A general mathematical model for a disease with an exposed
(latent) period and relapse is proposed. Such a model is
appropriate for tuberculosis, including bovine tuberculosis in
cattle and wildlife, and for herpes. For this model with a
general probability of remaining in the exposed class, the basic
reproduction number $\R_0$ is identified and its threshold
property is discussed. In particular, the disease-free equilibrium
is proved to be globally asymptotically stable if $\R_0<1$.
If the probability of remaining in the exposed class is assumed to
be negatively exponentially distributed, then $\R_0=1$ is a
sharp threshold between disease extinction and endemic disease. A
delay differential equation system is obtained if the probability
function is assumed to be a step-function. For this system, the
endemic equilibrium is locally asymptotically stable if
$\R_0>1$, and the disease is shown to be uniformly persistent
with the infective population size either approaching or
oscillating about the endemic level. Numerical simulations (for
parameters appropriate for bovine tuberculosis in cattle) with
$\mathcal{R}_0>1$ indicate that solutions tend to this endemic
state.
Citation: P. van den Driessche, Lin Wang, Xingfu Zou. Modeling diseases with latency and relapse[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 205-219. doi: 10.3934/mbe.2007.4.205
Abstract
A general mathematical model for a disease with an exposed
(latent) period and relapse is proposed. Such a model is
appropriate for tuberculosis, including bovine tuberculosis in
cattle and wildlife, and for herpes. For this model with a
general probability of remaining in the exposed class, the basic
reproduction number $\R_0$ is identified and its threshold
property is discussed. In particular, the disease-free equilibrium
is proved to be globally asymptotically stable if $\R_0<1$.
If the probability of remaining in the exposed class is assumed to
be negatively exponentially distributed, then $\R_0=1$ is a
sharp threshold between disease extinction and endemic disease. A
delay differential equation system is obtained if the probability
function is assumed to be a step-function. For this system, the
endemic equilibrium is locally asymptotically stable if
$\R_0>1$, and the disease is shown to be uniformly persistent
with the infective population size either approaching or
oscillating about the endemic level. Numerical simulations (for
parameters appropriate for bovine tuberculosis in cattle) with
$\mathcal{R}_0>1$ indicate that solutions tend to this endemic
state.
References