The regulation of an age-structured population by a fatal disease

A model describing the effect of a fatal disease on an age-structured population which would otherwise grow is presented and analysed. If the disease is capable of regulating host numbers, there is an endemic steady age distribution (SAD), for which an analytic expression is obtained under some simplifying assumptions. The ability of the disease to regulate the population depends on a parameter R(alpha), which is defined in terms of the given age-dependent birth and death rates, and where alpha is the age-dependent disease-induced death rate. If R(alpha) < 1 the endemic SAD is attained, while R(alpha) > 1 means the disease cannot control the population's size. The number R(0) is the expected number of offspring produced by each individual in the absence of the disease; for a growing population we require R(0) > 1. A stability analysis is also performed and it is conjectured that the endemic SAD is locally asymptotically stable whenever it is attained. This is demonstrated explicitly for a very simple example where all rates are taken as constant.

Thresholds and stability analysis of models for the spatial spread of a fatal disease

A simple two-class (susceptibles and infectives) model describing the dynamics of a fatal disease in a variable-size population is presented and analysed. Spatial dependence is introduced into the model by considering two different mechanisms for the geographic spread of the disease: nonlocal interaction between susceptibles and infectives, and migratory spread of the animals. The steady states and their stability for these spatially dependent models are deduced; no spatially heterogeneous steady states were possible. For nonlocal interaction, there were two spatially uniform steady states: the trivial state (no infectives or susceptibles), which was unstable, and the endemic state (constant proportion of the population infected), which was locally asymptotically stable. With migratory spread, the number of spatially uniform steady states was dependent on the boundary conditions imposed. With hostile (Dirichlet) boundary conditions, only the trivial steady state was possible and its local stability found to depend on the rate of diffusion of the total population. With no-flux (Neumann) boundary conditions, the steady states are the trivial and endemic states; these were unstable and locally asymptotically stable, respectively.

The population dynamics of nematode infections of ruminants: the effect of seasonality in the free-living stages

A previously published differential equation model for the dynamics of nematode infections of ruminants is modified to allow for time-dependent development and loss rates in the free-living stages. This converts the nonlinear autonomous system to one with periodic coefficients. Fourier transform methods are used to analyse the response of the modified system to these forced oscillations. For biologically reasonable parameter values, the annual pattern of parasitism is determined for the nonautonomous system, and compared with that for the autonomous system with periodic perturbations, and a system incorporating both effects. It is found that, whereas periodic perturbations due to management intervention determine the qualitative annual patterns of larval abundance and host infection, the seasonal dynamics of the larval stages change the magnitude of the adult worm burden.

The dynamics and control of bovine tuberculosis in possums

A model for the dynamics of an infectious disease in a wild animal population is analysed. The model incorporates susceptible, infected but not infectious, and infectious classes, with no recovery from disease, density-dependent regulation of host birth and death rates, and pseudo-vertical transmission. The existence and local stability of equilibrium levels are determined. Expressions for the intensity of three alternative control measures - the culling, sterilization, and vaccination of host animals - required to eradicate the disease are obtained. Finally, the model is applied to possible control measures against bovine tuberculosis infection in possums in New Zealand.

The population dynamics of nematode infections of ruminants: periodic perturbations as a model for management

We propose and analyse a model for the dynamics of directly transmitted nematode infections of ruminants which allows for the long-term effects of the annual removal of hosts. The model is a simple continuous-time formulation which captures the principal features of parasite transmission and the acquisition of immunity to infection by the host. Analysis of the simplest version of the model, which assumes a constant host population through time, indicates that its equilibrium is locally stable for feasible biological parameters. The regular removal of hosts involved in most management strategies is modelled in terms of periodic perturbations. The analysis indicates that the periodic removal of parasitized hosts corresponds effectively to a reduction in the basic reproductive rate of infection. We also demonstrate that the system exhibits a dramatic peak in parasite numbers during each grazing season at its dynamic equilibrium. This phenomenon (which is characteristically observed in real systems) is discussed in terms of the balance between acquired immunity and periodic perturbations of the parasite population.

