To reduce the number of new infectious cases and hence, to decline the overall morbidity of the disease, each infectious case should generate less than one new infectious case. The total number of secondary infectious cases generated by one infectious case is called as the effective reproductive number (R). Outbreaks start declining when the R drops below one due to the temporary exhaustion of the susceptible population. The total number of secondary cases generated by one primary case introduced into a fully susceptible population is called as the basic reproductive number (R_g). The total number of secondary cases generated by one primary case introduced into a population in which a proportion p has been vaccinated is called as the basic reproductive number under vaccination (R_0v. R_0v for a vaccine which provides lifelong immunity = [(1-p) R_:]. For eradication of a disease from the population R_g should be at least one or less than that. Therefore, the critical vaccination proportion (p_c)) to eradicate a disease from the population = (1 - 1/R_:) (Schrer and McLean, 2002).

Mathematical models are useful in predicting the dynamics of the disease outbreak with or without the introduction of a vaccination program during the outbreak. A population with a vaccination program contains four different group of animals: susceptible (S), vaccinated (V), infected (I), and recovered (R). The immunity of commercially available and commonly used alum-precipitated HS vaccines is from 4 to 6 months and that for the oil-adjuvant HS vaccines is up to 1 year (Verma and Jaiswal, 1998). Since, the duration of a HS outbreak is generally less than the immunity time period of any HS vaccine, the equations of our SVIR model are (Fig. 1):-

Where N is the total animal population (S+V+I+R), (1- ep) pN are the unvaccinated births, PIS are the infected animals, and pS are the deaths in susceptible.

Where eppN are the vaccinated births, and pV are the deaths in vaccinated.

Where PIS are the infected animals, yI are the recovered animals, and pI are the deaths in the infected.

Where yI are the recovered animals, and pR are the deaths in the recovered.

The model parameters used in the above transitions’ equations are calculated from a HS outbreak data from Murshidabad district, West Bengal, India (Mitra et al., 2013). The population at risk is assumed to be fixed where birth rate is equal to the death rate. The model parameters are described below in Table 1.

MATLAB and Simulink were used to create and run the model. Since, outbreaks of HS mainly occurred in the monsoon months from July to September (Gowrakkal et al., 2014), we ran our model for 90 days (3 months). With the aforementioned model parameters in Table 1 and following the posteriori model building approach by using the HS outbreak data (Mitra et al, 2013), we calculated the best possible R_g parameter (which is equal to 1.011) by comparing the total infectious fraction of animals at the end of the outbreak for different transmission parameter (fi) values. The graph (Fig. 2) below shows our described approach.

The population of cattle and buffalo at risk was 7120. The best possible transmission coefficient (ft) to imitate the outbreak was found to be 0.335 (Kermack and McKendrick, 1927). The efficacy of the HS vaccines in some of the experimental studies was found to be nearly 100% (Mohammad et al., 2007). Hence, we used the vaccine efficiency (E) = 0.99.

We used R₀= -Ln(1-I) /1 formula for the calculation of the basic reproduction number for various outbreaks of HS in Kazakhstan in May, 2016 (“Promed Post - ProMED-mail,” n.d.), in Israel in March, 2017 (“PRO/AH> Hemorrhagic septicemia - Israel: (HZ) bovine, OIE ~ Outbreak Watch,” n.d.), in Malaysia from 29^th September to 10^th October, 2003 (Bisht et al, 2004) and in Haryana state of India from 1995 to 1999 (Jindal et al, 2002), where I was the proportion infected at the end of the outbreak (Dietz , 1993).

Finally, we introduced a vaccination campaign by using a constant vaccination signal in the Simulink model starting in the first week the outbreak (Mitra et al, 2013) from the 5th day to the 20th day of the outbreak with a target of 90% of the population to be covered under the vaccination campaign during this period. Hence, the per day vaccination target was 6% of the total population.

The best possible transmission coefficient (fi) to imitate the outbreak was found to be 0.335 (Kermack and McKendrick, 1927). The best possible R₀ was calculated by comparing the total infectious fraction of animals at the end of the outbreak for different transmission parameter (fi) values. The best possible R₀ calculated by the posteriori model building approach by using the HS outbreak data (Mitra et al, 2013) was found to be 1.011.

