Computational and Mathematical Methods in Medicine

Volume 2019, Article ID 2658971, 15 pages

https://doi.org/10.1155/2019/2658971

## Optimal Control Analysis of Pneumonia and Meningitis Coinfection

Haramaya University, Department of Mathematics, Dire Dawa, Ethiopia

Correspondence should be addressed to Getachew Teshome Tilahun; moc.liamg@hcegmg

Received 19 June 2019; Revised 26 July 2019; Accepted 20 August 2019; Published 22 September 2019

Academic Editor: Michele Migliore

Copyright © 2019 Getachew Teshome Tilahun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we proposed a deterministic model of pneumonia-meningitis coinfection. We used a system of seven ordinary differential equations. Firstly, the qualitative behaviours of the model such as positivity of the solution, existence of the solution, the equilibrium points, basic reproduction number, analysis of equilibrium points, and sensitivity analysis are studied. The disease-free equilibrium is locally asymptotically stable if the basic reproduction number is kept less than unity, and conditions for global stability are established. Then, the basic model is extended to optimal control by incorporating four control interventions, such as prevention of pneumonia as well as meningitis and also treatment of pneumonia and meningitis diseases. The optimality system is obtained by using Pontryagin’s maximum principle. For simulation of the optimality system, we proposed five strategies to check the effect of the controls. First, we consider prevention only for both diseases, and the result shows that applying prevention control has a great impact in bringing down the expansion of pneumonia, meningitis, and their coinfection in the specified period of time. The other strategies are prevention effort for pneumonia and treatment effort for meningitis, prevention effort for meningitis and treatment effort for pneumonia, treatment effort for both diseases, and using all interventions. We obtained that each of the listed strategies is effective in minimizing the expansion of pneumonia-only, meningitis-only, and coinfectious population in the specified period of time.

#### 1. Introduction

Pneumonia, which can be categorized as one of the airborne diseases, claims for the death of millions of human beings through inhaling pathogenic organism, mainly *Streptococcus pneumoniae* [1]. These bacteria are also responsible for the cause of other diseases, such as meningitis, ear infections, and sinus infections. Pneumonia can affect human beings of all ages, from children to the elderly, and it becomes dangerous when the immunity level is lowered, as well as when it is coinfected with other diseases like meningitis [2]. Meningitis, an infection which covers the brain and spinal cord, is caused by both bacteria and virus. Bacterial infection of meningitis is the most common one, particularly, *Streptococcus pneumoniae*, *Haemophilus influenzae*, and *Neisseria meningitidis* are responsible for 80% cause of meningitis [3]. To control these diseases, a lot of scholars proposed different methods. In this aspect, mathematical models played a great role in proposing controlling strategies. Several scholars proposed different models to describe the dynamics of infectious diseases in the community. Some of them [4–8] proposed a mathematical model of pneumonia only, and the others [9–11] proposed a mathematical model of meningitis only. Few scholars like Tilahun et al. [12] proposed a mathematical model of pneumonia and typhoid fever coinfection using optimal control strategies. Moreover, Onyinge et al. and Akinyi et al. [13, 14] developed a mathematical model for coinfection of pneumonia with malaria and HIV. More recently, Tilahun [15] proposed a mathematical model of pneumonia and meningitis and investigated their coinfection using an SIR approach. However, to the best of our knowledge, no one has proposed a mathematical model by incorporating optimal control strategies for coinfection of pneumonia and meningitis. Therefore, this work is devoted in fulfilling this gap.

This paper is organized as follows. Section 2 presents the description of the model. Qualitative behaviour of the model is discussed in Section 3. In Section 4, the basic model is extended to optimal control analysis. In Section 5, numerical simulation of the optimality system is presented. A brief discussion and conclusion are presented in Section 6.

#### 2. Description of the Model

In this section, a deterministic mathematical model of pneumonia-meningitis coinfection is presented. The model is proposed using seven compartments with total population size denoted by . The compartment that has individuals who are healthy but able to be infected is denoted by . Individuals that are affected by pneumonia and can transmit the disease to others are denoted by . Similarly, meningitis-infected individuals’ compartment is denoted by , and coinfectious individuals’ compartment is represented by . Additionally, recovered /removed compartments from pneumonia, meningitis, and coinfection of both diseases are denoted by , and , respectively. Then, the total population is . Susceptible compartment increase by recruitment rate of *π* and also from Pneumonia recovered compartment with rate of δ_{1}, meningitis recovered compartment with rate of δ_{2} and from co-infectious recovered compartment with rate of δ_{3}. Force of infection of pneumonia and meningitis is and , respectively, where *a* is the contact rate of pneumonia and *b* is the contact rate of meningitis. Pneumonia-only recovered compartment is increased due to the recovery rate of pneumonia denoted by , and meningitis-only and coinfectious recovered compartments increase their number with a rate of recovery of and *σ*, respectively. In the coinfectious recovered/removed compartment, individuals either recovered only from pneumonia, meningitis, or from both diseases with a probability of or , respectively, where sigma, e, and g are any number between zero and one. The natural death rate is denoted by *μ* and pneumonia-causing death rate and meningitis-causing death rate are represented by and , respectively. All parameters described in this model are assumed as nonnegative. The above description of the model is plotted in Figure 1.