Population dynamics in echinococcosis and cysticercosis: regulation of Taenia hydatigena and T. ovis in lambs through passively transferred immunity

A comparison has been made of the interactions between passively transferred and actively acquired immunity in regulating populations of Taenia hydatigena and T. ovis. When ewes were grazed prior to parturition under a high infection pressure, immunity was transferred to their offspring for up to 8 weeks. A qualitative difference between the species was the destruction of larval T. ovis prior to their establishment ('pre-encystment immunity') and that of T. hydatigena after they had become established ('post-encystment immunity') in the challenged lambs. The major difference in terms of population regulation between the two parasites was that infection occurred with T. hydatigena but not with T. ovis in those lambs reared from birth for 16 weeks under high infection pressure. Passive, like active immunity, is a density-dependent constraint. It plays an important role in the population regulation of T. ovis, but not of T. hydatigena. This is discussed in terms of transmission in the natural environment, an hypothesis on humoral protection and the need to elucidate pathways of protection when immunization schedules are being evaluated for controlling the taeniid zoonoses.

Population dynamics in echinococcosis and cysticercosis: economic assessment of control strategies for Echinococcus granulosus, Taenia ovis and T. hydatigena

An official control programme against Echinococcus granulosus and Taenia hydatigena has been in operation in New Zealand for more than 28 years and against Taenia ovis for more than 18 years. This unique effort to control three metazoan parasites at the same time has led to a change from endemic to extinction status for E. granulosus but only a change from hyperendemic to endemic status for T. hydatigena and T. ovis. This has presented problems in determining the most cost-effective future control strategies. To facilitate this, a benefit/cost analysis of 20 options for the combined control of E. granulosus, T. hydatigena and T. ovis in New Zealand was undertaken. This showed that for E. granulosus a future change from the current non-targeted to a targeted approach is strongly indicated. For T. ovis 6 options were cost-effective using a discount rate of 10%. These were (1) a targeted control package using a vaccine in the non-targeted attack phase; (2) a targeted control package using a larvicide in the attack phase; (3) the transfer of all losses due to and responsibility for the control of T. ovis to the producer who administers a larvicide to sheep to be killed for dog food; (4) the transfer of all losses due to and responsibility for the control of T. ovis to the producer who administers praziquantel every 6 weeks to dogs; (5) and (6) two options involving the discontinuation of control. Control of T. hydatigena was assumed to be an incidental outcome of the policies for the other two parasites.

Population dynamics in echinococcosis and cysticercosis: evaluation of the biological parameters of Taenia hydatigena and T. ovis and comparison with those of Echinococcus granulosus

An evaluation has been made of the biological and epidemiological parameters that determine the basic reproductive rates of Taenia hydatigena and T. ovis. These host-parasite systems are characterized by (i) no overcrowding in either host; (ii) no parasite-induced mortality of either host; (iii) no density-dependent constraint in the definitive host, but a strong, rapidly mobilized, short-acting immunity in the intermediate host and (iv) egg production which, in the natural environment, is high enough to prevent superinfection. It is considered that tapeworms with these characteristics are more stable to fluctuations in environmental conditions and to control measures such as dog dosing, than species which have a low egg production and infectivity such as Echinococcus granulosus. Reciprocal immunity exists between T. hydatigena and T. ovis in sheep. Exposure to T. hydatigena suppresses infection by T. ovis, but not by E. granulosus. This has important epidemiological consequences where these parasites co-exist.

Population dynamics in echinococcosis and cysticercosis: mathematical model of the life-cycles of Taenia hydatigena and T. ovis

It is shown that under the conditions that prevailed in New Zealand in the late 1950s, Taenia hydatigena was hyperendemic, the life-cycle being regulated by a density-dependent constraint in the form of acquired immunity, and T. ovis was rare. The control measures that caused Echinococcus granulosus, which was endemic at the time, to decline towards extinction reduced T. hydatigena and T. ovis to endemic status only. A non-linear integrodifferential equation model, which was previously linearized to describe the life-cycle of E. granulosus in dogs and sheep in New Zealand, is used to describe the life-cycles of T. hydatigena and T. ovis. The model is then used to compare and contrast the population dynamics of these three species. The model is used to demonstrate that the endemic steady state is structurally unstable, and may be asymptotically unstable to small perturbations. It is also shown that despite the lower infection pressure experienced by the intermediate host in the endemic state, the numbers of larvae in sheep may be higher than in the hyperendemic state. Finally it is shown that the partial success of the control measures against T. hydatigena may have caused an increase in the numbers and prevalence of T. ovis larvae in sheep due to the reciprocal immunity between the two species.

