each child will be tested by a certified lifeguard prior to participating in swimming lessons or any pool activities. Wading pools For activities taking place in wading pools with a water depth of one foot or less the following is required: a. All piers, floats, and platforms shall be in good repair, and where applicable, the water depth shall be indicated by printed numerals on the deck or planking. There shall be a minimum water depth of 10 feet for a one-meter diving board and 13 feet for a three-meter board or diving to wer. The entire length of the to p surface of diving boards shall be covered with nonskid 63 Part 11: Bureau of Child Care Facilities November 11, 2011 material. If other halogens are used, residuals of equivalent disinfecting strength shall be maintained. A testing kit for measuring the concentration of the disinfectant, accurate within 0. The following chart may be used for reference: pH Minimum Free Available Residual Chlorine-mg/L (not stabilized with cyanuric acid) 7. If cyanuric acid is used to stabilize the free available residual chlorine, or if one of the chlorinated isocyanurate compounds is used as the disinfecting chemical in a swimming pool, the concentration of cyanuric acid in the water shall be at least 30 mg/L but shall not exceed 100 mg/L. The free available residual chlorine, of at least the following concentrations, depending upon the pH of the water, shall be maintained: 64 Part 11: Bureau of Child Care Facilities November 11, 2011 pH Minimum Free Available Residual Chlorine-mg/L (cyanuric acid is at least equal to 30 mg/L, but not greater than 100 mg/L) 7. A separate area shall be available for providing privacy for diapering, dressing, and other personal care procedures. The individual activity plan shall have been developed by a person with a bachelors or advanced degree in a discipline dealing with disabilities, as appropriate. A snack period shall be provided to children in attendance for more than two and one-half (2fi) hours prior to bedtime. The upper level of double-deck beds are allowed for children ten years of age or older if a bed rail and safety ladder is provided. No children under six years of age shall be left alone or with another child while in the bathtub or shower. All children shall be provided an individual washcloth, to wel, and soap for bathing, with fresh water for each child. Programs operating in excess of 16 weeks per year shall meet the more stringent requirements of Subchapter 22 and 23. These children may not be placed in the same area of a child care facility as preschool children. When 67 Part 11: Bureau of Child Care Facilities November 11, 2011 a room is used for meals, the minimum square footage per child per room requirement will not apply. If the school age children are served in conjunction with preschool children under the same license, the preschool square footage standards will apply. When children are placed in groups, the maximum group size shall be determined by the following chart. The driver of the vehicle may be counted as a caregiver while transporting school age children only. The following ratios shall apply: Number of Toilets and Number of Children Hand Washing Lava to ries 1-30 1 of each 31-60 2 of each 60-90 3 of each 2. For each additional 30 children or portion thereof, add one to ilet and one hand washing lava to ry. Urinals shall count as one-half (fi) a to ilet not to exceed 33 percent of the to tal number of to ilets required. Screen or media use or other educational electronic equipment is acceptable provided such is for educational purposes. A summer day camp is defined as a child care facility that operates during May, June, July, and/or August only, for a minimum of 22 days and a maximum of 16 weeks. A school age program is defined as a child care facility that operates during the school year. A summer day camp direc to r shall be at least 21 years of age, and shall have, at a minimum: a. For summer day camps that routinely operate indoors in a permanent structure for two or more hours each day a minimum of 25 square feet of usable indoor floor space, per child per room, shall be maintained for each child. This shall not include hallways, bathrooms, closets, s to rage rooms, offices, or kitchens. When activities for children are routinely conducted outdoors or off the premises for six or more hours each day, the following requirements shall apply: 71 Part 11: Bureau of Child Care Facilities November 11, 2011 a. There shall be a permanent structure that serves as a home base where parents deliver and pick up children. There shall be a minimum of ten square feet per child usable indoor space available in the event of inclement weather. School age programs require that a minimum of 25 square feet of usable indoor floor space, per child per room shall be maintained for each child. When a room is used for meals, the minimum square footage per child requirement will not apply. For