Measuring Infection Transmission in a Stochastic SIV Model with Infection Reintroduction and Imperfect Vaccine

Abstract

An additional compartment of vaccinated individuals is considered in a SIS stochastic epidemic model with infection reintroduction. The quantification of the spread of the disease is modeled by a continuous time Markov chain. A well-known measure of the initial transmission potential is the basic reproduction number R 0 , which determines the herd immunity threshold or the critical proportion of immune individuals required to stop the spread of a disease when a vaccine offers a complete protection. Due to repeated contacts between the typical infective and previously infected individuals, R 0 overestimates the average number of secondary infections and leads to, perhaps unnecessary, high immunization coverage. Assuming that the vaccine is imperfect, alternative measures to R 0 are defined in order to study the influence of the initial coverage and vaccine efficacy on the transmission of the epidemic.

Measures to assess a warning vaccination level in a stochastic SIV model with imperfect vaccine

“…We recall that in accordance with diphtheria and vaccine characteristics, it is possible to determine the appropriate vaccination level that provides herd immunity in the population. In a Markovian stochastic framework, vaccination coverage providing herd immunity can be determined in terms of the exact reproduction number $$R_{e0}$$ 30,31 . Applying this methodology to our choice of model parameters, we get that an initial coverage of 424 vaccinated individuals reduces viral transmission and prevents major outbreaks in the whole institution.…”

“…According to the above description, the involved model is a standard SIS/logistic one, with external infection and the additional feature that some of the population is initially vaccinated, with imperfect immunity. More specifically, mathematical model was introduced in the paper 31 as a compartmental model that, at any particular instant $$t$$, classifies individuals as susceptible $$(S)$$, vaccinated $$(V)$$, or infected $$(I)$$. Figure 1 represents the movement of individuals among the three epidemiological classes.…”

“…The matrix structure of the infinitesimal generator $$\mathbf {Q}=[q_{(v,i),(v^*,i^*)}]$$ of $$X$$ was fully described in Appendix A of our previous paper 31 and it shows a block bidiagonal representation, which is really appropriate for computational purposes.…”

“…In consequence, herd immunity is linked to appropriate estimates of the basic reproduction number 𝑅 0 , that measures the reproductive potential of a disease regarding population characteristics and also vaccine attributes, when available. [28][29][30][31][32] Public health objective of vaccination is to increase the level of herd immunity to keep the infection controlled or even eliminated from the population, and in the longer term to eradicate the infection world or region-wide. To achieve this goal, vaccination coverage must be optimal.…”

On First-Passage Times and Sojourn Times in Finite QBD Processes and Their Applications in Epidemics

“…Gaver et al [1]. They are capable of modeling, in a unified and algorithmically tractable manner, relevant systems, such as queueing models (Baumann and Sandmann [2]; Gun and Makowski [3]; Perel and Yechiali [4]; Ye and Li [5]), communication networks (Artalejo and Gómez-Corral [6]; Artalejo and Lopez-Herrero [7]), reliability and manufacturing systems (Chakravarthy and Gómez-Corral [8]; Moghaddass et al [9]), and epidemic models (Amador and Gómez-Corral [10]; Artalejo et al [11]; Economou et al [12]; Gamboa and Lopez-Herrero [13]), among others. The properties of finite QBD processes have been studied extensively in continuous and discrete time.…”