Miniproject
Modelling malaria epidemy using deterministic and stochastic methods
1 Motivation
We consider an extended epidemic model to simulate the dynamics of malaria epidemy, in- evolving the demography of both humans and malaria mosquitoes.
Malaria is a communicable disease and can lead to a dramatical decrease of the human population. Even though the disease has been investigated for hundreds of years, it still remains a major public health problem with 109 countries declared as endemic to the disease in 2008. There were then 243 million malaria cases reported, and nearly a million deaths - primarily of children under 5 years. In 2018, according to the World Health Organization, there have been 228 million cases of malaria and 405 000 death cases. Hence, malaria is responsible for the fifth greatest number of deaths due to infectious diseases and is the second leading cause of death in Africa behind AIDS. Therefore, a significant effort is put into constructing adequate mathematical models that will enable us to simulate the development of this epidemy and to study the mechanisms by which such malaria spread, to predict the future course of an outbreak and to evaluate strategies to control it.
Already in 1927, Kermack and McKendrick ([2]) created an epidemic model (also known as SIR model) for the following three compartments (seen as corresponding densities):
• susceptibles (S)- who have yet to contract the disease and become infectious,
• infectives (I) - who can pass on the disease to others,
• removed (R)- who have been infected but cannot transmit the disease for some reason.
The SIR model is of the form of a system of three coupled ordinary differential equations, as described in Lab1, Exercise 3.
We consider a different type of model, referred to as ’host-vector-host’ model, where the disease spreads indirectly through an intermediary (’vector’), a typical example of which is malaria.
Malaria is caused by infection with single-celled parasites of genus Plasmodium. The parasites are transmitted to humans through the bites of infected female mosquitoes (the vectors). The so-called Ross-MacDonald model, developed in 1910 ([3]) is the earliest attempt to quantitatively describe the dynamics of malaria transmission at a population level. We consider a more advanced model, described in [1]. It is again a deterministic differential
equation model of malaria, however it better differentiates the various groups in the human and mosquito populations, and is expected to enable a better description of the processes during the development and seise of a malaria epidemy.
The model divides the two population into the following groups:
Sh susceptible humans
Ih infected humans
Eh exposed humans, i.e., in a transition state, when the individuals who are infected are not able to pass on the infection to others
Rh recovered humans
Sm susceptible mosquitoes Im infectious mosquitoes Em exposed mosquitoes
(There are no ’recovered’ mosquitoes due to their short life time.) The model parameters are the following:
Λh Birth number of humans Λm Birth number of mosquitoes b Biting rate of the mosquito
βh Probability that a bite by an infectious mosquito results in transmission of disease to human
βm Probability that a bite results in transmission of parasite to a susceptible mosquito
µh Per capita rate of human mortality µm Per capita rate of mosquito mortality δh Disease induced death rate of human
δm Disease induced death rate of mosquito
αh Per capital rate of progression of humans from the exposed state to the infectious state
αm Per capital rate of progression of mosquitoes from the exposed state to the infectious state
r Recovery rate of humans
ω Per capital rate of loss of immunity in humans
νh Proportion of an antibody produced by human in response to the incidence of infection caused by mosquito
νm Proportion of an antibody produced by mosquito in response to the inci- dence of infection caused by human
The detailed description of the deterministic model is given in Section 2.
As in many cases, a deterministic model of the above type can be converted into a stochas- tic model in terms of reactions, as shown in Section 3.
Major aim
The major task of the miniproject is to compare the deterministic and the stochastic models from Sections 2 and 3 - their performance and sensitivity to certain parameters.
2 Description of the deterministic model
The deterministic model consists of the following coupled nonlinear ODEs:
The initial conditions and the parameter values are as follows
Sh(0) = 900 Λh = 20 δh = 0.05
Ih(0) = 50 Λm = 40 δm = 0.15
Eh(0) = 30 b = 0.075 αh = 0.6
Rh(0) = 20 βh = 0.3 αm = 0.6
Sm(0) = 900 βm = 0.5 νh = 0.5
Im(0) = 270 µh = 0.015 νm = 0.5
Em(0) = 30 µm = 0.02 ω = 0.02
r = 0.05
3 Description of the stochastic model
The ODE model (2) is converted to a system of reactions as follows:
4 Tasks
4.1 Perform numerical simulations using the ODE model
(ODE 1) Implement the ODE solver. You can use the uploaded Matlab script MalariaSym ODE 0.m, edit it correspondingly and use as a solver your implementation of Ralston’s method.
(ODE 2) Run simulations for the time interval [0, 100].
(ODE 3) Plot the solutions (say, in Figure 1) on one and the same plot (each curve in a different color or line type) and add a legend.
(ODE 4) In a separate figure (say, Figure 2) plot the total amount of humans and mosquitoes as a function of time.
(ODE 5) Repeat the simulation with new values of the parameters, Λm = 0.5 and νm = 0.15.
These values can be thought as a result of measures to distinguish the mosquitoes.
4.2 Perform numerical simulations using the SSA model
(SSA 1) Implement the SSA solver. The functions Malaria stoch.m and Malaria prop.m that define the reactions and the corresponding propensities, are given. You can use the uploaded Matlab script MalariaSym SSA 0.m and edit it correspondingly. When deter- mining the next reaction, use the code you have written in Assignment 3 (or some other suitable Matlab commands).
(SSA 2) Run simulations for the time interval [0, 100].
(SSA 3) Plot the solutions (say, in Figure 3) on one and the same plot (each curve in a different color or line type) and add a legend.
(SSA 4) In the same figure used for the ODE simulations, Figure 2, plot the total amount of humans and mosquitoes as a function of time.
(SSA 5) Repeat the simulation with parameters, changed as given in Section 4.1.
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