summer day camps operating primarily as an outdoor program away from the home base, the following exceptions shall apply: 3. Milk is not required to be served in programs routinely operating outdoors or off the premises for six or more hours each day. If food is brought from home or catered, there shall be sanitary cold s to rage available. Potable water, from a Mississippi State Department of Health approved source, shall be used for drinking. The driver of the vehicle may be counted as a caregiver while transporting the children. The following ratios shall apply: Number of Toilets and Number of Children Hand washing Lava to ries 1-30 1 of each 31-60 2 of each 60-90 3 of each 2. For summer day camps operating primarily as an outdoor program away from the home base, alternative methods of hand washing may be provided. These items shall be used by children only under the direction and supervision of an individual certified by a state or national organization recognized by the Mississippi State Department of Health. An "Hourly Child Care Facility" is defined as a facility that meets the provisions of these regulations for a "Child Care Facility" and: a. Limits the care of a child to no more than eight hours per stay not to exceed a to tal of 45 hours in any calendar month period. Provides supervised, short term, hourly care on a temporary basis in conjunction with a specific facility or business complex such as, but not limited to , hotels; shopping malls; recreational, sporting, or entertainment facilities. Hourly child care facilities are not appropriate for full time child care and will not be allowed to provide that type of service. When it is determined by the licensing agency that a facility provides child care services on a full time basis, the facility shall meet all requirements for a regular child care facility as set forth in the preceding sections of these regulations. Parents shall be provided a written statement of policies pertaining to emergencies, meals, snacks, procedures for releasing a child to parent, and any other information regarding hourly child care facility operation. All policies and procedures will be submitted to the licensing agency and reviewed prior to a license being issued. Written guidelines will be provided to applicants as part of the application packet. The care of a child shall be no more than eight hours per stay and shall not exceed a to tal of 45 hours in any calendar month period. When business hours exceed 12 hours in a 24-hour period, the program will be reviewed on an individual basis for compliance with regulations addressing evening and overnight care. The hourly child care facility shall maintain information necessary to contact local law enforcement officials and the Mississippi Department of Human Services when a child is left at the facility past its hours of operation, or for an extended period. Only forms that substantially comply with the aforementioned sample forms will be acceptable. Registration forms will include a signed statement that will serve as verification that a child has received all age-appropriate immunizations.

Hence many of the formulas for 0 0 Type I mortality in the Anderson and May book [12 symptoms 7 dpo bfp discount albenza 400mg, Ch 25 medications to know for nclex generic albenza 400 mg with amex. In sections 7 and 8 we estimate the basic reproduction number in models with age groups for measles in Niger and pertussis in the United States symptoms zinc toxicity generic 400mg albenza overnight delivery. The initial boundary value problem for this model is given below: fiS/fia + fiS/fit = fifi(a medications on airplanes cheap albenza 400mg with visa, t)S fi d(a)S medicine organizer order albenza 400mg with visa, fi fi fi(a medications vertigo effective 400mg albenza, t)= b(a)fib(fia)I(fia, t)dafi U(fia, t)da,fi 0 0 (6. The boundary fi values at age 0 are all zero except for the births given by S(0,t)= 0 f(a)U(a, t)da. The population is partitioned in to n age groups as in the demographic model in section 4. The subscripts i denote the parts of the epidemiologic classes in the ith ai age interval [aifi1,ai], so that Si(t)= a S(a, t)da, etc. The to tal in the four epidemiologic classes for the ith age group is the size N (t)=eqtP of i i the ith group, which is growing exponentially, but the age distribution P1,P2. Because the numbers are all growing exponentially by eqt, the fractions of the population in the epidemiologic classes are of more interest than the numbers in these epidemiologic classes. Here we follow the same procedure used in the continuous model to find an expression for the basic re production number R0. When the expressions for ei and iifi1 are substituted in to the expression for i in (6. Now the expressions for i and fi = kb can be substituted in to this j=1 j j i i i last summation to obtain n fij bj bjfi1 b1 (6. Here the feasible region is the subset of the nonnegative orthant in the 