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A transmission model of malaria with two delays is formulated. We calculate the basic reproduction number

Malaria is a mosquito-borne disease which is due to the four species of the genus Plasmodium, namely,

Sporozoites are injected into a human host, which are carried through the blood to the liver within 30 min [

The application of mathematics to study of infectious disease appear to have been initiated by Berniulli, 1760. Presently, a lot of mathematical models have been proposed to evaluate and compare control procedures and preventive strategies, and to investigate the relative effects of various sociological, biological, and environmental factors on the spread of diseases ([

The transmission process involves considerable time delay both in human and in mosquitoes due to the incubation periods of the several forms of the parasites [

Both the models in [

In order to study the impact of incubation periods in both human and mosquitoes on the basic reproduction number and the transmission dynamics of malaria over long periods, we propose a model based on SIRS in human population and SI for the mosquito vector population. Since the mosquito dynamics operates on a much faster time scale than the human dynamics and the turnover of the mosquito population is very high and the total size of vector population is largely exceeding the human total size, the mosquito population can be considered to be at a equilibrium with respect to changes in the human population. Hence, the total number of mosquito population is assumed to be constant.

We formulate an SIRS-SI model for the spread of malaria in the human and mosquito population with the total population size at time

Considering the assumption made above, the interaction between human hosts and the mosquito vector population with standard incidence rate is described as shown below:

System (

In order to reduce the number of parameters and simplify system (

Since

Define the basic reproduction number by

Equating the derivatives on the left-hand side to zero and solving the resulting algebraic equations. The points of equilibrium

The solution

When

When

When

If

Then, we can conclude the above results in the following theorem.

(1) The disease-free equilibrium

(2) If

(3) If

(4) If

It is easy to see the disease-free equilibrium

Linearizing the system (

If

We denote by

Usually, the disease can be controlled if the basic reproduction number is smaller than one. Nevertheless, Theorem

To check for this, the discriminant

Let

Thus, the following result is established.

The model (

Epidemiologically, because of the existence of backward bifurcation, whether malaria will prevail or not depends on the initial states.

The existence of the subthreshold condition

From the expression of

From Theorem

To obtain precise results, we first assume that malaria does not produce significant mortality (

Linearizing the system (

For the endemic equilibrium

It is clear that

When

When

We assume that

Clearly, If

When

When

If

In this paper, in order to study the impact of the incubation periods in human and mosquitoes, we formulate a model with two delays for the transmission dynamics of malaria.

Existence of equilibria is obtained under different conditions and their stabilities are analyzed too. We have also identified the basic reproduction number

The parameters

The reason inducing backward bifurcation in [

Malaria transmission can be affected by a lot of aspects. In this paper, we are trying to model the impact of temperature increasing on malaria transmission. In some extent, our results can provide a theoretical principle for allocating and using medical health resource reasonably, which can be applied in the practice of malaria prevention and control.

H. Wan is supported by NSFC (no. 11201236 and no. 11271196) and the NSF of the Jiangsu Higher Education Committee of China (no. 11KJA110001 and no. 12KJB110012). J.-a. Cui is supported by NSFC (no. 11071011) and Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality (no. PHR201107123).