The control of some childhood diseases has proven to be difficult

The control of some childhood diseases has proven to be difficult even in countries that maintain high vaccination coverage. of ordinary differential equations then extended to a system of partial differential equations to accommodate age structure. We derived analytic expressions for the steady states of the system and the final age distributions in the case of homogenous contact rates. The stability of these equilibria are determined by a threshold parameter for which = 1 yields the essential vaccination percentage a measure of herd immunity. Using this concept we can compare vaccines that confer the same level of herd immunity to the population but may fail at the individual level in different ways. For any fixed > 1 the leaky model results JNJ 1661010 in the highest prevalence of illness while the all-or-nothing and waning models possess the same stable state prevalence. The actual composition of a vaccine cannot be determined on the basis of steady state levels alone however the distinctions can be made by looking at transient dynamics (such as after the onset of vaccination) the mean age of infection the age distributions at stable state of the infected class and the effect of age-specific contact rates. of the vaccinees and ideal lifetime immunity to the remainder. Vaccine 3: Provides perfect safety to each vaccinee for an exponentially distributed “waning time ” after which the vaccinee becomes as vulnerable as unvaccinated individuals. The probability of immunity waning within a vaccinated host’s lifetime is JNJ 1661010 given by vaccine (influenced by the literature on malaria) to describe a vaccine that only exhibits failure in degree and vaccine for one that demonstrates failure in take. A vaccine that only displays failure in duration is called a vaccine. Farrington et al. [4] lists the vaccine for pertussis like a probably leaky vaccine those for measles and rubella as all-or-nothing vaccines and that for cholera like a waning vaccine. Vaccines 1 2 and 3 are examples of leaky all-or-nothing and waning vaccines respectively. The direct effects of these vaccines at the individual level are given in their descriptions but the indirect safety that they confer as a result of reduction in disease transmission (herd immunity) is not so very easily surmised from individual effects. It is important to examine this as it may possess significant implications in the control of the disease at the population level. Conversely it might also be possible to deduce info within the individual-level effects from human population data on disease incidence. Using a susceptible-infectious-recovered (SIR) JNJ 1661010 model having a vaccine component we can compare the dynamics of vaccines 1-3. Under the same assumptions on the disease it can JNJ 1661010 be shown the critical proportion of the model human population that needs to be vaccinated in order to drive the disease to extinction is the same for those three. Furthermore if the protection is managed below the essential ratio then the disease remains endemic in the model human population at a higher level for vaccine 1 while the levels for vaccines 2 and 3 are equivalent. These results follow from our investigation of the model in 2-4. Different transient JNJ 1661010 dynamics and contrasting age distributions of the infected class will also be discussed in these sections. Numerical simulations show that vaccines 1 and 3 may display a significant “honeymoon period ” a temporary period of very low disease incidence after the initiation of vaccination programs. On the other hand vaccine 2 appears to display a more stable transition from pre-vaccine to vaccine-era stable states. With this paper we present a systematic analysis of a general model of an imperfect vaccine. We begin by considering an unstructured human population described by a system of regular differential equations (ODEs) related to that EDNRB offered by McLean and Blower [14]. It is JNJ 1661010 assumed that the disease follows standard SIR dynamics. A vaccinated class is definitely added with contacts that represent the three avenues of vaccine failure. We also lengthen this into an age-structured model using the McKendrick-Von F?rster formalism [13 19 The age-structured equations consist of a system of partial differential equations (PDEs) which allows for age-specific transmission. We set up the well-posedness of this system find expressions for the disease-free and endemic stable claims and derive a threshold parameter (the basic reproduction quantity incorporating the effects of vaccination) that.