Stochastic Model of Pneumonia and Meningitis Co-infection Using Continuous Time Markov Chain Approach

: Pneumonia disease is a lung infection caused by Streptococcus pneumoniae . Meningitis is an infection of the meninges and cerebrospinal fluid caused by Streptococcus pneumoniae . Both diseases may occur at the same time. A mathematical model is needed to represent the spread of pneumonia and meningitis co-infection. This study aims to build the stochastic model of pneumonia and meningitis co-infection with CTMC, determine the transition and outbreak probability, and conduct simulations to assess the effect of increasing the contact rate on pneumonia (𝑎) and meningitis (𝑏) . Based on the computer simulation undertaken, it can be concluded that if 𝑎 was decreased while 𝑏 was set to be fixed, the probability of disease outbreak decreased. If 𝑎 was set to be fixed while 𝑏 was decreased, the probability of disease outbreak decreased. However, the latter is smaller than the previous. Similarly, if 𝑎 was increased while 𝑏 was set to be fixed, the probability of disease outbreak increased. If 𝑎 was set to be fixed while 𝑏 was increased, the probability of disease outbreak increased. However, the latter is smaller than the previous. Moreover, if both 𝑎 and 𝑏 were decreased, the probability of disease outbreak was equal to zero.


Introduction
Pneumonia is an infection in the lungs that affects the alveolar space explicitly (Lim, 2022).The infection can be transmitted by breathing in pathogenic microorganisms or inhalin (Cilloniz et al., 2016).The disease can claim the lives of millions of people through inhalation of pathogenic organisms.In 2015, as many as 920,000 children under the age of five died from pneumonia worldwide, or two children every minute and approximately 99% of these child deaths took place in developing countries (Watkins et al., 2017).Bacteria, fungi, and viruses can cause pneumonia, but the most common cause is Streptococcus pneumoniae (Leach & McLuckie, 2009).Pneumonia is particularly dangerous in people with weakened immune systems, such as infants and the elderly, or when infected with other diseases, such as meningitis (Feldman & Anderson, 2019;Kotola & Mekonnen, 2022).
Meningitis is an infection of the meninges and cerebrospinal fluid surrounding the brain and the spinal cord (Howlett, 2012).The disease affects all age groups, and children under the age of 5 are particularly at risk.In 2017, 290 thousand meningitis-related deaths and 5 million new cases of meningitis were reported globally.(W.H.O., 2021).Bacteria or viruses can cause meningitis.One of the bacteria that commonly causes meningitis is Streptococcus pneumoniae (Tambunan, 2019).
The same bacteria can cause pneumonia and meningitis, so a person can get both diseases simultaneously (Zhang et al., 2018).Approximately 4% to 20% of patients infected with meningitis may develop pneumonia.In other words, both disorders occur at the same time.Based on age, 20% are elderly, while younger patients vary between 4% and 10%.When grouped by the pathogen causing the infection, patients infected with meningitis due to S. pneumoniae have the highest pneumonia infection rate, which is about 18% -22% (Brueggemann et al., 2021;Figueiredo et al., 2020).
Mathematical modelling is essential in representing the dynamics of infectious disease co-infection and its control.Mathematical models for infectious diseases are divided into two types, namely deterministic models and stochastic models (Dadlani et al., 2020;& Li, 2018).Research on the spread of disease co-infection using deterministic models has been widely conducted.However, there are still relatively few studies of infectious disease co-infection using stochastic models.Stochastic models are needed to account for variation and uncertainty in an epidemic (Nurlazuardini et al., 2016).In their study, Allen & Lahodny (2012) found that outbreak opportunity information is very useful in epidemic models, and the value of the disease-free opportunity obtained by the branching process is almost the same as in numerical studies.The disease-free probability can be used to determine the persistence or extinction of a disease (Maliyoni, 2021).
Therefore, a Continuous Time Markov Chain (CTMC) stochastic model is developed in this study, referencing the model introduced by (Tilahun, 2019a).This approach was chosen because infection can occur at any time.In addition, this study also modified the recovery compartment in the co-infection of pneumonia and meningitis.