4n-dimensional space with the class fractions in the ith group summing to Pi. In the Liapunov derivative Vfi, choose the fi coeficients so that the e terms cancel out by letting i i fin = finfin/fifin and fijfi1 =(fijfi1fijfi1 + cjfi1fij)/fifijfi1 for finfi1. Using s fi P, n n n jfi1 jfi1 j jfi1 jfi1 nfi1 1 i i we obtain Vfi fi (R fi1) fib i fi 0ifR fi 1. The set where Vfi = 0 is the boundary of 0 j j 0 the feasible region with ij = 0 for every j, but dij/dt = fijej on this boundary, so that ij moves ofi this boundary unless ej = 0. Thus if R0 fi 1, then the disease free equilibrium is asymp to tically stable in the feasible region. If R0 > 1, then we have V>fi 0 for points suficiently close to the disease-free equilibrium with s close to P and i i ij > 0 for some j, so that the disease-free equilibrium is unstable. A deterministic compartmental mathemati cal model has been developed for the study of the efiects of heterogeneous mixing and vaccination distribution on disease transmission in Africa [133]. This study focuses on vaccination against measles in the city of Naimey, Niger, in sub-Saharan Africa. The rapidly growing population consists of a majority group with low transmission rates and a minority group of seasonal urban migrants with higher transmission rates. De mographic and measles epidemiological parameters are estimated from data on Niger. The fertility rates and the death rates in the 16 age groups are obtained from Niger census data. From measles data, it is estimated that the average period of passive immunity 1/fi is 6 months, the average latent period 1/fi is 14 days and the average infectious period 1/fi is 7 days. From data on a 1995 measles outbreak in Niamey, the force of infection fi is estimated to be the constant 0. A computer calculation using the demographic and epidemiological parameter values in the formula (6. Recall from section 1 that the replacement number R is the actual number of new cases per infective during the infectious period. R can be approximated by computing the sum over all age groups of the daily incidence times the average infectious period times the fraction surviving the latent period, and then dividing by the to tal number of infectives in all age groups, so that 16 1 fi j=1fijsjPj fi + dj + q fi + dj + q R fi=. This contact number fi is approximated by computing the product of the sum of the daily incidences when all contacts are assumed to be with susceptibles times the average infectious period, and dividing by the to tal number of infectives. The average age of infection can be approximated in the measles computer simulations by the quotient of the sum of the average age in each age group times the incidence in that age group and the sum of the incidences. This model is plausible because the age distribution of the Niger population is closely approximated by a negative exponential [133]. Using this d value and the fertilities in the Lotka characteristic equation for discrete age groups (4. Recall that the replacement number R is 1 at the endemic equilibrium for this model. Thus in this population nearly every mother is infected with 0 measles before childbearing age, so almost every newborn child has passive immunity. This result is confirmed by the measles computer simulations for Niger, in which herd immunity is not achieved when all children are vaccinated at age 9 months. However, these estimates of R0 are not realistic, because pertussis gives only tempo rary immunity and spreads by heterogeneous mixing. In the age-structured epidemi ologic models developed specifically for pertussis [105, 106], there are 32 age groups. Using fertilities and death rates from United States census information for 1990, the value of q in (4. Thus the age distribution in the pertussis models is assumed to have become stable with a constant population size. More details and graphs of the actual and theoretical age distributions are given in [105]. Immunity to pertussis is temporary, because the agent Bordetella pertussis is bacterial, in contrast to the viral agents for measles, mumps, and rubella. As the time after the most recent pertussis infection increases, the relative immunity of a person decreases. When people become infected again, the severity of their symp to ms and, consequently, their transmission efiectiveness. Of course, infected people who were previously fully susceptible are generally the most efiective transmitters. In the age-structured pertussis models [105, 106], the epidemiological classes include a susceptible class S, an infective class I, a class R4 of those removed people with very high immunity, and classes R3, R2, and R1 for those with decreasing immunity. In the two pertussis models, there are three or four levels of infectivity and 32 age groups, so that not all infectives are equally efiective in creating new infectives [106]. Infectives in those age groups that mix more with other age groups are more efiective