Together, the extensions of the basic model illustrate that the finding of increased attack rates due to a variable duration of immunity is generic, and is expected to be found in even more complex models
Together, the extensions of the basic model illustrate that the finding of increased attack rates due to a variable duration of immunity is generic, and is expected to be found in even more complex models. Open in a separate window Fig. seasons results in a disproportionately large outbreak in a subsequent season. Importantly, variation in the duration of immunity increases the average infection attack rate when the vaccination coverage is around the outbreak threshold. Specifically, in a tailored age-stratified model with a realistic reproduction number (in the previous season, i) vaccination, ii) the yearly influenza epidemic, and iii) the loss SX-3228 of SX-3228 immunity during the inter-epidemic period by virus evolution and demographic turnover. As these processes largely take place sequentially and on different timescales, we model these processes sequentially. In the model, vaccination is applied before the epidemic, and loss of immunity takes place after the epidemic in the inter-epidemic period. Schematically, the model looks as follows: =?(1???and are the vaccination coverage (the proportion of individuals who are vaccinated) and vaccine efficacy (the proportion of vaccinated individuals that is protected against infection), respectively. Thus, a fraction is protected against infection. Throughout, we vary and take =?=?s(1???exp(?is the fraction of the population that retains its immunity. Loss of immunity is caused by CRF2-S1 the reduction of cross-protection between the current strain and strains circulating previous seasons. Throughout, we explore the impact of annual variation in the rate at which immunity is lost due to antigenic drift of the virus (i.e. the effect of variability in from a realistic Beta(5,2) distribution that has a mean of 5/7. In this manner, SX-3228 the deterministic scenario arises naturally as a special case of the stochastic scenario when the parameters of the beta distribution tend to infinity. In both scenarios, immunity lasts 3.5 years on average in both scenarios. Combining the above, we obtain the following set of equations that map the susceptibility in year to the susceptibility and set =?(1???+?is the proportion of vaccinated individuals that is protected, while with a leaky vaccine, we define to be the probability that vaccinated individuals are protected in a single exposure that would have led to transmission to an unvaccinated individual. Unless stated otherwise, we take and are the attack rates of susceptible and vaccinated individuals, and is the overall attack rate. As in the basic model, each year a fraction 1?of the vaccinated and partially immune individuals lose immunity. Combining the above, we obtain the following map linking and to with elements is given by is the (symmetric) contact rate between individuals in age groups and is the proportion of the population in age group SX-3228 [40]. The proportionality parameter is used to scale such that the dominant eigenvalue equals 1 and in the next season are given by as in Eq. 5. To calculate the peak prevalence in the age-stratified model (Eq. 5), we consider the age-stratified SIR-model with next-generation matrix (given by, math xmlns:mml=”http://www.w3.org/1998/Math/MathML” display=”block” id=”M48″ overflow=”scroll” mrow msub mrow mi /mi /mrow mrow mi i /mi /mrow /msub mo = /mo mi /mi munder mrow mo mathsize=”big” /mo /mrow mrow mi j /mi /mrow /munder msub mrow mi g /mi /mrow mrow mtext mathvariant=”italic” ij /mtext /mrow /msub mfenced close=”)” open=”(” separators=”” mrow msubsup mrow mi I /mi /mrow mrow mi j /mi /mrow mrow mo ( /mo mn 1 /mn mo ) /mo /mrow /msubsup mo + /mo msubsup mrow mi I /mi /mrow mrow mi j /mi /mrow mrow mo ( /mo mn 2 /mn mo ) /mo /mrow /msubsup /mrow /mfenced mo , /mo /mrow /math and em i /em , {em j /em children,. Following [35], we take em /em =2.5 (day ?1) and em /em =1.1 (day ?1). Furthermore, we take em /em = em /em em R /em 0/2, and other parameters are as specified earlier. Results Variation in the duration of immunity increases the height of epidemic peaks To investigate the impact of variation in the duration of immunity on the epidemic dynamics, we compare scenarios with and without SX-3228 variation in the duration of immunity. For the moment, we consider the case without vaccination. If the duration of immunity is constant, the number of individuals who are infected over an epidemic is balanced exactly by the number of individuals added to the susceptible pool during the inter-epidemic period by waning of immunity. As a consequence, we observe regular annual epidemics in this scenario (Fig.?1). When we take the more realistic assumption that the duration of immunity is variable, and vary the fraction of the population that loses its protection in each year,.