Method
This research is a literature study with a mathematical approach regarding the co-infection of pneumonia and meningitis.The steps of this study are as follows: Modify (Nigmatulina, 2009;Tilahun (2019) deterministic model of pneumonia and meningitis coinfection into a stochastic model using the CTMC approach and change the pneumonia-cured and meningitis-cured compartments into pneumonia-cured but still infected with meningitis and meningitis-cured but still infected with pneumonia compartments; Determine the transition probability and the outbreak probability; Perform numerical simulations using ℎ 11.3 and  −  software to determine the following: (a).The effect of changing the pneumonia contact rate () when the meningitis contact rate () is fixed; (b).The impact of changing the meningitis transition rate () when the pneumonia transition rate () is fixed (Asamoah et al., 2020).

Mathematical Modification Model
The modification of the pneumonia and meningitis co-infection model is done by changing the variables   and   to    and    as the subpopulation cured of meningitis and still infected with pneumonia and the subpopulation cured of pneumonia and still infected with meningitis, respectively (Tilahun, 2018).Therefore, this model has seven variables, namely (),   (),   (),   (),    (),    (), and ().
The assumptions used in this study are as follows: Birth and death rates are equal in each subpopulation; there is no migration within each subpopulation; Individuals in the susceptible () subpopulation have two possibilities to become infected with the disease, namely to become infected with pneumonia with an infection rate of  1 (  ) and to become infected with meningitis with an infection rate of  2 (  ), where   The differential equation obtained from the above diagram is as follows: : Transition rate of recovered individuals to resusceptibility The overall pneumonia and meningitis co-infection model was separated into several sub-models, namely the pneumonia-only model and the meningitis-only model.This is useful to gain a deeper understanding of the dynamics and interactions of pneumonia and meningitis (Tilahun, 2019b) & (Kotola et al., 2022).

Pneumonia Disease Model
Based on the system of differential equations in the modified model, the pneumonia model with   =   =    =    =  and   =   =  as follows:

Disease-Free Fixed Point
The disease-free fixed point is obtained by solving  0 = (, 0,0)

Basic Reproduction Number
The basic reproduction number (ℜ 0 ) is necessary to determine the potential spread of disease in the population.The calculation of the basic reproduction number from the equation is obtained by using the nextgeneration matrix, which is based on the subpopulations that cause infection only.ℜ 0 is defined as the largest eigenvalue of the matrix .The matrix  is the product of two matrices written  =  − , where  is the infection rate increase matrix, while  is the rate at which infection moves evaluated at a fixed point (Diekmann et al., 2010) & (Delamater et al., 2019).The basic reproduction number for pneumonia (ℜ 0 ) to determine the potential spread of pneumonia in the population is obtained as follows (Adeniyi & Oluyo, 2018).

Expected value of many infected individuals (𝑚 𝑝 )
Stochastically, an outbreak occurs when the expected value of the number of infected individuals () is greater than 1.The determination of the outbreak probability and disease-free probability can be obtained by using a branching process with a probabilitygenerating function (pgf).Pgf for   with initial value Furthermore, the expected value of the number of pneumonia-infected individuals is obtained

Basic Reproduction Number
After obtaining the disease-free fixed point, the next step is to find the basic reproduction number of meningitis (ℜ 0 ) to determine the potential spread of meningitis in the population.The calculation of the basic reproduction number is obtained by using the nextgeneration matrix, so the value of ℜ 0 is obtained as follows: Expected value of many infected individuals (  ) Pgf for   with initial value   (0 Furthermore, the expected value of the number of meningitis-infected individuals is obtained

Pneumonia and Meningitis Co-infection Model Disease-Free Fixed Point
The disease-free fixed point is obtained by solving the system of equations of the overall co-infection model  0 = (, 0,0,0,0,0,0,0)

Basic Reproduction Number
The basic reproduction number (ℜ 0 ) is obtained using the next generation matrix based on the subpopulations that cause infection only, namely   ,   ,   ,    , dan    .Therefore, the differential equation system used is as follows.

