Mathematical Model for the Dynamics of Neisseria Gonorrhea Disease with Natural Immunity and Treatment Effects

Abstract

Neisseria gonorrhea infection; a sexually transmitted disease, is caused primarily by a type of germ; a bacteria called neisseria gonorrhea. The infection is a major public health challenge today due to the high incidence of infections accompanied by a dwindling number of treatment options especially in developing and underdeveloped countries. In this paper, we developed a mathematical model for the transmission dynamics of neisseria gonorrhea infection and studied the effect of natural immunity and treatment as the only available control interventions on the spread of the disease in a population. We computed the model disease-free equilibrium and analyzed its local and global stability in a well-defined positively invariant and attracting set Ω using the next-generation matrix plus linearization method and the comparison theorem respectively. The disease-free equilibrium was proved to be both locally and globally asymptotically stable if $R_0<1$ and unstable if $R_0>1$. We conducted sensitivity analysis of parameters in the basic reproduction number $R_0$ using the normalized forward sensitivity index method. Results of the analysis revealed that $R_0$ decreases with increase in treatment and natural immunity rates. The results of the numerical simulations carried out using MATLAB R2012B showed that there is increase in new infections due to increased contact with infected individuals in the susceptible population and that, with increased treatment rate and controlled death due to the disease in the population, neisseria gonorrhea infection would be wiped out within 300 days of the treatment intervention.

Optimal control dynamics of Gonorrhea in a structured population

“…Using a heterogeneous population model, we transformed the model developed by [5] in this paper. We calculated the gonorrhoea reproduction number and the gonorrhea-free equilibrium point of the model.…”

“…Developing new therapies is a crucial strategy for combating the threat posed by N. gonorrhoeae that is resistant to several common antibiotics [4] . Adamu et al [5] conducted iteratively a theoretical investigation of gonorrhea patterns and gave some ideas that help understand the dynamics of gonorrhea. Most investigations on N. gonorrhoeae infection have used a straightforward mechanistic approach for the pathogen's spread in the population [6] .…”

“…The authors of [13] performed a sensitivity analysis to determine how various factors–such as condom effectiveness, effective contact rate, condom compliance, progression rate, and treatment rate–affected their gonorrhea reproduction number, . Adamu et al [5] created a deterministic mathematical framework for the spread of N. gonorrhoeae infection and investigated how the only known control interventions-natural immunity and treatment-affected the disease's spread in a population. However, a conceptual model must take gonorrhea's specific epidemiologic characteristics into account.…”

“…Several mathematical models have employed optimal control in a single population (see, [14] , [15] , [16] , [17] ). Therefore, this study builds on the work of [5] by developing a novel mathematical model with optimal control considering the heterogeneous (crisscrossed) population. Crisscrossed models are a slight extension of the generalized population model to account for the fact that gonorrhea can be transmitted through sexual activity between males and females, where the female infectives transmit the disease to a male susceptible and vice versa.…”