Towards global control of cystic and alveolar hydatid diseases

Control programmes against Echinococcus granulosus in its dog-sheep transmission cycle (Fig. 1) have been successful in many parts of the world'. In contrast, the related E. multilocularis presents a much more complex problem for control authorities. Unlike E. granulosus, the life cycle of E. multilocularis predominantly involves sylvatic hosts (e.g. rodents and foxes) (Fig. 2) and the control of wild life echinococcosis presents a formidable challenge to ecologists and epidemiologists. This review contrasts the two parasites, explaining why E. granulosus in its domestic dog-sheep life cycle has been so responsive to control, and examines the prospects for control of E. multilocularis.

Population dynamics in echinococcosis and cysticercosis: comparison of the response of Echinococcus granulosus, Taenia hydatigena and T. ovis to control

A comparative study has been made of the progress in the control of ovine echinococcosis caused by Echinococcus granulosus and the ovine cysticercoses caused by Taenia hydatigena and T. ovis in New Zealand. The methods of control included an educational and a dog-dosing programme. In the Styx field trial, the Otago/Southland Surveillance Programme and the national control programme, E. granulosus declined towards extinction. In contrast, this same control effort transformed the cysticercoses from hyper-endemic to endemic status with focal epidemics. In this unstable state, the density-dependent constraint was negligible and superinfection occurred.

Population dynamics in echinococcosis and cysticercosis: biological parameters of Echinococcus granulosus in dogs and sheep

The numerical distributions of Echinococcus granulosus in an experimental dog population are described. At all dose rates of protoscoleces from 10 to 175000 the distribution of worms was over-dispersed. Host age had no effect. There was a direct proportionality between the infective-stage density and rate of infection, and between the latter and the index of clumping. The worm burdens were significantly higher in the proximal than distal portions of the small intestine. Lengths of the 3- and 4-segmented worms increased from 4 to 10 and 4 to 8 weeks of age, respectively. Thereafter apolysis was asynchronous and could not be determined. Eggs were first detected in the faeces at 6 weeks and the mean age at oogenesis was 7.26 weeks. Retarded growth of the whole population of worms was observed in some dogs. For the first few infections, worm burdens varied widely in the same dog, but by the 6th infection 50% of the dog population had developed a relative insusceptibility to infection. Growth or oogenesis of the worms were not affected. A short-acting immune response was artificially induced in some dogs following the parenteral injection of activated embryos of E. granulosus, Taenia hydatigena, T. ovis, T. multiceps, T. pisiformis and T. serialis. The response affected either the number of worms established, growth or oogenesis or all three parameters. There was a strong positive correlation between numbers and lengths of worms in dogs with acquired and induced immunity, indicating that no 'crowding' effects were involved. In sheep populations the mean number of cysts which established was directly proportional to the number of eggs given, implying that there was no negative feedback mechanism operating at this stage of the life-cycle. The distribution of the larval population in sheep was over-dispersed and the index of clumping increased with the size of the egg dose from 25 to 2500 eggs. Protoscoleces were first observed in cysts at 2 years and the proportion producing them increased with age, with an estimate of 50% of cysts containing protoscoleces at 6.29 years. No deaths were observed in dogs or sheep even when high parasite burdens were present, implying that E. granulosus does not regulate the population of its hosts.

