Formulation of a vector SIS malaria model in a patchy environment with two age classes

We formulate an SIS model describing transmission of highland malaria in Western Kenya. The host population is classified as children, age 1- 5 years and adults, above 5 years. The susceptibility and infectivity of an individual depend on age class and residence. The large scale system with 6 n equations is reduced into a compact form of 3 n equations by a change of variables. Then 3 n equations are vectorialized using the matrix theory to get a one dimension, compact form of the system, equation in n 3  . Using Vidyasagar theorem [1], the graph of the reduced system is shown to be strongly connected and the system is a monotone dynamical system. This means that circulation of malaria parasites among the species and among the patches is strongly connected, hence transmission is sustained. We show that for the n -dimensional age structured system the positive orthant is positively invariant for all positive values of the


Introduction
In Kenya, malaria is the leading cause of morbidity and mortality. It accounts for 30% of all outpatient attendances and 19% of all admissions to health facilities. The infection ranges from intense in the lowland to endemic in the highlands causing havoc to the public health system. About 20% of all deaths in children under five result from malaria (Ministry of Health- Kenya, 2006). People living around the lake, the coast, and the Western highlands epidemic-prone districts have 20% risk to be infected.
In the Western Kenya highlands, the risk of infection is 70%. The climate and topography influence the epidemic magnitudes. Drainage quality and rainfall determine vector breeding. The U-shaped valleys are broad and with slow moving rivers with poor drainage. This favor mosquitoes and high malaria infection and incidence. The V-shaped valleys with narrow bottoms and fast flowing rivers with good drainage are less favorable to mosquitoes. The plateaus are flat, but have good drainage. Their ecosystem resembles the V-shaped valleys without dams. The terrain can modify the transmission of malaria. The V-shaped and the U-shaped valleys are separated from each other. In Kenya malaria is a "traveling disease". 80% percent of the people treated for malaria in Kibera had travelled out of Nairobi. The neighboring estate is inhabited by people who originate from Lake Victoria region where the disease is widespread. By migrating to Nairobi, they are less exposed to malaria, thus loose the semi-immunity they used to have. This is why they contract it easily when traveling upcountry.
Children between 1 and 5 years of age are easily infected [2]. They are not bitten in same way as adults, [3] [4]. Most deaths occur in infants and parasitemia levels of infected individuals decrease with age [5]. The large areas affected by malaria make the spraying of every house impossible, [6]. Hyman [7], formulated a general differential susceptibility and differential infectivity model to prove that the disease free equilibrium is globally stable when 0 1 ≤  and unstable otherwise. When there exists an endemic equilibrium, which solutions approach asymptotically. The model can only be used for direct transmission. Pongsumpun [8] modelled the influence of age structure in an SIS model for dengue hemorrhagic fever (DHF). They showed that age structure reduces the periods of oscillations on the susceptible and infected human population and on the infected mosquito population. The difference with us is the metapopulation setting, the differentiated patch, age susceptibility and infectivity. Gao and Ruan [9] examined how population dispersal affects malaria spread between patches. The residents migrate to other patches instead of making short visits as we shall assume. There is no age structure. They showed that travel can lead the disease to become endemic in both patches, even though the disease dies out in each isolated patch. Auger [10] modified Ross [11] model to n patches without vectors migration. They model assumed that susceptibility and infectivity are similar in all patches.
Motivated by the work of Auger [10] and Pongsumpun [8], we formulate an age structured model of malaria with susceptibility and infectivity depending on residence patch.
The region is subdivided into homogenous patches, 1, 2, , i n =  and we distinguish children, aged 1-5, from adults, over 5 years of age. The two age groups are allowed to visit other patches other than their residences. Children move with the same rate as adults. Mosquitoes fly between patches less than 2 km apart, the approximate distance a mosquito can travel, Lutambi [12] and Kelly [13].

The Model
is the total vector population in patch i . Migration of hosts is allowed in and out of the patch with an assumption that those who visit return to their resident patches (i.e. they make short visits sufficient for transmission to take place). The total host population on patch i is defined as and its dynamics by ∑ is the total number of residents of patch i at time t [14].
We define the vectors C N , (2) and the matrix A M analogously. Equation (1) becomes The total population of mosquitoes V satisfies, If all the patches are sufficiently distant from each other, then mosquitoes do not migrate and The dynamics of the total host and vector populations on patch i is where N ∆ is the vector defined by The total host and vector populations are constant in all the patches. The parameter i a depends on the host age group in each patch. We also assume that the recovery rate i γ does not depend on age [5] but on how quickly the host receives treatment and history of their malaria infection. For patches The term represents the recruitment of newborns into the susceptible children's class, while caters for vector mortality. We will assume in the sequel that for any index i The rationale for this assumption is that the rate of recovering, for an adult or a child, is considerably greater than the mean sojourn time in the compartment of childhood. Actually

