Mathematical Modeling for Hospitalization due to Temperature Variations

Sudden changes in temperature occur due to ecological disturbances which results into global warming, volcanic eruption, depletion of ozone layer etc. This has become a crucial environmental and social issue all over the world. Changes in temperature influences transmission pattern for several diseases which leads to increase in the death rate. This research focusses on hospitalization due to temperature variations. Here, class of exposed individuals are divided into two types: asymptomatic individuals who do not show any symptoms and symptomatic individuals which shows symptoms. The indoor or outdoor medications are the remedial steps to get cured. Severe dehydration, blood pressure fluctuations etc. need hospitalization. The rate at which the individual gets hospitalized is scrutinized using SEIRS model. The system of non-linear ordinary differential equations is formulated for the given model and the reproduction number is then computed using next generation matrix method which connotes the recovery rate of an individual. Stability and numerical simulations are carried out.


Introduction
Temperature is a measure of the warmth or coldness of an object or substance with reference to some standard value. In the proposed model, temperature refers to ambient temperature. Change in temperature can occur at any location for any period. On this basis, it is classified as in Figure 1. Periodic temperature variation is the continuous change of temperature during a day, week, month, year or any period, whereas a horizontal temperature variation is the change of solar energy with respect to equatorial region and latitude. Vertical temperature variation is divided according to four spheres namely troposphere, stratosphere, mesosphere and thermosphere. This model is based on periodic temperature variations. During the last few years, the association between extreme temperatures and health issues has been frequently investigated. Unstable changes in temperature disturbs the weather we experience and our basic necessities resulting in increased health issues. The sternness of these health risks (hospitalization rate) depends on factors such as age, gender, individual's strength to sustain temperature change and monetary status. It will also vary based on location, sensitivity and adapting capability of an individual. Health effects may include gastrointestinal illness like vomiting, diarrhoea, effects on the body's nervous and respiratory systems, liver and kidney damage, etc. Some health issues may show some symptoms, others may not. Those individuals who show symptoms such as dehydration, low blood pressure, respiratory problems, heat stroke etc. are known as symptomatic individuals and those who do not show any symptoms such as any type of infection are asymptomatic individuals. Proper intime treatment is required to control these health issues. If the condition becomes critical and cannot be controlled by taking indoor medications, they get hospitalized. This research is focused on transmission rate of any health issue due to temperature variations on an individual from susceptible stage to hospitalised stage.
A lot of survey is done about effects of climate change on human health such as in book "Global climate change and health: an old story writ large" (McMichael, 2003) which contains information about every issue related to climate change and human health: Risks and responses. It is also reviewed (Wu et al., 2016) about the empirical evidence and human adaptation due to climate change on human infectious diseases. One of the study (Paaijmans et al., 2010) showed the influence of climate on malaria transmission which depends on daily temperature variation. The impact of rainfall and temperature variation on diarrheal prevalence in sub-saharan Africa was studied (Bandyopadhyay et al., 2012). A dynamic model of air pollution, health and population growth was examined using system dynamics: a study on Tehran-Iran (Shahgholian and Hajihosseini, 2009). Direct and indirect effect of climate change was analysed on human health (McMichael et al., 1997). Analysis was done for greater temperature variation within a day associated with increased emergency hospital admissions for asthma (Qiu et al., 2015). The temperature variation and emergency hospital admissions were reported for stroke in Brisbane, Australia, 1996-2005(Wang et al., 2009. The impact was studied Zell et al. (2008) for global warming on viral diseases. Also, the basics of SEIR model were discussed and applied it for vector borne diseases to analyse steady state conditions. Also, after finding threshold parameter 0 R it was shown that the disease will spread only if its value exceeds 1 Gupta, 2013, 2014). A mathematical model was formulated for water pollutants with stability analysis (Shah et al., 2018). A preliminary mathematical model was also computed for the transmission of dengue, chikungunya and zika (Isea and Lonngren, 2016). Similarly, a system of ordinary differential equations was designed to simulate the transmission of infection between humans and Aedes albopictus (Yakob and Clements, 2013).
The mathematical model for hospitalization rate is shown in Section 2 followed by stability at equilibrium points in Section 3. Numerical simulation is then computed in Section 4, concluding the model finally in Section 5.