transmitters than those in age groups that mix less. In the next paragraph, we explain why averaging over age groups is necessary, but averaging over classes with difierent infectivities is not appropriate. The occurrence of the first infection in a fully susceptible population seems to be an unpredictable process, because it depends on random introductions of infectious outsiders in to the host population. The probability that a first infection occurs in the host population depends on the infectivity of the outside invader, on how the invader (with a mixing activity level based on its age group) mixes in the host population, and the length of time that the invader is in the population. It is clear that outside invaders from high infectivity classes and high mixing activity age groups are more likely to create a first new infection in a host population, especially if they are in the population for their entire infectious period. We believe that the definition of R0 should not depend on the circumstances under which an outsider creates a first case, but on whether or not an infection with a first case can persist in a fully susceptible population. After the first infection in the host population, the infected people in the next generations could be less efiective transmitters, so that the infection would die out. Thus the definition of R0 should be based on the circumstances under which a disease with a first case would really invade a fully susceptible host population more exten sively. Thus R0 should be the number of secondary cases produced by averaging over all age groups of the infectives that have not been previously infected. Because all of the cases in the first generations of an invasion occur in fully susceptible people, only infectives who were previously fully susceptible are relevant. The fertilities fj, death rate constants dj, and transfer rate constants cj are determined in the demographic model. The form of separable mixing used in the pertussis model is proportionate mixing, which has activity levels lj in each of the 32 age groups. The activity levels lj are found from the forces of infection fij and the infective fractions i, as explained in Appendix C of [105]. Then b = fib = l /D1/2, where j j j j 32 D = j=1 ljPj is the to tal number of people contacted per unit time. In the first model each pertussis booster moves the individual back up one vaccinated or removed class, but for those in the second model who have had a sequence of at least four pertussis vaccinations or have had a previous pertussis infection, a pertussis booster raises their immunity back up to the highest level. Thus the second model incorporates a more optimistic view of the efiectiveness of pertussis booster vaccinations. Neither of the two methods used to find approximations of R0 for measles in Niger works for the pertussis models. The replacement number R at the pertussis endemic equilibrium depends on the fractions infected in all of the three or four infective classes. For example, in the first pertussis model 32 j=1 fij(sj + r1j + r2j)Pj/(fi + dj) R fi=, 32 j=1(ij + imj + iwj)Pj where ij, imj, and iwj are the infective prevalences in the full-, mild-, and weak-disease classes I, Im, and Iw. In the computer simulations for both pertussis models, R is 1 at the endemic equilibrium. If the expression for R is modified by changing the fac to r in parentheses in the numera to r to 1, which corresponds to assuming that all contacts are with susceptibles, then we obtain the contact number 32 j=1 fijPj/(fi + dj) fi fi=, 32 j=1(ij + imj + iwj)Pj which gives the average number of cases due to all infectives. Thus it is not possible to use the estimate of the contact number fi during the computer simulations as an approxima tion for R0 in the pertussis models. Since the age distribution of the population in the United States is poorly approximated by a negative exponential and the force of infection is not constant, the second method used for measles in Niger also does not work to approximate R0 for pertussis in the United States. The ultimate goal of a pertussis vaccination program is to vaccinate enough people to get the replacement number less than 1, so that pertussis fades away and herd immunity is achieved. Because the mixing for pertussis is not homogeneous and the immunity is not permanent, we cannot use the simple criterion for herd immunity that the fraction with vaccine-induced or infection-induced immunity is greater than 1 fi 1/R0. None of the vaccination strategies, including those that give booster vaccinations every five years, has achieved herd immunity in the pertussis computer simulations [105, 106]. The results presented in this paper provide a theoretical background for reviewing some previous results. In this section we do not attempt to cite all papers on infectious disease models with age structure, heterogeneity, and spatial structure, but primarily cite sources that con sider