Numerical Simulation
Simulations were conducted to see the effect of pneumonia contact rate () on pneumonia and meningitis co-infection and meningitis contact rate () on pneumonia and meningitis co-infection.The assumed initial values of the subpopulations were (0) = 820,   = 100,   = 50,   = 30,    = 0, and    = 0.The parameter values used include  0 ,  The value of  by determining the dominant eigenvalue of matrix  is defined as () =  below: Stochastically, an outbreak occurs when the value of  > 1 (Allen & Lahodny, 2012).Furthermore, the expected number of infected individuals () is defined as the dominant eigenvalue of the matrix , i.e.,  = (), where  is a nonnegative matrix whose elements are Below are simulation results with changes in the parameters  and , including the value of ℜ 0 ,  and the probability of an outbreak, as indicated in Table 2. Table 2 shows a certain probability that an outbreak will occur when ℜ 0 > 1 and  > 1.The higher the pneumonia contact rate () or meningitis contact rate (), the higher the probability of an outbreak.
When the value of  = 0.6 and  = 0.9 results in a matter of ℜ 0 = 3.644,  = 1.569, and the probability of an outbreak is 1 − 7.34 × 10 −9 .The following is a sample path when the values of  = 0.6 and  = 0.9. Figure 3 shows that co-infection of pneumonia and meningitis is endemic.If the value of  was set to be fixed while the value of  was decreased, then ℜ 0 = 1.96 and  = 1.325 were obtained.Probability of a disease outbreak was 1 − 1.74 × 10 −4 .If the value of  was set to be fixed while the value of  was increased, then ℜ 0 = 6.478 and  = 1.732 were obtained.Probability of disease outbreak was 1 − 8.71 × 10 −11 .The following is a sample path when the value of  was set to be fixed, while the value of  was changed.
Figure 4 shows that when  = 0, co-infection (  ) will disappear from the population in about 15 months and the outbreak is due to the dominant pneumonia infection. = 0.2,   will disappear in about 46 months and the outbreak is due to pneumonia infection.When  = 1.6,   becomes endemic.
If the value of  was decreased while the value of  was set to be fixed, the value of ℜ 0 = 3.644 and  = 1.569 was obtained and the probability of a disease outbreak was 1 − 3.51 × 10 −5 .Similarly, if the value of  was increased while the value of  was set to be fixed, the same values of ℜ 0 and  was obtained as previous.nevertheless, the probability of a disease outbreak was 1 − 1.24 × 10 −11 .The following is a sample path when the value of  was changed and the value of  was set to be fixed.
Figure 5 shows that when  = 0,   disappears from the population in approximately 13 months and the outbreak occurs due to the dominant meningitis infection.When  = 0,2 causes   to disappear from the population in approximately 28 months and the outbreak occurs due to meningitis infection only.When  = 1,0, it causes   to be endemic.
If both values of  and  were decreased to  =  = 0.2, then ℜ 0 = 0.810,  = 0.895, and the probability of a disease outbreak was zero.In this case, pneumonia, meningitis and their co-infections would disappear from the population within a particular time.When both values of  and  were increased to  = 1.0 and  = 1.6, then ℜ 0 = 6.478 and  = 1.732, and the probability of a disease outbreak was 1 − 1.66 × 10 −13 .
Figure 6 shows that if the values of  and  were decreased to 0.2,   will disappear in the population in about 25 months and no outbreak will occur.If the values of  and  was increased to  = 1.0 and  = 1.6, then   becomes endemic.(c)

Conclusion
In this study, a deterministic SIRS model of pneumonia and meningitis co-infection was developed into a stochastic CTMC model of pneumonia and meningitis co-infection.Based on this study, information on the expected value of the number of infected individuals and information on the probability of an outbreak can be obtained.Both the increasing of pneumonia contact rate () or meningitis contact rate (), will increase the probability of disease outbreak.Based on the computer simulation undertaken, it can be concluded that if the value of  was decreased while the value of  was set to be fixed, the probability of disease outbreak decreased.If the value of  was set to be fixed while the value of  was decreased, the probability of disease outbreak decreased.However, the disease outbreak probability of the latter is smaller than the previous.Similarly, if the value of the value  was increased while  was set to be fixed, the probability of disease outbreak increased.If the value of  was set to be fixed while the value of  was increased, the probability of disease outbreak increased.Nevertheless, disease outbreak probability of the latter is smaller than the previous.Moreover, if both values of  and  were decreased, the probability of disease outbreak was equal to zero.In other words, the disease will disappear from the population within a certain period.

Figure 2 .
Figure 2. The modified model diagram (the red arrows were added to improve the original model) = 0, thus obtained as follows: Disease-Free Fixed PointThe disease-free fixed point is obtained by solving the equation in the meningitis only model with   = 0, thus obtained as follows: ) = 0,   () = 0,   = 0, so the following disease-free fixed point is obtained:

Table 2 .
Numerical simulation result