Population dynamics in echinococcosis and cysticercosis: mathematical model of the life-cycle of Echinococcus granulosus

A mathematical model of the life-cycle of Echinococcus granulosus in dogs and sheep in New Zealand is constructed and used to discuss previously published experimental and survey data. The model is then used to describe the dynamics of transmission of the parasite, and the means by which it may be destabilized. It is found that under the conditions that prevailed in New Zealand during the late 1950s, at the time of surveys of this parasite, the dog-sheep life-cycle was not regulated by any effective density-dependent constraint. In contrast there was evidence for an effective acquisition of immunity to reinfection by cattle. The long time to maturity of the cyst in sheep, together with the practice of feeding aged sheep to dogs, provides a time delay in the intermediate host. By comparison, the time to maturity of the adult stage in dogs is short, but it is of sufficient magnitude to be a key factor in the destabilization of the system by a regular dog-dosing programme. The model used to describe the life-cycle is a linear integrodifferential equation of the Volterra type. Such equations are intrinsically unstable in that a small perturbation in parameters can drive a previous equilibrium solution to zero. At the time of the surveys, the value of the basic reproductive rate, R0, was close to 1, and it has since been reduced below 1 by control measures.

An accurate demand feeder for fish, suitable for microcomputer control

This paper describes an easily constructed and inexpensive demand feeder. The feeder is driven by an AC synchronous motor and gearbox and is suitable for microcomputer control. It will operate with inexpensive commercially available pelleted fish food, and will consistently deliver a single pellet for each operation of the motor. The components and materials can be purchased for approximately 23 pounds.

Control of echinococcosis/hydatidosis: present status of worldwide progress

Progress in national and provincial control programmes directed against Echinococcus granulosus is reviewed. The results show that this parasite, because of the low basic reproductive rate in its domestic life-cycle, is unstable and is amenable to control or eradication by currently available methods. The difference between control and eradication is defined and some suggestions for extension of control are discussed.

Stability in cyclic epidemic models

A general formulation for a family of cyclic epidemic models with density-dependent feedback mechanisms and removed classes is presented. A parameter, lambda, related to the basic reproductive rate determines the asymptotic behaviour of solutions of the model. It is shown that if lambda less than 1 the trivial solution is globally stable, and if lambda greater than 1 it is conditionally stable. The results are applied to a set of differential equations that has been used to model the life cycle of a parasite that has two hosts.

Vaccination of sheep against Taenia ovis. The response to various dose rates of antigens obtained by incubation of oncospheres in vitro

Activated oncospheres of Taenia ovis were incubated in vitro for 14 days and the antigens produced were collected and concentrated. Tenfold serial dilutions of the antigens were emulsified with Freund's incomplete adjuvant and injected intramuscularly into sheep. When subsequently challenged with T ovis eggs, these sheep displayed a level of immunity that was related to the amount of antigen injected. The calculated dose-response curve indicated that for the vaccine to be 90 per cent effective, the secretion from approximately 4600 T ovis oncospheres need to be injected into each sheep.

Effects of hypothalamic lesions on the food intake of the goldfish (Carassius auratus)

Goldfish were trained to perform an operant in response in order to obtain food, thereby allowing the food intake to be accurately determined. The normal daily food intake was established for each fish before it was given a sham operation or bilateral hypothalamic lesions. It was then observed over a period of up to 60 days. Lesions of the lateral areas of the hypothalamus were found to cause cessations of operant feeding of up to 60 days duration, and cessations of feeding on manually presented food of up to 35 days. Lesions of the anterior-medial areas were followed by cessationsof operant feeding of up to 26 days. The post-lesioning food intake of fish lesioned in the posterior-medial area was similar to that of the control fish. No increases in operant feeding were observed after lesioning. The results are considered to be consistent with a genuine aphagia induced by lesions of the lateral hypothalamus.

Behavioural effects of electrical stimulation of the hypothalamus of the goldfish (Carassius auratus)

Food-satiated goldfish were stimulated in tanks containing food on the surface of the water and on the bottom of the tank. Various components of the feeding response were seen, and complete feeding was observed, which was elicited from sites in the postero-medial hypothalamus. Evidence is put forward to support the supposition that the full response was due to a change in feeding motivation, rather than to changes in sensory or motor factors. Other types of behaviour, such as fin movements and swimming, could not be identified as part of any normal ethological sequence. However, vertical movements and gas-spitting responses indicated a possible hypothalamic control of buoyancy. None of the behaviours observed was seen to change in a social context.