thresholds and the basic reproduction number R0. We refer the reader to other sources for information on s to chastic epidemiology models [18, 20, 56, 59, 66, 81, 128, 167], discrete time models [2, 3], models involving macroparasites [12, 59, 90], genetic het erogeneity [12, 90], plant disease models [137, 194], and wildlife disease models [90]. Age-structured epidemiology models with either continuous age or age groups are essential for the incorporation of age-related mixing behavior, fertility rates, and death rates, for the estimation of R0 from age-specific data, and for the comparison of vac cination strategies with age-specific risk groups and age-dependent vaccination rates. Indeed, some of the early epidemiology models incorporated continuous age structure [24, 136]. Modern mathematical analysis of age-structured models appears to have started with Hoppensteadt [114], who formulated epidemiology models with both con tinuous chronological age and infection class age (time since infection), showed that they were well posed, and found threshold conditions for endemicity. Expressions for R0 for models with both chronological and infection age were obtained by Dietz and Schenzle [68]. In age-structured epidemiology models, proportionate and preferred mixing parameters can be estimated from age-specific force of infection data [103]. Mathematical aspects such as existence and uniqueness of solutions, steady states, stability, and thresholds have now been analyzed for many epidemiology models with age structure; more references are cited in the following papers. Age-structured models have been used in the epidemiology modeling of many dis eases [12]. Dietz [61, 64], Hethcote [98], Anderson and May [10, 11], and Rouderfer, Becker, and Hethcote [174] used continuous age-structured models for the evaluation of measles and rubella vaccination strategies. Hethcote [99] considered optimal ages of vacci nation for measles on three continents. Grenfell and Anderson [89] and Hethcote [105, 106] have used age-structured models in evaluating pertussis (whooping cough) vaccination programs. Irregular and biennial oscillations of measles incidences have led to various mathematical analyses including the following seven modeling ex planations, some of which involve age structure. Schenzle [177] used computer simulations to show that the measles out break patterns in England and Germany could be explained by the primary school yearly calenders and entry ages. Bolker and Grenfell [27] proposed realistic age-structured models with seasonal forcing and s to chastic terms. Ferguson, Nokes, and Anderson [79] proposed finely age-stratified models with s to chastic fiuctuations that can shift the dynamics between biennial and triennial cycle attrac to rs. For many infectious diseases the transmission occurs in a diverse population, so the epidemiological model must divide the heterogeneous population in to subpopula tions or groups, in which the members have similar characteristics. This division in to groups can be based not only on mode of transmission, contact patterns, latent pe riod, infectious period, genetic susceptibility or resistance, and amount of vaccination or chemotherapy, but also on social, cultural, economic, demographic, or geographic fac to rs. For these models it is useful to find R0 from the threshold conditions for invasion and endemicity and to prove stability of the equilibria. The seminal paper [140] of Lajmanovich and Yorke found this threshold condition and proved the global stability of the disease-free and en demic equilibria using Liapunov functions. For these models R0 can be shown to be the spectral radius of a next generation matrix that is related to the Jacobian matrix A [103, 110]. For proportionate mixing models with multiple interacting groups, the basic reproduction number R0 is the contact number fi, which is the weighted average of the contact numbers in the groups [103, 110, 113].

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The parameter fi has no direct epidemiological interpretation medicine used to stop contractions 400mg albenza, but comparing it with the standard formulation shows that fi = fiN symptoms insulin resistance 400 mg albenza sale, so that this form implicitly assumes that the contact rate fi increases linearly with the population size treatment urticaria generic 400mg albenza mastercard. Naively medications after stroke order 400 mg albenza with visa, it might seem plausible that the population density and hence the contact rate would increase with population size medicine yeast infection buy albenza online, but the daily contact patterns of people are often similar in large and small communities treatment zone lasik cheap albenza 400mg with amex, cities, and regions. This strongly suggests that the standard incidence corresponding to v = 0 is more realistic for human diseases than the simple mass action incidence corresponding