Reduced System
It turns out that system given by Equation (5) can be rewritten in a triangular form so we need the following theorem to reduce such a system and thus study a smaller system.
If * x is globally asymptotically stable (GAS) in n  for the system ( ) , and if * y is GAS in n  , for the system ( ) * , y g x y ′ = , then ( ) * * , x y is (locally) asymptotically stable for (6). Moreover if all the trajectories of (6) are forward bounded, then ( ) * * , x y is a GAS for (6).
To apply the above Vidyasagar theorem we would need to prove the stability analysis of first equation, then we would only have to test the stability of the infection equation.
From Equation (5) if we add the first and third equation together we get For a matrix M the stability modulus, that we will denote ( ) We have now implying that this Metzler matrix is non singular, which in turn implies the opposite, that is, its inverse is nonnegative [15] [16]. Therefore the equilibrium of this linear system is given by and is globally asymptotically stable. A similar result is obtained for the adult population with an equilibrium denoted by

Reduction Process
We will now give different expressions for the equation of our system. Depending on the case at hand we will use the most convenient form to give the properties of this system and the corresponding proofs.
This system is clearly triangular if we consider the first variables

By application of
Theorem (3.1), the stability analysis of (7) is now reduced to the stability analysis of the system  T   2  21  22  2  2  21  22  2 , , , , , , , T   1  2  1  2 , , , , , , , (8) can be written, in a vectorialized way, as I  N  N  I I  I  M I   I  N  N  I I  I  I  M I   I  I  I  N  V I  I For another variable change, we set ( ) Rewriting system (9) in terms of x , y and z , we have where we define the matrices Finally, we will make a final "vectorization" of the system  and  are nonnegative matrices and we claim that − +    is a stable Meztler matrix. Indeed the stability modulus of − is negative and where the minimum is taken over the components of the 3 positive vectors.
We have Since the matrices involved are Metzler matrices, this implies the following inequality for the corresponding stability modulus . Using the preceding matrices and the vector ( ) , , X x y z = we rewrite Equation (10) in a compact form as This system evolves on the unit cube of 3n  .

Basic Properties of the Model
For any index i we shall use the classical notations: x y < if x y ≤ and i i x y ≤ for some i and we write

Proposition 4.1 (Positively Invariant Set)
The unit cube is positively invariant for system (10).

Proof
To show the invariance of the unit cube K , under the flow of the system (10), it suffices to show that each of the faces of the cube cannot be crossed.
On the patch i we have ( ) ( ) implying that 0 x = cannot be crossed from positive to negative.   We utilise the theory of monotone dynamical systems introduced by [17] [18], developed further in [19] and applied in [10].
System (11) is monotone if its Jacobian is a Metzler matrix on the unit cube. The Jacobian of system (11) is given by The Jacobian ( ) J X is clearly a Metzler matrix since 0 1 X ≤ ≤ , which implies that the system is cooperative in the unit cube.
Next, we show that the Jacobian ( ) J X is an irreducible matrix in the set [ ] 3 0,1 n . This will imply strong monotonicity of the system in the interior of the unit cube. In this set the diagonal terms of ( ) diag 1 X − are positive, the same property is satisfied for  , then the connectivity of the associated graph of ( ) is reduced to the connectivity of  .
It is well known that a matrix is irreducible if its associated graph is strongly connected. Then only the off diagonal terms are concerned. Then it is sufficient to prove that the matrix ( ) ( ) x y z in the general digraph to obtain a new digraph which represents movements between the patches. To prove the strong connectedness, it is sufficient to prove that this contracted digraph is strongly connected.
To the matrix    , there is a corresponding digraph between the patches. This graph is equivalent to the associated digraph of C  . Since and its corresponding multigraph on the graph of patches. This oriented multigraph represents actually the circulation of malaria parasites between the patches. Malaria parasites can be transported either by child, adults or mosquitoes.
Since we have assumed that the matrix is irreducible, our graph is strongly connected, consequently the Jacobian is irreducible. We note that we are in a context of multiple species (i.e., childs, adults and mosquitoes) in a metapopulation model as conceptualized in [20].
Our assumption simply means that the circulation of parasites is strongly connected. : NB The establishment of the equilibrium solution and study of stability for the disease free and the endemic equilibrium will be tackled in a forth coming article.

Conclusions
In this study, we formulated an infinite model representing the spread of malaria in a heterogenous population classified. This population is classified as children (0 -5 years) and adults (above 5 years). The infectivity and susceptibility of the population to highland malaria depend on: 1) the age class and; 2) the type of ecosystem the individual inhabits. The study captures the role played by the age of the individual and the ecosystem heterogeneity in malaria epidemically spreading in the regions called patches. Hats are defined depending on their topography. Model properties are described to show that the solution set exists in the positive orthant which is positively invariant. The system is reduced to a single equation in 3n  , which is strongly connected and strongly monotone. The large scale system has been reduced using vectorialization into a compact form which is easier to study. The establishment of equilibrium solution and stability study will be done in a forthcoming article.