A Stochastic Model for Malaria Transmission Dynamics

Abstract

Malaria is one of the three most dangerous infectious diseases worldwide (along with HIV/AIDS and tuberculosis). In this paper we compare the disease dynamics of the deterministic and stochastic models in order to determine the effect of randomness in malaria transmission dynamics. Relationships between the basic reproduction number for malaria transmission dynamics between humans and mosquitoes and the extinction thresholds of corresponding continuous-time Markov chain models are derived under certain assumptions. The stochastic model is formulated using the continuous-time discrete state Galton-Watson branching process (CTDSGWbp). The reproduction number of deterministic models is an essential quantity to predict whether an epidemic will spread or die out. Thresholds for disease extinction from stochastic models contribute crucial knowledge on disease control and elimination and mitigation of infectious diseases. Analytical and numerical results show some significant differences in model predictions between the stochastic and deterministic models. In particular, we find that malaria outbreak is more likely if the disease is introduced by infected mosquitoes as opposed to infected humans. These insights demonstrate the importance of a policy or intervention focusing on controlling the infected mosquito population if the control of malaria is to be realized.

Mathematical Modeling of the Infectious Diseases: Key Concepts and Applications

“…In the numerical solution presented by Ngeleja, et al [21], In the absence of interventions, the parameter that has the greatest effect on the dynamics of transmission to humans and rodents is the β infectivity rate in the flea population, suggesting that the most effective strategies should focus on vector control, with the infected populations of humans and rodents being negligible parameters, as well as the environment in the transmission dynamics. This result is similar to those from Mbogo, et al [22], that developed a comparative analysis of deterministic and stochastic models with the structural dynamics susceptible and infected (SI) for…”

“…That is, the deterministic approach provides an overall insight about the disease spreading in a fast way, whereas the stochastic framework provides statistical insights into the transmission events providing a range of possible epidemic scenarios [22]. Deterministic models tend to present results that do not undergo major changes due to fluctuations in the population, but undergo significant changes if the parameters inserted in the differential equations are modified; in this context, the stochastic models are more responsive to quantitative changes both in the populations and subpopulations, as well as in the modeling parameters, making it important to emphasize that there are several ways of working probabilities in stochastic processes as addressed above, which makes these models too complex and difficult to interpret [22,30,35].…”

“…That is, the deterministic approach provides an overall insight about the disease spreading in a fast way, whereas the stochastic framework provides statistical insights into the transmission events providing a range of possible epidemic scenarios [22]. Deterministic models tend to present results that do not undergo major changes due to fluctuations in the population, but undergo significant changes if the parameters inserted in the differential equations are modified; in this context, the stochastic models are more responsive to quantitative changes both in the populations and subpopulations, as well as in the modeling parameters, making it important to emphasize that there are several ways of working probabilities in stochastic processes as addressed above, which makes these models too complex and difficult to interpret [22,30,35]. Moreover, both deterministic and stochastic models share the challenge in representing the natural history of infectious diseases, considering sources of infection, transmission routes, incubation period, infection and transmissibility periods, treatment, and development of natural immunity, which are parameters that can be implemented in mathematical systems through three ways: i) Using data already described in the literature [36], ii) Empirically through estimations with basis on epidemiological data [37,38], or iii) Estimated by computer programs using statistical methods such as root mean square error on epidemiological data [17].…”

“…Highlighting the model's parameter were evaluated using the maximum-likelihood adopting 95% of confidence interval, then sets of parameters generated by latin hypercube sampling were computed assuming twice the difference of log-likelihood values that was X 2 distributed with degrees of freedom with values equal to the number of estimated parameters, then 700 simulations were run by the gillespie's first reaction method, and in each simulation, the partial rank correlation coefficient was computed to analyze the influence of each parameter in the epidemic size. [21,22,29,45], by the exponential growth rate method and maximum likelihood method [17], or by bayesian statistics [46]. Moreover, the R0 is a parameter used to evaluate the efficacy of interventions such as quarantine, mask-wearing, vaccination, washing hands in hospital sets, among others; where if the intervention decreases the R0 to values smaller than 1, it is considered effective, and uneffective if it does not change the R0 [7,[47][48][49].…”

Modelling the Transmission Dynamics of Tuberculosis in the Ashanti Region of Ghana

“…Chowell et al and Mbogo et al argued that increasing the infection rate implies increasing the number of infectious individuals, while increasing the exposed rate simply suggests decreasing the exposed or incubation period. Nonetheless, increasing the recovery rate means a decline in the infectious period [39,40].…”