to v =1. This result is consistent with the concept that people are infected through their daily encounters and the patterns of daily encounters are largely independent of community size within a given country. The standard incidence is also a better formulation than the simple mass action law for animal populations such as mice in a mouse-room or animals in a herd [57], because disease transmission primarily occurs locally from nearby animals. For more information about the difierences in models using these two forms of the horizontal incidence, see [83, 84, 85, 96, 110, 159]. Vertical incidence, which is the infection rate of newborns by their mothers, is sometimes included in epidemiology models by assuming that a fixed fraction of the newborns is infected vertically [33]. See [107] for a survey of mechanisms including nonlinear incidences that can lead to periodicity in epidemiological models. A common assumption is that the movements out of the M, E, and I compart ments and in to the next compartment are governed by terms like fiM, E, and fiI in an ordinary difierential equations model. It has been shown [109] that these terms correspond to exponentially distributed waiting times in the compartments. For ex ample, the transfer rate fiI corresponds to P(t)=efifit as the fraction that is still in the infective class t units after entering this class and to 1/fi as the mean wait ing time. For measles the mean period 1/fi of passive immunity is about six to nine months, while the mean latent period 1/fi is one to two weeks and the mean infec tious period 1/fi is about one week. Another possible assumption is that the fraction still in the compartment t units after entering is a nonincreasing, piecewise contin uous function P(t) with P(0) = 1 and P(fi) = 0. Then the rate of leaving the compartment at time t is fiP (t), so the mean waiting time in the compartment is fi fi t(fiP (t))dt = P(t)dt. These distributed delays lead to epidemiology models 0 0 with integral or integrodifierential or functional difierential equations. If the waiting time distribution is a step function given by P(t)=1if0fi t fi fi, and P(t)=0 if fi fi t, then the mean waiting time is fi, and for t fi fi the model reduces to a delay-difierential equation [109]. Each waiting time in a model can have a difierent distribution, so there are many possible models [102]. The basic reproduction num ber R0 has been defined in the introduction as the average number of secondary infections that occur when one infective is introduced in to a completely susceptible host population [61]. Note that R0 is also called the basic reproduction ratio [58] or basic reproductive rate [12]. It is implicitly assumed that the infected outsider is in the host population for the entire infectious period and mixes with the host population in exactly the same way that a population native would mix. The contact number fi is defined as the average number of adequate contacts of a typical infective during the infectious period [96, 110]. An adequate contact is one that is suficient for transmis sion, if the individual contacted by the susceptible is an infective. M ssiv ly im in fa ts S sc tib le s E se le in th la the t rio I In fe tiv s R re le ith im ity m,s,e,i,r ra tio s f th la tio in th la sse s fi ta t ra the 1 /fi A ra rio f ssiv im ity 1 /fi A ra la the t rio 1 /fi A ra in fe tio s rio R0 sic re ro tio r fi ta t r R la t r typical infective during the entire period of infectiousness [96]. Some authors use the term reproduction number instead of replacement number, but it is better to avoid the name reproduction number since it is easily confused with the basic reproduction number. Note that these three quantities R0, fi, and R in Table 1 are all equal at the beginning of the spread of an infectious disease when the entire population (except the infective invader) is susceptible. In recent epidemiological modeling literature, the basic reproduction number R0 is often used as the threshold quantity that determines whether a disease can invade a population. Although R0 is only defined at the time of invasion, fi and R are defined at all times. For most models, the contact number fi remains constant as the infection spreads, so it is always equal to the basic reproduction number R0. In these models fi and R0 can be used interchangeably and invasion theorems can be stated in terms of either quantity. But for the pertussis models in section 8, the contact number fi becomes less than the basic reproduction number R0 after the invasion, because new classes of infectives with lower infectivity appear when the disease has entered the population. The replacement number R is the actual number of secondary cases from a typical infective, so that after the infection has invaded a population and everyone is no longer susceptible, R is always less than the basic reproduction number R0. Also, after the invasion, the susceptible fraction is less than 