R₀ for the outbreak in bovines in Kazakhstan in May, 2016 (“Promed Post - ProMED-mail,” n.d.) was found to be 1.003. R₀ for the outbreak in cattle in Israel in March, 2017 (“PRO/AH> Hemorrhagic septicemia - Israel: (HZ) bovine, OIE ~ Outbreak Watch,” n.d.) was found to be 1.116. R₀ for the outbreak in buffaloes and cattle in Malaysia from 29^th September to 10th October 2003 (Bisht et al, 2004) was found to be 1.206. The Haryana state of India faced 26 HS outbreaks in buffaloes and cattle from July 1995 to June 1999 out of which 15 outbreaks occurred in winter, 6 in the rainy season and the remaining 5 occurred in the summer season (Jindal et al, 2002). The R₀ values calculated for all the Haryana outbreaks in buffaloes and cattle ranged from 1.01 to 2.96 which are presented in the table below (Table 2).

During the 90-day course of the outbreak, the proportion of the susceptible animals decreased from 0.99 to 0.97 and the proportion of the infectious decreased from 0.0005 to 0.0003. 0.01 proportion of the animals recovered during this duration of the outbreak. The critical proportion of the animals to be vaccinated (1- 1/R_g) to stop the outbreak was found to be 0.005.

After the introduction of the vaccination campaign from the 5^th to the 20^th day of the outbreak, the proportion of the susceptible animals declined from 0.99 to 0.4 on the last day of the campaign. The proportion of the infectious declined from 0.0005 to 0 with in a time span of first 33 days of the outbreak.

The delineation of effectual control programs for HS is often impeded by the lack of the basic sero-epidemiological information. In this study of mathematical model designing for HS, we used an outbreak data from Murshidabad district, West Bengal, India (Mitra et al., 2013) to imitate the outbreak. This helped us in the better understanding of the HS disease dynamics during the outbreak and in finding the critical intervention times for vaccination of the remaining susceptible animals for better control of the outbreak. We used the parameters calculated from the outbreak data to formulate our model for HS to further predict the impact of vaccination programs on HS outbreak control. We assumed homogeneous mixing of animals for HS transmission during the outbreak.

Our results show that the R_g calculated for the data of various HS outbreaks in Kazakhstan in May, 2016 (“Promed Post - ProMED-mail,” n.d.), in Israel in March, 2017 (“PRO/AH> Hemorrhagic septicemia - Israel: (HZ) bovine, OIE ~ Outbreak Watch,” n.d.), in Malaysia from 29^th September to 10^th October, 2003 (Bisht et al., 2004), and in Haryana state of India from July 1995 to June 1999 (Jindal et al., 2002) ranged from 1.003 to 2.96. The above¬calculated lower values of R_g for HS are may be due to the quicker treatment and vaccination interventions in its outbreaks. Hence, these small R₀ values also point towards the fact that on time treatment and vaccination campaigns during the outbreaks are always beneficial for the control of this deadly disease. The disease dynamics also vary with the use of different types of HS vaccines based on the efficacy and duration of immunity provided by the vaccine and this is a matter of further research. Using the results for efficacy and duration of immunity of the HS vaccines from the vaccination studies in natural environment instead of the small experimental vaccination studies, can be more reliable. Further analysis of the contact patterns between animals can shed more light on the epidemiology and spread of HS in a specific area. More precise demographic, biological, and epidemiological information and real-time data about HS can be helpful in designing more robust and informative mathematical models for this disease. A combination of factual and theoretical research is needed for better understanding of HS and for animal health policy planning. More complex stochastic models can be used to analyze both endemic cycle and outbreak episodes of HS in endemic animal populations of India.

The success of introduction of a vaccination campaign in controlling an HS outbreak is variable if vaccination is introduced at early, mid or late phases of the outbreak. We also conclude that animal vaccination modeling for eradication of HS by mass regional or nationwide vaccination campaigns can be understood even by simpler mathematical models and these models can be useful in assessing alternate strategies for control of HS and furthermore, in guiding HS related epidemiological research.