SIRS epidemics with individual heterogeneity of immunity waning (preprint)

We analyse an extended SIRS epidemic model in which immunity at the individual level wanes gradually at exponential rate, but where the waning rate may differ between individuals, for instance as an effect of differences in immune systems. The model also includes vaccination schemes aimed to reach and maintain herd immunity. We consider both the informed situation where the individual waning parameters are known, thus allowing selection of vaccinees being based on both time since last vaccination as well as on the individual waning rate, and the more likely uninformed situation where individual waning parameters are unobserved, thus only allowing vaccination schemes to depend on time since last vaccination. The optimal vaccination policies for both the informed and uniformed heterogeneous situation are derived and compared with the homogeneous waning model (meaning all individuals have the same immunity waning rate), as well as to the classic SIRS model where immunity at the individual level drops from complete immunity to complete susceptibility in one leap. It is shown that the classic SIRS model requires least vaccines, followed by the SIRS with homogeneous gradual waning, followed by the informed situation for the model with heterogeneous gradual waning. The situation requiring most vaccines for herd immunity is the most likely scenario, that immunity wanes gradually with unobserved individual heterogeneity. For parameter values chosen to mimic COVID-19 and assuming perfect initial immunity and cumulative immunity of 12 months, the classic homogeneous SIRS epidemic suggests that vaccinating individuals every 15 months is sufficient to reach and maintain herd immunity, whereas the uninformed case for exponential waning with rate heterogeneity corresponding to a coefficient of variation being 0.5, requires that individuals instead need to be vaccinated every 4.4 months.

Extending SIRS epidemics to allow for gradual waning of immunity (preprint)

SIRS epidemic models assume that individual immunity (from infection and vaccination) wanes in one big leap, from complete immunity to complete susceptibility. For many diseases immunity on the contrary wanes gradually, something that's become even more evident during COVID-19 pandemic where also recently infected have a reinfection risk, and where booster vaccines are given to increase immunity. This paper considers an epidemic model allowing for such gradual waning of immunity (either linear or exponential waning) thereby extending SIRS epidemics, and also incorporates vaccination. The two versions for gradual waning of immunity are compared with the classic SIRS epidemic, where the three models are calibrated by having the same \emph{average cumulative immunity}. All models are shown to have identical basic reproduction number $R_0$. However, if no prevention is put in place, the exponential waning model has highest prevalence and the classic SIRS model has lowest. Similarly, the amount of vaccine supply needed to reach and maintain herd immunity is highest for the model with exponential decay of immunity and lowest for the classic SIRS model. consequently, if truth lies close to exponential (or linear) decay of immunity, expressions based on the SIRS epidemic will underestimate the endemic level and the critical vaccine supply will not be sufficient to reach and maintain herd immunity. For parameter choices fitting to COVID-19, the critical amount of vaccine supply is about 50% higher if immunity wanes linearly, and more than 150% higher when immunity wanes exponentially, as compared to the classic SIRS epidemic model.

Analysis of a stochastic coronavirus (COVID-19) Lévy jump model with protective measures

This paper studied a stochastic epidemic model of the spread of the novel coronavirus (COVID-19). Severe factors impacting the disease transmission are presented by white noise and compensated Poisson noise with possibly infinite characteristic measure. Large time estimates are established based on Kunita’s inequality rather than Burkholder-Davis-Gundy inequality for continuous diffusions. The effect of stochasticity is taken into account in the formulation of sufficient conditions for the extinction of COVID-19 and its persistence. Our results prove that environmental fluctuations can be privileged in controlling the pandemic behavior. Based on real parameter values, numerical results are presented to illustrate obtained results concerning the extinction and the persistence in mean of the disease. [ABSTRACT FROM AUTHOR] Copyright of Stochastic Analysis & Applications is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Bayesian forecast of the basic reproduction number during the Covid-19 epidemic in Morocco and Italy

In a Covid-19 susceptible-infected-recovered-dead model with time-varying rates of transmission, recovery, and death, the parameters are constant in small time intervals. A posteriori parameters result from the Euler-Maruyama approximation for stochastic differential equations and from Bayes’ theorem. Parameter estimates and 10-day predictions are performed based on Moroccan and Italian Covid-19 data. Mean absolute errors and mean square errors indicate that predictions are of good quality. [ABSTRACT FROM AUTHOR] Copyright of Mathematical Population Studies is the property of Routledge and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)