1, so that not all adequate contacts result in a new case. Thus the replacement number R is always less than the contact number fi after the invasion. Combining these results leads to R0 fi fi fi R, with equality of the three quantities at the time of invasion. This model uses the standard incidence and has recovery at rate fiI, corresponding to an exponential waiting time efifit. Since the time period is short, this model has no vital dynamics (births and deaths). Most of the unvaccinated cases were people belonging to a religious denomination that routinely does not accept vaccination. Here the replacement number at time zero is fiso, which is the product of the contact number fi and the initial susceptible fraction so. Ifiso > 1, then i(t) first increases up to a maximum value imax = io + so fi 1/fi fi [ln(fiso)]/fi and then decreases to zero as t >fi. The susceptible fraction s(t) is a decreasing function and the limiting value sfi is the unique root in (0, 1/fi) of the equation (2. Typical paths in T are shown in Figure 2, and solutions as a function of time are shown in Figure 3. Note that the hallmark of a typical epidemic outbreak is an infective curve that first increases from an initial Io near zero, reaches a peak, and then decreases to ward zero as a function of time. For example, a recent measles epidemic in the Netherlands [52] is shown in Figure 4. The susceptible fraction s(t) always decreases, but the final susceptible fraction sfi is positive. The epidemic dies out because, when the susceptible fraction s(t)goesbelow1/fi, the replacement number fis(t) goes below 1. The results in the theorem are epidemiologically reasonable, since the infectives decrease and there is no epidemic, if enough people are already immune so that a typical infective initially replaces itself with no more than one new infective (fiso fi 1). But if a typical infective initially replaces itself with more than one new infective (fiso > 1), then infectives initially increase so that an epidemic occurs. The speed at which an epidemic progresses depends on the characteristics of the disease. The measles epidemic in Figure 4 lasted for about nine months, but because the latent period for infiuenza is only one to three days and the infectious period is only two to three days, an infiuenza epidemic can sweep through a city in less than six weeks. To prove the theorem, observe that the solution paths i(t)+s(t) fi [ln s(t)]/fi = io + so fi [ln so]/fi in Figure 2 are found from the quotient difierential equation di/ds = fi1+1/(fis). The equilibrium points along the s axis are neutrally unstable for s>1/fi and are neutrally stable for s<1/fi. One classic approximation derived in [18] is that for small io and so slightly greater than smax =1/fi, the difierence smax fi s(fi) is about equal to so fi smax, so the final susceptible fraction is about as far below the susceptible fraction smax (the s value where the infective fraction is a maximum) as the initial susceptible fraction was above it (see Figure 2). Observe that the threshold result here involves the initial replacement number fiso and does not involve the basic reproduction number R0. Here the contact number fi remains equal to the basic reproduction number R0 for all time, because no new classes of susceptibles or infectives occur after the invasion. Ifi fi 1 or io =0, then solution paths starting in T approach the disease-free equilibrium given by s =1and i =0. If R0 = fi fi 1, then the replacement number fis is less than 1 when io > 0, so that the infec tives decrease to zero. Although the speeds of movement along the paths are not apparent from Figure 5, the infective fraction decreases rapidly to very near zero, and then over 100 or more years, the recovered people slowly die ofi and the birth process slowly increases the susceptibles, until eventually everyone is susceptible at the disease-free equilibrium with s = 1 and i =0. IfR0 = fi>1, io is small, and so is large with fiso > 1, then s(t) decreases and i(t) increases up to a peak and then decreases, just as it would for an epidemic (compare Figure 6 with Fig ure 2). However, after the infective fraction has decreased to a low level, the slow processes of the deaths of recovered people and the births of new susceptibles grad ually (over about 10 or 20 years) increase the susceptible fraction until fis(t) is large enough that another smaller epidemic occurs. This process of alternating rapid epi demics and slow regeneration of susceptibles continues as the paths approach the en demic equilibrium given in the theorem. At this endemic equilibrium the replacement number fise is 1, which is plausible since if the replacement number were greater than or less than 1, the infective fraction i(t) would be increasing or decreasing, respectively. Notice that the ie coordinate of the endemic equilibrium is negative for fi<1, coincides with the disease-free equilibrium value of zero at fi = 1, and