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Dengue disease is the most common vector borne infectious disease transmitted to humans by infected adult female Aedes mosquitoes. Over the past several years the disease has been increasing remarkably and it has become a major public health concern. Dengue viruses have increased their geographic range into new human population due to travel of humans from one place to the other. In the present paper, we have proposed a multi patch SIR-SI model to study the host-vector dynamics of dengue disease in different patches including the travel of human population among the patches. We have considered different disease prevalences in different patches and different travel rates of humans. The dimensionless number, basic reproduction number R0 which shows that the disease dies out if R0 < 1 and the disease takes hold if R0 ≥ 1, is calculated. Local and global stability of the disease free equilibrium are analyzed. Simulations are observed considering the two patches only. The results show that controlling the travel of infectious hosts from high disease dominant patch to low disease dominant patch can help in controlling the disease in low disease dominant patch while high disease dominant becomes even more disease dominant. The understanding of the effect of travel of humans on the spatial spread of the disease among the patches can be helpful in improving disease control and prevention measures. In the present study, a patch may represent a city, a village or some biological habitat.

Dengue disease is regarded as a serious infectious disease. The four serotypes of viruses DEN 1 to DEN 4 are responsible for the disease. It is one of the re-emerging diseases in tropical and subtropical countries. A person infected by one of the four serotypes of dengue viruses will never be infected again by the same serotype, but the person loses immunity to other serotype of viruses and becomes more susceptible in developing dengue hemorrhagic fever [

There have been many mathematical studies to understand the dynamics of infectious diseases. Mathematical models can help in providing guides and suggestions for the control of the disease to the concerned authorities. Kermack and McKendrick introduced an SIR model to study the transmission of infectious diseases [

Emerging and re-emerging diseases like dengue disease spread very quickly due to the travel of infective human population from one region to the other. They spread the disease in new regions. Different spatial models have been developed to study infectious diseases. Arino and Driessche [

Lee and Castillo-Chavez [

For the formulation of the model, we divide human population in three classes, susceptible, infective and recovered. Let

The SIR-SI Model for

Symbols | Description |
---|---|

death rate in host population | |

death rate in vector population | |

recovery rate of host population | |

transmission probability from vector to host | |

transmission probability from host to vector | |

biting rate of vector | |

recruitment rate of host population | |

recruitment rate of vector population | |

travel rate of susceptible, infective, recovered host population from patch j to patch i, |

where,

The total host and vector population sizes in all n-patches in time t is

Theorem 1. The system of Equations (2.1) has a unique disease free equilibrium point.

Proof: A disease free equilibrium (DFE) for the system of Equations (2.1) is a steady state solution of the system where

In disease free situation,

In matrix form,

where,

Here,

Now, we show that the disease free equilibrium is unique. From the system of Equations (2.1), in disease free situation:

For the host populations only:

i.e.,

where,

For vector populations only:

i.e.,

where,

Here, the matrix C has positive column sums and each non-diagonal element is negative. So, the matrix C is an irreducible and non-singular M-matrix. Again, since C is an irreducible non-singular M-matrix, C must have positive inverse, i.e.,

Also, the matrix D is a diagonal matrix with positive diagonal elements. So, there exists

Basic reproduction number

We use next generation matrix method [

where,

Here, the matrix

Also,

In fact,

Theorem 2. If R_{0} < 1, then the disease free equilibrium is locally asymptotically stable and unstable if R_{0} > 1.

Proof: Let

Matrix J is triangular. So, the eigenvalues of J are those of the partition matrices

Matrices C and D (matrices defined in Theorem 1) are non-singular M matrices. So, spectral abscissa,

Hence, the matrix J will have eigenvalues all with negative real parts if the matrix

If

Theorem 3. If R_{0} < 1, then the disease free equilibrium is globally asymptotically stable and unstable if R_{0} > 1.

Proof: Since,

Consider the linear system

The system of Equations (3.2) can be written as

where,

i.e., Eigenvalues of

i.e.,

Since all the variables in the system of Equations (2.1) are non-negative, the use of Comparison theorem [

From the system of Equations (2.1), we have

i.e.,

Here,

Again, as

In matrix form

Here, matrices C and D are non-singular M-matrices, all their eigenvalues lie in left half plane. Therefore, if

Matrix C is an irreducible, non-singular M-matrix. So, the matrix C has positive inverse.

for all

Thus, as

We have

and

Basic Reproduction Number

where,

We considered the case of two patches and computed basic reproduction number

When the susceptible hosts come in contact with infectious mosquitoes, hosts get infected. So, the population size of infected hosts increases (

Changes in basic reproduction number

dominant patch are high. Thus, we should increase the travel rates of hosts from high disease dominant patch to the low disease dominant patch to bring the disease under control.

Basic reproduction number

In this section, the dynamics of the host population is observed with the restriction of the travel of symptomatic hosts from one patch to the other patch.

Restricting the travel of symptomatic hosts from low disease dominant patch to high disease dominant patch

(

Similarly, when

Dynamics of infected host populations are observed in

In the present work, we have studied the effect of travel of humans on the transmission dynamics of dengue disease. We discussed the disease transmission dynamics between n-patches by subdividing vector population in susceptible and infectious class and host population in susceptible, infectious and recovered class.

We defined the multi-patch basic reproduction number

Travel of human from one place to another place affects the whole dynamics of the dengue disease transmission. We have shown that traveling of infected human changes the less disease dominant patch to high disease

dominant patch. Also, restricting the travel of infected hosts helps in controlling the disease. Basic reproduction number is seen higher when there is higher travel rate from low disease dominant patch to the high disease dominant patch. The basic reproduction number is seen lowered when there is higher travel rate from high dominant disease patch to the low disease dominant patch. Thus, we can control the disease in low disease dominant patch by restricting the travel of infected hosts from high disease dominant patch.

Ganga Ram Phaijoo,Dil Bahadur Gurung, (2016) Mathematical Study of Dengue Disease Transmission in Multi-Patch Environment. Applied Mathematics,07,1521-1533. doi: 10.4236/am.2016.714132