Mathematical Model
The hospitalization rate due to temperature variations is studied in this mathematical model. It is based on SEIRS-model. This model consists of five compartments, namely, susceptible individual


The rate at which susceptible individuals gets exposed 0.5 2  The rate at which exposed individuals are asymptomatic 0.01 3  The rate at which Using parametric values, the mathematical model shown in Figure 2 is formulated. The system of non-linear differential equations is as follows: Adding the above differential equations of system (1), we have Therefore, the feasible region of the model is 5 ( , , , , ) : .
. Now, we calculate the threshold called basic reproduction number 0 R using next generation matrix method (Diekmann et al., 2009).

Stability
In this section, for all equilibrium points local as well as global stability is discussed.

Local Stability
As all eigenvalues of above matrix are negative. Therefore, equilibrium point of system is stable.  Here, all the eigen values are negative. So, 1 Y is locally asymptomatically stable.

Global Stability
Here, we discuss the global stability behavior of the equilibrium 0 Y , 1 Y and * Y by Lyapunov's function.
Theorem 3.2.1 (Stability at 0 Y ): The disease-free equilibrium point 0 Y is globally asymptotically stable.
Theorem 3.2.2 (Stability at 1 Y ): The unaffected equilibrium point 1 Y is globally asymptotically stable.

Proof: Consider the Lyapunov function
Here, we denote . Therefore, * Y is globally stable. Next, we validate the model by numerical simulation.

Numerical Simulation
In Figure 3, transmission pattern of humans are observed which are affected due to temperature variations. One can see that if any individual experience sudden temperature change while traveling from one place to another, they gets exposed immediately. Some individuals do not show symptoms for two days and after approximately nine days they begins indicating symptoms. It can also be observed that after approximately eight days six individuals again gets susceptible out of which two individual gets exposed.     Figure 4 and 5, one can observe that increase in rate results in increase of symptomatic individuals. Figure 4 shows that increase in 3  by 1% results in increase of symptomatic individuals by 8%. Figure 5 implies that 64% increase of symptomatic individuals will be observed if there is an increase in rate of 4  by 10%. It also connotes that during exposed or asymptomatic stage if proper precautions or medication are not taken then increase in symptomatic individuals will be observed. Figure 6. Effect of the rate at which symptomatic individuals moves to hospitalization Figure 6 shows that 8% increase in rate of symptomatic individuals results in increase of hospitalized rate from 8 individuals to 26 individuals. So, increase in the rate of symptomatic individuals is directly proportional to hospitalization.  Figure 7 shows that 2% individuals are susceptible out of which 11% gets exposed. From that 11% exposed individuals, 37% are asymptomatic individuals and 28% are symptomatic out of which 22% gets hospitalized. Figure 8 shows that at every equilibrium point the value of reproduction number is less than 1. Therefore, the mathematical model is stable for the given parametric values.

Conclusion
Using system of nonlinear ordinary differential equations, a mathematical model is formulated to study the hospitalization rate due to impact of temperature variations on human health. System is locally and globally asymptotically stable at all equilibrium points. A basic reproduction number has been evaluated at an equilibrium point and is equal to 0.3732 which shows that approx. 37% of the individuals in the society gets hospitalized due to temperature variations. To validate the model, numerical simulations has been carried out which interprets that with the increase in rate of exposed and asymptomatic compartments, increase of susceptible individuals are observed which further results in increased hospitalization rate and the mathematical model is stable at every equilibrium point. Hospitalization rate can be controlled by taking proper precautions at the early stages. As we know better health helps to face physical, mental and social challenges. So, it is essential to become hale and hearty human as it makes people live longer and more productive resulting in decrease in hospitalization rate. Hence, every individual in the society should take proper care of their body and respond immediately if symptoms are observed by taking proper medications.