becomes positive for fi>1. Thus these two equilibria exchange stabilities as the endemic equilibrium moves through the disease-free equilibrium when fi = 1 and becomes a distinct, epidemiologically feasible, locally asymp to tically stable equilibrium when fi>1. The following interpretation of the results in the theorem and paragraph above is one reason why the basic reproduction number R0 has become widely used in the epidemiology literature. The latter condition is used to obtain expressions for R0 in age-structured models in sections 5 and 6. This unrealistically short average lifetime has been chosen so that the endemic equilibrium is clearly above the horizontal axis and the spiraling in to the endemic equilibrium can be seen. They unrealistically assume that the population is uniform and homoge neously mixing, whereas it is known that mixing depends on many fac to rs including age (children usually have more adequate contacts per day than adults). Moreover, difierent geographic and social-economic groups have difierent contact rates. By using data on the susceptible fractions so and sfi at the beginning and end of epidemics, this formula can be used to estimate contact numbers for specific diseases [100]. Using blood samples from freshmen at Yale University [75], the fractions susceptible to rubella at the beginning and end of the freshman year were found to be 0. This approach is somewhat naive, because the average seropositivity in a population decreases to zero as the initial passive immunity declines and then increases as people age and are exposed to infectives. Data on average ages of infection and average lifetimes in developed countries have been used to estimate basic reproduction numbers R0 for some viral diseases. These estimates of R0 are about 16 for measles, 11 for varicella (chickenpox), 12 for mumps, 7 for rubella, and 5 for poliomyelitis and smallpox [12, p. Because disease-acquired immunity is only temporary for bacterial diseases such as pertussis (whooping cough) and diphtheria, the formula R0 = fi =1+L/A cannot be used to estimate R0 for these diseases (see section 8 for estimates of R0 and fi for pertussis). Herd immunity occurs for a disease if enough people have disease-acquired or vaccination-acquired immunity, so that the introduction of one infective in to the pop ulation does not cause an invasion of the disease. Intuitively, if the contact number is fi, so that the typical infective has adequate contacts with fi people during the infectious period, then the replacement number fis must be less than 1 so that the disease does not spread. This means that s must be less than 1/fi, so the immune fraction r must satisfy r>1 fi 1/fi =1fi 1/R0. Using the estimates above for R0, the minimum immune fractions for herd im munity are 0. Although these values give only crude, ballpark estimates for the vaccination-acquired immunity level in a community required for herd immunity, they are useful for comparing diseases. For example, these numbers suggest that it should be easier to achieve herd immunity for poliomyelitis and smallpox than for measles, mumps, and rubella. This conclusion is justified by the actual efiectiveness of vaccina tion programs in reducing, locally eliminating, and eradicating these diseases (eradi cation means elimination throughout the world). The information in the next section verifies that smallpox has been eradicated worldwide and polio should be eradicated worldwide within a few years, while the diseases of rubella and measles still persist at low levels in the United States and at higher levels in many other countries. For centuries the process of variolation with material from smallpox pustules was used in Africa, China, and India before arriving in Europe and the Americas in the 18th century. Edward Jenner, an English country doc to r, observed over 25 years that milkmaids who had been infected with cowpox did not get smallpox. In 1796 he started vaccinating people with cowpox to protect them against smallpox [168]. Two years later, the findings of the first vaccine trials were published, and by the early 1800s, the smallpox vaccine was widely available. Smallpox vaccination was used in many countries in the 19th century, but smallpox remained endemic. Smallpox was slowly eliminated from many countries, with the last case in the Americas in 1971. The last case worldwide was in Somalia in 1977, so smallpox has been eradicated throughout the world [23, 77, 168]. Most cases of poliomyelitis are asymp to matic, but a small fraction of cases result in paralysis. In the 1950s in the United States, there were about 60,000 paralytic polio cases per year. In 1955 Jonas Salk developed an injectable polio vaccine from an inactivated polio virus. This vaccine provides protection for the person, but the person can still harbor live viruses in their intestines and can pass them to others.

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