A Stochastic Model for Kala-azar Transmission Dynamics in Libo Kemkem, Ethiopia


 The objective of this paper is to analyse and demonstrate the dynamics of Kala-azar infected group using stochastic model, particularly using simple SIR model with python script over time. The model is used under a closed population with N = 100, transmission rate coefficient β = 0.09, recovery rate γ = 0.03 and initial condition I(0) = 1. In the paper it is discussed how the Kala-azar infected group behaves through simple SIR model. The paper is completed with stochastic SIR model simulation result and shows stochasticity of the dynamics of Kala-azar infected population over time. Fig. 2 below depicts continuous fluctuations which tells us the disease evolves with stochastic nature and shows random process.Subject: Infectious Disease, Global Health, Health Informatics and Statistical and Computational Physics


Background of the Study
Vector-borne diseases(VBD) are infections transmitted by the bite of infected arthropod(insect) species, such as mosquitoes, ticks, triatomine bugs, sand flies, and black flies [1]. Arthropod vectors are coldblooded (ectothermic) and thus especially sensitive to climatic factors. Weather influences survival and reproduction rates of vectors, in turn influencing habitat suitability, distribution and abundance; intensity andtemporal pattern of vector activity (particularly biting rates) throughout the year; and rates of development, survival and reproduction of pathogens within vectors [2].
"Kala-azar" (or Indian Visceral Leishmaniasis for "black fever") are a group of diseases with very complex dynamics caused by more than 20 species of the protozoan genus Leishmania that are transmitted between humans and other mammalian hosts by the vector phlebotomine sand-flies [3]. There are two major clinical forms of leishmaniasis, namely Cutaneous Leishmaniasis (CL) and Visceral Leishmaniasis (VL) depending on the species of leishmania responsible and the immune response to infection [4].
The cutaneous form tends to heal spontaneously leaving scars. Depending on the species of leishmania responsible it may evolve into diffuse cutaneous leishmaniasis, recidivate leishmaniasis, or mucocutaneous leishmaniasis, with disastrous aesthetic consequences for the patient.
Visceral leishmaniasis has two major forms leishmania donovani, and leishmania infantum. VL shows signs of fever, weight loss, splenomegaly, and anaemia and is the most severe form and fatal in almost all cases if left untreated. It may cause epidemic outbreaks with a high mortality rate. A varying proportion of visceral cases may evolve into a cutaneous form known as Post Kala-azar Dermal Leishmaniasis (PKDL), a nonfatal stage of infection dermatological symptoms, & which requires lengthy and costly treatment [5].
The disease endangers some 350 million people in 98 countries, most of them in the poorer regions of the globe [6]. An estimated 200,000 to 400,000 new cases of VL occur worldwide each year. And from this, greater than 90% of VL human cases occur in six countries, namely Bangladesh, Brazil, Ethiopia, India, South Sudan and Sudan [7]. Eastern Africa has the second highest number of VL cases, after the Indian Subcontinent. The disease is endemic in Eritrea, Ethiopia, Kenya, Somalia, Sudan, South Sudan, and Uganda [8].
In Ethiopia, the first case of VL was documented in 1942 in the lower Omo plains, the southwestern part of the country [9]. The disease has spread to become endemic in many parts of the country. It is prevalent mostly in lowland, arid areas, and the parasite involved is mainly leishmania donovani, with an estimated annual incidence of more than 4,000 cases [10]. Most important endemic foci include the Humera and Metema plains in the northwest [11], the Omo plains, the Aba Roba focus, and the Weyto River Valley in the southwest of the country [12].
In the country, VL mainly occurs in the arid and semi-arid areas; however, recent reports indicate spreading of the disease to areas where it was previously non-endemic [13]. In 2003, an outbreak of VL occurred in highland areas of the Libo Kemkem district, in the Amhara regional state [14].
VL is becoming a growing public health threat; the spatial distribution and burden of VL is up surging year after year [15]. VL-HIV co-infection is rising in Ethiopia, and it poses a new and difficult challenge to VL control effort [16]. VL-HIV co-infection is characterized by a number of complexities, including challenging diagnosis, increased drug toxicity, and poor treatment response. Leishmaniasis in Ethiopia was formerly overseen by the Ministry of Health (MoH), but after the MoH underwent a large reorganization in 2007, a national leishmaniasis task force was established with the aim of eliminating VL by 2015 [17]. Efforts made by the MoH so far to control VL are not withstanding its upsurge and, hence, VL is developing both on a spatial and temporal basis. Considering its recent upsurging, I thoroughly reviewed and analysed the previous works done, and I also propose a way forward to tackle this disease.

Statement of the problem
Conducting epidemiology research that deals with the vector-borne disease has a long history [18].
Plenty of these studies used mathematical models. In this paper I focused on the analysis of the dynamics of Kala-azar disease using stochastic model.

Models and method
In this paper the dynamics of Kala-azar disease is modelled using Stochastic SIR model. The model is developed with the data collected from Addis Zemen primary hospital report and it is simulated with python programming software.

