An Age-structured Model for the Spread of Buruli Ulcer:Analysis and Simulation in Ghana

Despite the advances in medical research on its treatment and intensive public education on prevention 4 and control, the Buruli ulcer (BU) continues to be a major public health problem that continues to 5 overwhelm authorities in Ghana. Ghana is the second most endemic country after the Ivory Coast at the 6 global level. While it is common knowledge in literature that the disease can affect people of all ages, the 7 mode of transmission is still evasive. The studied model is expressed as a system of hyperbolic (first 8 order) partial differential equations. We first, employ a representation from the method of characteristics 9 and a fixed point argument and also prove the existence and uniqueness of solutions to the nonlinear 10 system. We establish the mathematical well-posedness of the time evolution problem using the 11 semigroup theory approach. We then determine the basic reproduction ratio R0. Then we present 12 a numerical scheme to model the dynamics of BU. The simulation results showed that Mycobacterium 13 ulcer has peak period of spread and reduced subsequently. 14 15


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Buruli ulcer, also known as Bairnsdale ulcer is a chronic, indolent, and necrotizing disease of the 19 skin tissue caused by Mycobacterium ulcerans (M. ulcerans) [5]. The disease usually begins as a 20 painless nodule or papule and may progress to massive skin ulceration [9]. It also appears that 21 different modes of transmission occur in different geographical and epidemiological setting [18]. 22 Though the disease can affect people of all ages, children less the 15 years of age are particularly 23 more vulnerable in many tropical and subtropical countries [12]. Buruli ulcer causes serious pain 24 as well as permanent physical damage. The physical signs visually mark the individual and 25 deprive them of societal standards of beauty. Additionally, physical deformities may prevent the 26 individual from participating in any economic and social activities.

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The study of Buruli ulcer continues to be an important problem in mathematical epidemiology as 29 outbreaks of M. ulcerans continues to pose a public health challenge [1]. The mode of 30 transmission of the ulcer is not well understood, however residence near aquatic environment has 31 been identified as a risk factor for the ulcer in Africa [12,13]. The modes of transmission vary 32 with geographical and epidemiological settings [18]. In Africa, it is estimated that almost 30, 000 33 cases were reported between 2005 and 2010 [14]. Buruli ulcer is a severe, disfiguring disease 34 which affects all age groups but particularly children less the 15 years of age in many tropical 35 and subtropical countries [17]. The disease has emerged over the past two or more decades, and [16] for nonlinear age structured models. Barbu [15] developed mathematical theory behind 67 age-structured populations [3] and studied nonlinear age-dependent population and predator prey 68 dynamics. 69 The increasing mathematical complexity of biological issues, nonlinearities and age structure in 70 biological models, has brought about new dimension of analyzing them. One of these powerful 71 tools is method semi-groups of linear and nonlinear operators in Banach spaces. 72 This paper sought to develop an age-structured BU model and provide some theoretical and 73 numerical analysis of the model. The system differential equations along with initial and 74 boundary conditions that form the disease model will be discussed. We will further prove the 75 existence and uniqueness of the solutions in 1 L and L  to our PDE system using the fixed point 76 theory on a representation derived from the method of characteristics. Finally the numerical 77 simulations and its implications will be discussed.
For the above equations 1 7 week days   is the coefficient introduced to balance the units of age a  The boundary and initial conditions 137 Buruli ulcer disease does not transmit vertically from parent to infants and therefore we can infer 138 that children have some immunity. In this regard, newborns will appear in the R class in SIR 139 model. This is significantly different from most , , S I R model. We translate this consideration to 140 state the boundary conditions.
where the fecundity function f is stated as The fecundity function (.) f is stated here in units of per year for easier readability and assumes 147 that from age 15 to 40 years a woman will give generally give birth to three children, since is the largest age allowed for the simulation [25].

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The initial conditions are stated as  155 We assume that all the parameters are nonnegative, i.e 0, 0, 0, 0

Abstract Cauchy problem formulation
The parameters fulfil the following assumptions.

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Abstract Cauchy problem 160 In this section we seek to deal with quantitative properties of (2a)-(2d) as in [25,26]. In order to 161 undertake this, we consider the Banach spaces. Characterize the space of functions Endowed by the norm One obtains that ( ) u t satisfies the following abstract Cauchy problem together with the initial data We also take into account the positive cones is the unique integrated solution of (2.21a) with initial data y , Proof: Let us take into consideration that for each N centered at 0 . One gets the existence of maximal 193 Also let us consider the quantity the total population at time t . Then it satisfies the differential inequality Thus the map ( ) t Q t  cannot blow up in finite time and the global existence result follows. Let 198 us in addition, notice that, from this inequality one gets One the other hand one has This completes the proof of the result.

Equilibria and their stabilities
208 It is easy to demonstrate the following set is positively 209 invariant for the system (2a-d) System of equation (2a-d) always has the disease free equilibrium To simplify 212 expressions, we introduce the following notations

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This equilibrium satisfies the following equations

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Solving the second and fourth equations of (2.4b)-(2.4d) respectively, leads  3.0 Existence of the solution to the state system by method of characteristics 244 We determine solution of the system applying the method of characteristics [8]. By using Banach  The method of characteristics to determine the existence of a solution of system state was 261 applied. The Banach contraction mapping principle to prove the existence and uniqueness of 262 solution was applied. In order to find solution representation for the system (2a--d), the following 263 notation for the right hand sides of the partial differential equation (PDE S ) was created:       By the fact W is sufficiently minimal, then the above 308 estimate is less than or equal to 2B.

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The constants D1 and D2 hinge upon the coefficients and the parameters in the model. Also for 312 D to be sufficiently small, we get the estimates above and hence, the L maps Y into Y.

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Note that for the contraction property, for 314 315 1, 2, 3 i  we take into account this contraction property was taken into account where  is a constant (that is the wave speed), and t and x denote time and space, respectively.

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The forward time/backward space scheme [17] for the above model is expressed as   In Figure 2 above, we show the dynamics in the total population, susceptible population, infected 411 population, and recovered population over time. We note here, the decrease in the susceptible 412 population, which is attributed to humans who died of natural causes during the period-line of 413 the simulation. Furthermore, we notice an increase in recovered population, which is partly due 414 to antibiotic and partly due to natural recovery of MU by humans.

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This however, takes sometime for human to lose immunity to get back to susceptible class and is 416 governed by the rate of waning of immunity. In Figure 2b we see that the infection reduced with

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An age-structured model can model the infection pathway of Buruli ulcer more accurately since 448 the risk for contracting the disease has something to do with the age of a human being [1]. We 449 observe that introducing age as another independent variable encompasses solving a system of