Analytical threshold and stability results on age-structured epidemic models with vaccination. (English) Zbl 0657.92008
The paper studies an age-structured mathematical model of diseases (measles, rubella and mumps) taking into account the effect of a vaccination program. The model is a three-compartmental age-structured model. The population is divided into susceptible, infected and immune individuals and vaccination terms of susceptible individuals are introduced.
First, the general case of a vaccination term of the form f(a)X (f is a function of the age a and \(X=X(a,t)\) is the age density of susceptibles of age a at time t) is studied. It is shown that if a threshold value \(\rho\leq 1\) then there is just one equilibrium solution with no disease present. This solution is unstable if \(\rho <1\), whereas it is locally stable but not asymptotically stable if \(\rho =1\). If the threshold value \(\rho\) is greater than 1, there are two possible equilibria: one with no disease present that is stable and one with disease present.
Then, existence and behaviour of equilibrium solutions are studied in four different special cases of vaccination: 1) vaccination of a constant proportion p of susceptibles at age A; 2) vaccination of a constant proportion of all ages; 3) vaccination of all ages at a constant rate; 4) vaccination of a fixed number at a constant age.
In case 1) the equilibrium solution can be locally stable or unstable. This fact depends on the sign of the real part of the roots of a particular equation and this occurs both in case of constant death rate and in case of step death rate. In case 2) the equilibrium solution with disease present is always stable if the death rate is constant. With a step death function the equilibrium solution is locally stable or locally unstable according to the sign of the real part of the roots of a particular equation.
In case 3) the vaccination term is not of the type f(a)X but \(pI(X(a)>0)\) and corresponds to vaccination which acts with an effect independent of the number of the susceptible individuals remaining. In the model with constant death rate, if \(\rho\leq 1\) there is just one equilibrium solution with no disease present. If \(\rho >1\) there are two possible equilibrium solutions: one with disease present and one with no disease present. In the model with step death function, if \(\rho <1\) there is just one equilibrium solution corresponding to no disease present. If \(p>1\) there are two possible equilibrium solutions: one with disease present and one with no disease present. In case 4) the equilibrium solution and the threshold value are given explicitely and also with a general age dependent death rate.
First, the general case of a vaccination term of the form f(a)X (f is a function of the age a and \(X=X(a,t)\) is the age density of susceptibles of age a at time t) is studied. It is shown that if a threshold value \(\rho\leq 1\) then there is just one equilibrium solution with no disease present. This solution is unstable if \(\rho <1\), whereas it is locally stable but not asymptotically stable if \(\rho =1\). If the threshold value \(\rho\) is greater than 1, there are two possible equilibria: one with no disease present that is stable and one with disease present.
Then, existence and behaviour of equilibrium solutions are studied in four different special cases of vaccination: 1) vaccination of a constant proportion p of susceptibles at age A; 2) vaccination of a constant proportion of all ages; 3) vaccination of all ages at a constant rate; 4) vaccination of a fixed number at a constant age.
In case 1) the equilibrium solution can be locally stable or unstable. This fact depends on the sign of the real part of the roots of a particular equation and this occurs both in case of constant death rate and in case of step death rate. In case 2) the equilibrium solution with disease present is always stable if the death rate is constant. With a step death function the equilibrium solution is locally stable or locally unstable according to the sign of the real part of the roots of a particular equation.
In case 3) the vaccination term is not of the type f(a)X but \(pI(X(a)>0)\) and corresponds to vaccination which acts with an effect independent of the number of the susceptible individuals remaining. In the model with constant death rate, if \(\rho\leq 1\) there is just one equilibrium solution with no disease present. If \(\rho >1\) there are two possible equilibrium solutions: one with disease present and one with no disease present. In the model with step death function, if \(\rho <1\) there is just one equilibrium solution corresponding to no disease present. If \(p>1\) there are two possible equilibrium solutions: one with disease present and one with no disease present. In case 4) the equilibrium solution and the threshold value are given explicitely and also with a general age dependent death rate.
Reviewer: S.Totaro
Keywords:
epidemiology; age-structured mathematical model of diseases; measles; rubella; mumps; vaccination program; three-compartmental age-structured model; threshold value; locally stable; asymptotically stable; equilibrium solutionsReferences:
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