Model Formulation
In the SIR epidemic model of [19], the population is divided into three categories of individuals: S the susceptible individuals, I the infected individuals and R the recovered and immune individuals. The parameters of the model are denoted by β and γ.
 β is the transmission rate coefficient of infections in a time period (with β greater or equal to zero) and its value can be easily determined from the solutions of deterministic model.  γ is the recovery rate (with γ greater or equal to zero).
A compartmental diagram in Fig.1 below illustrates the relationship between the three classes(S, I and R).
Arrows denotes possible changes and transition probabilities.

Model Assumptions
The SIR Model is used in epidemiology to compute the amount of susceptible, infected and recovered people in a population. This model is an appropriate one to use under the following assumptions[20]: 1. The population is fixed (100 populations in this case).
2. The only way a person can leave the susceptible group is to become infected. The only way a person can leave the infected group is to recover from the disease. Once a person has recovered, the person received immunity.
3. Age, sex, social status, and race do not affect the probability of being infected.
4. There is no inherited immunity. 5. The member of the population mix homogeneously (have the same interactions with one another to the same degree).

Equations of the model
Assume the time variable is continuous, t ∈ [0, ∞) and the states S(t), I(t) and R(t) are continuous random variables, that is, The SIR epidemic process is bivariate, And where S(t) = is the number of susceptible individuals to VL at time t.
R(t) = is the number of recovered individuals from VL at time t. and N is the total population size.
The random variable I(t) has an associated probability density function (pdf), ( , ), For the stochastic SIR epidemic model, it can be shown that the pdf satisfies a forward Kolmogorov differential equation. The coefficient of ( , ) in the first term on the right side of the preceding equation, ] is the infinitesimal mean and the coefficient of ( , ) in the second term,  Table 1 below.

Change
Probability Event The vector ∆ (1) = (−1,1) represents the change of one individual from population in to (the infection of one individual) during ∆ with a probability that is proportional to [23]. And the vector ∆ (2) = (0, −1) represents a recovery of a population with probability 2 = ∆ .
Let ∆ ( ) = (∆ , ∆ ) , then the expectation vector µ and the variance covariance matrix of ∆ ( ) to order ∆ are Where, The drift and diffusion terms determine the change in number of infections over time [26].
The ratio ⁄ is the proportion of contacts by one infected individual and ( ⁄ ) the number of contacts by the infected population. The ratio 1 ⁄ denotes the average length of the infection period [27].
As [28] by applying Euler's method of systems, we can solve the deterministic model From these equations, we can discover how the different groups will act as → ∞.
From equation (12) of the first term, we understand that the susceptible group will decrease over time and approach zero. From the third term of equation (12), we also understand that the recovered group increase and will approach over time. However, in the second term of the equation it is difficult to understand easily how the infected group behaves.
The solutions to the deterministic differential equations are: where +1 , +1 and +1 are the number of susceptible, infected and recovered people at time ( + 1). ∆ is a small change in time, and will be equal to one.
In SIR model assumptions, birth rate and death is zero (b = d = 0). A key threshold outcome of an epidemic model is typically determined by the basic reproductive number, often denoted by R 0 . It is defined as the total number of secondary infections caused by single infected individuals. This threshold property provides important information about the potential of disease and impact of control mechanism. An epidemic will outbreak if and only if this number is greater than one else the disease is die out. The basic reproduction number is defined [29] as follows: From equation (16) then I (t) decreases monotonically to zero (disease-free equilibrium) and the disease eventually disappears from the population.

Stochastic Simulation Algorithm
A version of the ℎ_ algorithm is used to code this model as detailed in the procedure below: Step 1. Initialize the infected population of the system 0 = 1; Step 2. Calculate the transition rates for the given population of the system; Step 3. Simulate the time, until the next transition by drawing from an exponential distribution; Step 4. Simulate the transition type by drawing from the discrete distribution. Generate a random number from a uniform distribution and choose the transition as follows: choose transition 1 (do infection else: choose transition 2 (do recovery); Step 5. Update the new time and the new system population; Step 6. Iterate steps 2-5 until ≥

Result and discussion
In this work, stochastic model is formulated and analysed for SIR model with a closed population made up of 100 people that mix homogeneously. Since the disease is highly infectious, everyone will eventually become infected. From the population how many people will be infected by the disease in each state at a given period of time which is collected from Addis Zemen primary hospital report(in Libo Kemkem district) shows the infected population increases at the middle months of the year and it is shown in Table 2 below. This is because of the sand fly highly incubated in those months in the study area. At the beginning, 99 persons were susceptible to the disease, and the one had been infected by the disease. A period also lasts 29 days, an estimated average duration of Kalazar infection obtained from the report. In these cases, everyone recover in one period of 29 days (the average period of infectiousness at   (13), with ∆ = 1 to get the following = − +1 (18) Using this equation we can get for each period. The values are calculated for each period and the average value is obtained to 0.09 as shown in the Table 3 below.

Conclusion
It can be concluded SDE SIR Model is used in the modelling of infectious diseases by computing the amount of people in a closed population that are susceptible, infected, or recovered at a given period of time. In this work a stochastic SIR model without the presence of birth and death rate in the population is applied to analyse the transmission of Kala-azar disease. I have shown the stochasticity (random nature) of the dynamics of Kala-azar infected population using SIR model. The disease is endemic in the study area (R 0 > 1). Reproductive number, can be reduced through an increase in the recovery rate through medication [31].