A more realistic mathematical model of malaria is introduced, in which we not only consider the recovered humans return to the susceptible class, but also consider the recovered humans return to the infectious class. The basic reproduction number
Malaria is caused by a parasite called
Many epidemic models have been analyzed mathematically and applied to specific diseases [
Recently, Li et al. [
Chitnis et al. [
Motivated by these works, in this paper, we propose a more realistic mathematical model of malaria, in which we assume that the recovered humans return to the susceptible class and relapse. The basic reproductive number
The organization of this paper is as follows. In the next section, a mathematical model of malaria with relapse is formulated. In Section
In this section, we introduce a mathematical model of malaria with relapse. Because hosts might get repeatedly infected due to not acquiring complete immunity so the population is assumed to be described by the SIRS model. Mosquitoes are assumed not to recover from the parasites so the mosquito population can be described by the SI model. The total number of population at time
The parameters description of malaria model.
|
From an infectious human to a susceptible mosquito, transmission rate in mosquitoes |
|
From a recovered human to a susceptible mosquito, transmission rate in mosquitoes. |
|
From an infectious mosquito to a susceptible human, transmission rate in humans |
|
The total size of human population |
|
The total size of mosquito population |
|
Natural birth and death rate of humans |
|
Treatment rate |
|
Recovery rate |
|
Relapse rate |
|
Natural birth and death rate of mosquitoes |
|
The number of mosquitoes per individual |
Transfer diagram of the model (
Notice that from (
For system (
If
Under the given initial conditions, it is easy to prove that the solutions of the system (
In the second case, we have
In the third case, we have
The model (
The model has a disease-free equilibrium given by
Let
For system (
The linearised system (
In the following, we prove that when
For system (
We introduce the following Lyapunov function [
If
It follows from system (
We using the persistence theory of dynamical system to show the uniform persistence of the disease when
Let
By this lemma, we can show the uniform persistence of disease when
In system (
We set
Noting that
To illustrate the analytical results obtained above, we give some simulations using the parameter values in Table
The parameters values of malaria model.
|
From an infectious human to a susceptible mosquito, transmission rate in mosquitoes |
|
[ |
|
From a recovered human to a susceptible mosquito, transmission rate in mosquitoes |
|
[ |
|
From an infectious mosquito to a susceptible human, transmission rate in humans |
|
[ |
|
The total size of human population | Estimated | |
|
The total size of mosquito population |
|
[ |
|
Natural birth and death rate of humans |
|
[ |
|
Treatment rate |
|
[ |
|
Recovery rate | Estimated | |
|
Relapse rate | Estimated | |
|
Natural birth and death rate of mosquitoes |
|
[ |
|
The number of mosquitoes per individual | 1-2 | [ |
The relationship between
The relationship between
Finally, for showing the effect of relapse and recover rate to the basic reproduction number, we give the relation between
An ordinary differential equation for the transmission of malaria is formulated in this paper. The model exhibits two equilibria, that is, the disease-free equilibrium and endemic equilibrium. By constructing Lyapunov function and persistence theory of dynamical system, it is shown that if
On behalf of all the authors, Hai-Feng Huo declares that there is no conflict of interests regarding the publication of this paper.
This work was partially supported by the NNSF of China (10961018), the NSF of Gansu Province of China (1107RJZA088), the NSF for Distinguished Young Scholars of Gansu Province of China (1111RJDA003), the Special Fund for the Basic Requirements in the Research of University of Gansu Province of China, and the Development Program for Hong Liu Distinguished Young Scholars in Lanzhou University of Technology.